Emilie Cyril

Char- Lotte Claire

Lucie

Maxime, Nazih Marie Maud

Sébastien, Taieb Nicolas Pierre

Keywords: Sequential Monte Carlo, Interacting Particle System, Variance estimation, Genealogy, Central Limite Theorem, Rare-event simulation, Random Forests, Online learning

First and foremost I would like to thank my supervisor Arnaud Guyader. Throughout the past three years, his helpful advices and encouragements are invaluable to me. His strict supervision will guide me in the rest of my life, encouraging me to always pursue a higher standard. I thank him for letting me know the beautiful world of Interacting Particle System, and I feel lucky to have had the chance to dive into it for the past three years. Apart from research, I am also very grateful for his help on my daily life. At the same time, I want to thank my supervisor Tony Lelièvre. His kindness and encouragement have always been so inspiring, especially during the most di cult time of my Ph.D. career. As a former statistics student, I thank him for taking so much time to explain the ideas in Molecular Dynamics to me, and in a extremely clear and insightful way. In particular, I want to thank him for letting me be brave enough to study the asymmetric resampling scheme, which is one of the main contributions of this thesis. I do not know if I could have another chance to take such a long journey in mathematics in the future, and this is de nitely a precious experience of my life. For both of my supervisors, I appreciate that they have helped me in so many ways, and I learned so much from them.

I want to thank Gérard and Raphaël for helping me to nish this thesis, especially during the di cult time of COVID-19. I feel so lucky to be able to start a new journey in my career with both of them, and I am sure that there will be a lot of fruitful collaborations in the future.

I consider myself very fortunate to

2.3

The rst family of [2] 0:6 such that (2.7) is veri ed. . . . . . . . . . . . . . . . . . . 2.4 The second family of [2] 0:6 such that (2.7) is veri ed. . . . . . . . . . . . . . . . . 2.5 An illustration of the duality formula recursively applied to a mini IPS of +1 = 6 levels with 5 particles at each level. Every straight black or dotted arrow within the original IPS represents a Markov transition , and the black twisted ones pointing to the particles in the coupled particle block represent the Feynman-Kac transition kernels , . The red dotted bending arrows are identities. The indices of the original particles in the coupled particle block are [2] 0 = (4, 5), [2] 1 = (2, 4),

[2] 2 = (2, 3), [2] 3 = (3, 5), [2] 4 = (1, 5) and [2] 5 = (1, 2). . . . . . . . . . . . . . . . 2.6 The coupled particle block tracked by the event de ned by (2.28). . . . . . . . . 2.7 An illustration of the duality formula recursively applied to a mini IPS of +1 = 6 levels with 5 particles at each level. Every straight black or dotted arrow within the original IPS represents a Markov transition , and the black ones within the coupled particle block represent the Feynman-Kac transition kernels , .

The red dotted bending arrows are identities. The indices of the original particles in the coupled particle block are [2] 0 = (3, 5), [2] 1 = (3, 4), [2] 2 = (2, 5), [2] 3 = (1, 2), [2] 4 = (2, 5) and [2] A.1 An illustration for the estimation of AMS method along with 95% con dential interval. The number of particles varies from 100 to 50000 and = 0.01. . . . A.2 An illustration for the estimation of asymptotic variance with 95% con dential interval. The number of particles = 10 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators. . . A.3 An illustration for the estimation of asymptotic variance with 95% con dential interval. The number of particles = 50 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators. . . A.4 An illustration for the estimation of asymptotic variance with 95% con dential interval. The number of particles = 100 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators. . . A.5 An illustration for the estimation of asymptotic variance with 95% con dential interval. The number of particles = 200 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators. . . A.6 An illustration for the estimation of asymptotic variance with 95% con dential interval. The number of particles = 500 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators. . .

Chapter 1 Introduction 1.1 General background on Sequential Monte Carlo

Generally speaking, Sequential Monte Carlo methods (SMC) are a set of simulation-based methods aiming at sampling a sequence of probability or nite measures, usually connected by some nonlinear operators in high dimensional state spaces. They are among the most widely used tools in computational statistics, chemistry, nance and many other disciplines. It seems that the term "Sequential Monte Carlo" was rst coined by Liu and Chen in [START_REF] Liu | Sequential Monte Carlo methods for dynamic systems[END_REF], where two typical problems are discussed: Bayesian missing-data problem and state-space models. In the latter case, SMC methods are also called Particles Filters, and this term was rst proposed by Del Moral [START_REF] Moral | Nonlinear ltering: interacting particle solution. Markov Process[END_REF]. The most basic, and also the most famous Particle Filter was proposed a little bit earlier by Gordon, Salmond and Smith in [START_REF] Gordon | Novel approach to nonlinear/non-gaussian bayesian state estimation[END_REF], which is often referred to as Bootstrap Filter. Beyond Filtering and Bayesian Inference, SMC methods also provide powerful tools in rare-event simulation problems. A typical family of algorithms are called Subset Simulation, or Multilevel Splitting methods. The original idea was brought in the 1950s by Kahn and Harris [START_REF] Kahn | Estimation of particle transmission by random sampling[END_REF] and Rosenbluth and Rosenbluth [START_REF] Rosenbluth | Monte carlo calculation of the average extension of molecular chains[END_REF] to study particle transmission energies and molecular polymer conformations. More recently, a more detailed study was conducted by Au and Beck [START_REF] Au | Estimation of small failure probabilities in high dimensions by subset simulation[END_REF][START_REF] Au | Subset simulation and its application to seismic risk based on dynamic analysis[END_REF] for estimation of small failure probabilities in high dimensional setting. A more re ned version of this algorithm, called Adaptive Multilevel Splitting (AMS), was proposed by Cérou and Guyader in [START_REF] Cérou | Adaptive multilevel splitting for rare event analysis[END_REF]. Generalized Adaptive Multilevel Splitting methods (gAMS) were introduced later in [BGG + 16] by Bréhier, Gazeau, Goudenège, Lelièvre and Rousset, which are designed to solve some typical rare-event simulation problems in Molecular Dynamics. These are the models at the core of this thesis.

In this section, we provide some of the most important motivating examples of the SMC methods mentioned above, as well as the basic mathematical language used in this thesis. The reader is referred, for example, to [START_REF] Doucet | Sequential Monte Carlo methods in practice[END_REF] and [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF] for a wider list of applications and theoretical analysis. In particular, a good survey for the recent development of AMS methods can be found in [START_REF] Cérou | Adaptive multilevel splitting: Historical perspective and recent results[END_REF].

Particle Filters and Hidden Markov Models

The ltering problem consists in estimating the internal states in some unobserved dynamical systems when partial observations are made, and random perturbations are non-negligible in the observers as well as in the underlying dynamical system. The main objective is to compute and/or sample from the posterior distributions of the states of some Markov chain, given some noisy and partial observations. In this section, we present a basic setting of Hidden Markov Models (HMM, cf. Figure 1.1) and the original algorithm of Bootstrap Filter to illustrate the basic idea of the implementation of Particle Filters. We assume that ( ; ≥ 0) is a time-homogeneous Markov chain taking values in R , with initial distribution 0 ( ) , where denotes the Lebesgue measure on R and 0 ( ) is a probability density function. At each time step ≥ 1, we suppose that

L ( | -1 = -1 ) = ( | -1 )
,

where ( | -1 ) denotes the probability transition density function of the underlying Markov chain. The observations ( ; ≥ 0) take values in R and can be di erent from . We suppose that ( 0 , 1 , . . . , ) are conditionally independent given ( 0 , 1 , . . . , ). Moreover, we assume that

L ( | = ) = ( | ) .
A standard model that allows this structure can be reformulated as follows:

= ( -1 ) + = ℎ( ) + .

We assume that and ℎ are some known functions and ( ; ≥ 0) and ( ; ≥ 0) are independent sequences with known probability density functions. We remark that when and ℎ are linear functions, and when and are Gaussian random variables, the problem can be analytically solved by the famous Kalman Filter, also known as linear quadratic estimation. However, when and ℎ are nonlinear, the analytical solutions are intractable in general. This is one of the main motivations of the Monte Carlo-based numerical methods such as Particle Filters. A typical goal of HMM is to estimate recursively in time the posterior density ( | 0 , 1 , . . . , -1 ) given 0 = 0 , 1 = 1 , . . . , -1 = -1 . In other words, for any test function , we are interested in estimating the integral -1 ) .

( ) := ∫ ( ) ( | 0 , 1 , . . . ,
Thanks to Bayes formula, we have ( 0 , 1 , . . . , | 0 , 1 . . . , -1 ) = ( 0 , 1 , . . . , -1 | 0 , 1 , . . . , ) ( 0 , 1 , . . . , ) ( 0 , 1 , . . . , -1 ) .

By the conditional independence in the HMM construction, it is easily checked that

( 0 , 1 , . . . , -1 | 0 , 1 , . . . , ) = -1 =0 ( | ),

GENERAL BACKGROUND ON SEQUENTIAL MONTE CARLO

which also yields ( 0 , 1 , . . . , -1 ) = ∫ ( 0 , 1 , . . . , )

-1

=0

( | ) 0 1 . . . .

Therefore, the integral of interest becomes

( ) = ∫ ( ) -1 =0 ( | ) ( 0 ) =0 ( | -1 ) 0 1 . . . ∫ -1 =0 ( | ) ( 0 ) =0 ( | -1 ) 0 1 . . . .
The idea of Particle Filters is to simulate particles at each iteration of the algorithm. Then, by applying the importance sampling and bootstrap resampling steps, the quantity of interest ( ) can be estimated recursively in time by exploiting the corresponding empirical measures. More concretely, the mechanism of the Bootstrap Filter is the following:

(i) Initialization:

Set = 0;
For = 1, 2, . . . , , sample ( ) 0 ∼ 0 ( 0 ).

(ii) Importance weights calculations:

For = 1, 2, . . . , , calculate importance weights ( ) = ( ) .

(iii) Resampling step:

Resample with replacement particles

(1)

, (2) , . . . , ( ) from the set (1) , (2) , . . . , ( ) according to the importance weights (1) , (2) , . . . , ( ) .

(iv) Importance sampling step:

Set = + 1;
For = 1, 2, . . . , , sample ( ) ∼ ( ) -1 ; Go to step (ii).

By replacing the posterior distribution with its empirical version, the integral of interest can therefore be estimated by ˆ ( ) := 1 =1 ( ( ) ).

As shown above, the implementation of Particle Filter is straightforward and can be applied to a very large class of state-space models. The price to pay for this simplicity is the relatively expensive computational cost. For this topic, the reader is referred to [START_REF] Doucet | Sequential Monte Carlo methods in practice[END_REF] for more detailed analysis and possible variance reduction techniques that can be applied to increase the performance in real-world applications.

Subset Simulation and rare-event simulation

A rare event is an event with non-zero, but extremely small probability. To give an idea, this probability can be lower than 10 -20 . Generally speaking, when the probability of interest is extremely small, it is nearly impossible to have a single sample such that the event of interest occurs within a reasonable simulation time through a crude Monte Carlo approach, let alone to collect a handful of realizations to conduct any meaningful estimation. One typical example in real-world applications is the risk management in insurance, in which case it is important to estimate accurately the probability of some kind of catastrophe. Another example is when we are interested in simulating some certain transition paths between metastable states of a stochastic dynamical system. It can be a transition procedure that is not rare at the macroscopic scale. However, when the associated dynamic can only be simulated with a very small time step, the event of interest then becomes rare in the simulation timescale.

A classic solution for rare-event simulation problems is to apply Important Sampling (IS) techniques. If the underlying probability distribution is , we draw samples according to another auxiliary distribution , and weight each observation = by the Radon-Nikodym derivative ( ) = ( )/ ( ). When the auxiliary distribution is well chosen, Importance Sampling methods may dramatically decrease the variance of the estimated probability. As a consequence, sample sizes needed to conduct reliable estimation may also be dramatically reduced. The readers are referred to [START_REF] Robert | Monte Carlo Statistical Methods[END_REF] for a discussion on Importance Sampling techniques in general, and to [START_REF] Bucklew | Introduction to Rare Event Simulation[END_REF] for the context of rare-event estimation. However, the e ciency of IS methods depends highly on the choice of the auxiliary distribution, and it may completely fail for bad choices even compared to naive Monte Carlo. A good choice of auxiliary distribution requires strong knowledge and careful analysis on the underlying probability measure and the rare event of interest, which is not always tractable.

In this thesis, we mainly discuss another possible approach, that is referred to as Subset Simulation, or Multilevel Splitting. Compared to IS methods, fewer information is needed to conduct reliable estimation and inference, at the cost of a heavier computational cost. Some comments on the comparison of these two approaches can be found in the survey [START_REF] Cérou | Adaptive multilevel splitting: Historical perspective and recent results[END_REF], which will not be discussed in this thesis. In this section, we present a static rare-event simulation problem in order to have a big picture in mind. A more detailed setting in Molecular Dynamics is provided in the following sections. Let be some Polish space, namely, a separable completely metrizable topological space. As is shown in Figure 1.2, we assume that there exists a decreasing sequence of events

= -1 ⊃ 0 ⊃ 1 ⊃ 2 ⊃ • • • ⊃ -1 ⊃ = * ,
where * denotes the rare event of interest. Let ( ) be a probability measure on , and the goal is to estimate the probability * := ( * ). Let be a random variable with distribution ( ).

Thanks to Bayes formula, we have

P ( ∈ * ) = =0 P ∈ ∈ -1 . (1.1)
The basic idea of Subset Simulation is to estimate the probabilities

P ∈ ∈ -1
term by term, and the nal estimator is therefore constructed as the product of these estimators. The intuition behind this formulation is that each term of the conditional probability is not so small compared to the probability of the rare event of interest, and one expects that when they are estimated separately, the variance of the estimation can thus be reduced. In order to achieve this, it is required to be able to simulate according to some nontrivial transition kernel ( , ) at each time step ≥ 0 such that P = P , where P denotes the law of given that ∈ , namely,

P ( ) = 1 ∈ ( )/ ( ).
A standard construction is given by Metropolis-Hastings techniques (cf. Figure 1.3). More precisely, we assume that ( ) is absolutely continuous w.r.t. some reference probability measure on , and there exists an -reversible Markov kernel on the state space , that is ∀( , ) ∈ × , ( ) ( , ) = ( ) ( , ).

A Metropolis-Hastings kernel can then be constructed:

( , ) := ( , )1 ( )1 ( ) + ( ) 1 \ ( ) + 1 ( ) ( , \ ) .
Since is assumed to be -reversible, standard calculations ensure that is indeed Pinvariant. Roughly speaking, the Markov kernel only accepts the transitions within the event ; all the other kinds of transitions are rejected. In practice, this Markov transition kernel may be applied several times to ensure a certain level of acceptance rate in order to ensure decorrelation between the initial condition and the new sample. Namely, one may replace by ( ) de ned by (i) Initialization:

Set = = 0;
For = 1, 2, . . . , , draw 0 ∼ ( ).

(ii) Absorbing time:

Denote by the set of the indices of the surviving particles at level , i.e.,

:= ∈ [ ] ∈ ,
with [ ] := {1, 2, . . . , }.

Stop the algorithm if = and then set = ( -1) ∨ 0.

(iii) Selection:

For ∈ , let = .

For ∉ , uniformly draw a random index α in and let = α .

(iv) Mutation: Set = + 1 and = + 1; For = 1, 2, . . . , , sample ∼ ( -1 , );

Stop the algorithm if = ;

Otherwise, go to step (ii).

At each iteration of the algorithm, each term in (1.1) is estimated separately by

P ∈ ∈ -1 ≈ # × 1 ≥ .
Therefore, the estimator of * is de ned by * :=

# ∈ * -1 =0 # × 1 ≥ .
Note that it is possible that the particle system degenerates before the desired time horizon , namely < , when the sequence of subsets ( ; 1 ≤ ≤ ) or the transition kernels ( ; 1 ≤ ≤ ) are poorly designed. Theoretically speaking, one may encounter a situation where Subset Simulation is worse than naive Monte Carlo. Hence, in practice, it is of crucial importance to construct properly and wisely the sequence of subsets. This is one of the key motivation of AMS methods, where these subsets are generated "on the y" and in an optimal way. Let us provide a slightly modi ed version of the AMS algorithm introduced in [START_REF] Cérou | Sequential Monte Carlo for rare event estimation[END_REF][START_REF] Cérou | Fluctuation analysis of adaptive multilevel splitting[END_REF]. The setting will be referred to as static AMS in the following sections.

Adaptive Multilevel Splitting

We suppose that the rare event of interest is de ned by * := { ( ) > * } , where is a random variable on with distribution ( ) and : ↦ → R is a function that can be computed for each input value , but is explicitly inaccessible or too complex to be studied analytically. The goal is to estimate the probability * := P ( ( ) > * ) .

We call * the level of interest and this problem can be reformulated in the Subset Simulation framework by de ning := ( ) > for a xed sequence of levels

-∞ = -1 < 0 < 1 < • • • < -1 < = * .
The idea of static AMS is to estimate a sequence of adaptive levels when generating the particle system at each iteration of the algorithm, by xing a minimum number of particles * (or * ( ) which depends on the number of particles ) to kill at each resampling step. Accordingly, the mutation kernel in the previous case is replaced by a sequence of Markov kernels indexed by the level . More precisely,

( , ) := ( , )1 { ( ) > } 1 { ( ) > } + ( ) 1 { ( ) ≤ } + 1 { ( ) > } ( , -1 ((-∞, ]) .
The construction is then identical to the previous case. Now, let us present the dynamic of static AMS as follows.

(i) Initialization:

Set = = 0;
For = 1, 2, . . . , , draw 0 ∼ ( ).

(ii) Level calculation:

Order the particles at iteration such that

( (1) ) ≤ ( (2) ) ≤ • • • ≤ ( ( -1) ) ≤ ( ( ) ),
where denotes a permutation on [ ]; Set = ( ( * ) );

Stop the algorithm if > * .

(iii) Absorbing time:

Denote by the set of the indices of the surviving particles at iteration , = ∈ [ ] ( ) > .

Stop the algorithm if = and then set = ( -1) ∨ 0.

(iv) Selection:

For ∈ , let +1 = ;

For ∉ , uniformly draw a random index α ∈ and let = α .

(v) Mutation: Set = + 1 and = + 1;

For ∈ [ ]\ , sample ∼ -1

( -1 , );

Go to step (ii).

Similarly, the estimator of the rare-event probability * is de ned by

* := # ( ) > * -1 =0 # × 1 > * .
We remark that the algorithm above is indeed di erent from the algorithm discussed in [START_REF] Cérou | Fluctuation analysis of adaptive multilevel splitting[END_REF] due to some major di erence in the resampling scheme. In the AMS framework discussed in [START_REF] Cérou | Fluctuation analysis of adaptive multilevel splitting[END_REF], the particles on the same level are killed according to a new sequence of uniform random variables, in order to ensure that there are exactly * particles to be killed at each iteration.

To the best of our knowledge, the adaptive method presented above has not been rigorously studied in theory. Partial results will be given later under the Asymmetric SMC framework in some special cases, that is, when the image of is a nite set and * is set to be 1. Roughly speaking, this algorithm can be regarded as a static version of the gAMS algorithm introduced in [BGG + 16]: at each iteration , the resampled particles perform a mutation step according to the adaptive kernel and the surviving particles stay at the same site. Only the construction of the Markov kernel is di erent. This resampling scheme is di erent from the mechanism of Subset Simulation introduced earlier in this section: in Subset Simulation, all the particles mutate after each resampling step, while in the AMS, only the killed particles mutate. The intuition of this modi cation is to reduce the computational cost brought by the mutation kernel at each resampling step. Imagine that * is set to be very small compared to , say, * = 1. Most of the particles will survive at each resampling step. If we only execute the mutation to the resampled particles, the computational cost will be dramatically reduced. This resampling scheme will be referred to as asymmetric resampling scheme, namely, the surviving particles and the resampled particles mutate w.r.t. di erent Markov kernels. This di erence will be addressed several times in the following sections as it is one of the core problems discussed in this thesis.

In the same spirit as for the static AMS framework studied in [START_REF] Cérou | Fluctuation analysis of adaptive multilevel splitting[END_REF], we expect that with the same regularity assumptions, when the number of particles to kill * ( ) is set to be for a pre xed ratio ∈ (0, 1), the asymptotic behavior of the algorithm presented above is identical to a "limit" xed-level Multilevel Splitting models with *

0 = -1 ( ) (1 -) ≤ * 1 = -1 ( ) (1 -2 ) ≤ • • • ≤ * -1 = -1 ( ) (1 -) ≤ * ,
where -1 ( ) (1-) denotes the (1-)-quantile of ( ). In this limit case, we have equal transition probabilities at each level, namely,

P ( ) > * ( ) > * -1 = .
It is well understood that this balanced case is the optimal one in terms of asymptotic variance for the xed-level models. The interested reader is referred to Section 3.5 of [START_REF] Cérou | Adaptive multilevel splitting: Historical perspective and recent results[END_REF] and reference therein for details.

Feynman-Kac particle models

For the theoretical analysis of the algorithms presented above, we use the language of Feynman-Kac particle models mainly developed by Del Moral and his co-authors (see, e.g., [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF]), which provides a powerful and elegant approach to reformulate di erent models under the SMC framework. To give an idea, we start with a brief introduction on the connection between Feynman-Kac particle models and the Feynman-Kac semigroup. Let ( ; ≥ 0) be a Markov process taking values in R with initial distribution 0 and transition function , . For a Borelian function : R ↦ → R, the Feynman-Kac semigroup is de ned by

∀ ∈ B (R ), , (
, ) := E 1 ( ) exp ∫ ( ) = .
Revealed by Feynman-Kac formula, the Feynman-Kac semigroup is strongly connected to the solutions of parabolic Partial Di erential Equations. If one is interested in simulating according to this semigroup numerically, it is then necessary to consider the discretization in time. Taking = 0, for any > 0, we consider a time discretization

0 = 0 < 1 < 2 < • • • < -1 < = .
Under proper regularity assumptions, the Feynman-Kac kernel 0, can be approximated by 0, ( ,

) ≈ E 1 ( ) -1 =0 exp ( ) ( +1 -) 0 = ,
where the Markov chain ( ; ≥ 0) is a discrete-time approximation of the continuous-time Markov process ( ; ≥ 0). With a slight abuse of notation, we denote := . The Markov kernel from -1 to is denoted by . In addition, we also denote

( ) := exp ( ) ( +1 -) .
As a consequence, we have 0, ( , ) ≈ 0, ( ,

) := E 1 ( ) -1 =0 ( ) 0 = .
Similarly, for each < , , ( , ) can be approximated by

, ( , ) ≈ , ( , ) := E       1 ( ) -1 = ( ) =       . (1.2)
The semigroup , de ned above is called a discrete-time Feynman-Kac semigroup. More generally, given a Markov chain ( ; ≥ 0) with initial distribution 0 and transition kernels ( ; ≥ 1), as well as a sequence of bounded nonnegative potential functions ( ; ≥ 0), we call , de ned in (1.2) an element of the Feynman-Kac semigroup and we call ( , ) := -1, ( , ) = -1 ( ) ( , ) a Feynman-Kac kernel. The measure de ned by

( ) := 0 0, ( ) = E 1 ( ) -1 =0 ( )
is called the Feynman-Kac terminal measure at time . We also de ne the corresponding normalized version by

( ) := ( ) (1) = E 1 ( ) -1 =0
( )

E -1 =0 ( ) .
Remark that is well-de ned if and only if

∀ ≥ 0, E =0 ( ) > 0,
which is assumed in the following. The elegant part of this formulation is that even for some a priori di erent applications, such as HMM and rare-event simulation, the essentials of the problems are identical. More precisely, in HMM, if we take ( ) := ( | ), the prediction problem of estimating ( ) can be reformulated as to estimate ( ). Similarly, for rare-event simulation, if we consider ( ) := 1 ( ), the estimation of * can be reformulated as the estimation of (1 * ). Hence, in order to study the properties of the stochastic algorithms, which are essentially the same as explained in the next section, there is no need to return to the details of the models.

McKean interpretation and Interacting Particle System

Now, we give a brief introduction on how we can simulate according to the Feynman-Kac measures and . First, we remark that

= -1 =0 ( ).
(1.3) By de nition, it is obvious that the unnormalized measures -1 and are connected by the Feynman-Kac kernel , i.e., = -1 . A natural question to ask is whether the sequence of probability measures ( ; ≥ 0) can also be connected by some transition kernels. We consider the McKean-type kernel , ( , ) indexed by some entry probability measure such that ( -1 ) > 0 and de ned by

, ( ,
) := -1 ( ) ( , ) + (1 - -1 ( )) ( ) ( -1 )
,

where is a constant in [0, 1/ -1 ∞ ]. Standard calculations give -1 , -1 = .
By "McKean kernel" we mean that the de nition of such kernel depends on the probability measure of the previous step, and a discrete-time stochastic process that is connected by McKean kernels is referred to as a McKean chain. Unlike the Markov case, the transition of the normalized measures from -1 to depends on the probability measure -1 itself. Therefore, it is not possible to simulate directly according to , -1 . A natural solution is to construct an Interacting Particle System (IPS), such that at step -1, particles X n-1 := ( 1 -1 , 2 -1 , . . . , -1 ) samples an approximation of -1 . Then, by exploiting the empirical measure

-1 := 1 =1 -1 , (1.4)
we are able to sample from the approximated kernel , -1 . Similarly, by replacing each normalized measure by its particle approximation in (1.3), we de ne

:= -1 =0 ( ).
In this way, an IPS is therefore constructed, along with the particle approximations of ( ; ≥ 0) and ( ; ≥ 0). Now, let us provide the dynamic of the Feynman-Kac IPS.

(i) Initialization:

Set = 0 and = 0; Sample X 0 ∼ ⊗ 0 .
(ii) Absorbing time:

Stop if ( ) = 0 and then set = ( -1) ∨ 0.

(iii) Markov transition:

Set = + 1 and = + 1; Sample X p ∼ =1 , -1 ( -1 , •); Go to step (ii).
By de nition, it is obvious that (X ; ≥ 0) is a Markov chain with absorption taking values in . Roughly speaking, the IPS can be seen as a numerical technique that uses a Markov chain de ned in a high dimensional space to approximate a McKean chain in low dimensional space. Another remark is that all the details in the mechanism of resampling are encoded in the design of the McKean kernel , . For example, in HMM discussed in Section 1.1.1, the Bootstrap Filter corresponds to the case where = 0. This resampling scheme is often referred to as multinomial resampling scheme. The setting is thus referred to as Multinomial SMC. More concretely, when = 0, we have

, ( , ) = ( ) ( -1 )
.

Let -1 denote the empirical measure de ned in (1.4). It is then easy to verify that

, -1 ( , ) = =1 -1 ( -1 ) =1 -1 ( -1 ) ( -1 , ).
In practice, in order to sample -1 according to this kernel, we rst sample randomly a parent index based on the values of the potential functions

-1 ∼ =1 -1 ( -1 ) =1 -1 ( -1 )
, which corresponds to a multinomial selection mechanism. Then, based on the parent index, a Markov transition step is executed and often referred to as a mutation step:

∼ ( -1 -1 , •).
For the Subset Simulation discussed in Section 1.1.2, the mechanism of the algorithm corresponds to the case = 1. In this case, ∞ = 1 since = 1 . Di erent from the previous case where = 0, each particle performs a Bernoulli survival test based on its potential:

-1 ∼ -1 ( -1 ) 1 + 1 --1 ( -1 ) 0 .
Then, given the results of the survival tests ( -1 , 1 ≤ ≤ ), the killed particles, namely, the particle such that -1 = 0, execute a multinomial resampling. Finally, all the particles, i.e., the surviving ones and the resampled ones, mutate according to the Markov kernel . In this way, by executing the mechanism encoded in the McKean kernel , , we are able to construct an Interacting Particle System (IPS), along with the particle approximations ( ; ≥ 0) and ( ; ≥ 0) of the Feynman-Kac measures. Both of these cases are intensely studied in the nonadaptive context and many theoretical results, such as consistency and asymptotic normality of and , can be found in the literature such as [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF][START_REF] Moral | Mean eld simulation for Monte Carlo integration[END_REF].

Genealogy, survival history and variance estimation

We call the parent indices ( ; 1 ≤ ≤ , ≥ 0) the genealogy of an IPS and we call the results of the Bernoulli survival tests ( ; 1 ≤ ≤ , ≥ 0) the survival history of an IPS. These structures provide additional information on the uctuation of the particle system. In particular, we are interested by the variance estimation problem of the Feynman-Kac particle models. Chan and Lai [START_REF] Chan | A general theory of particle lters in hidden Markov models and some applications[END_REF] have shown that one can derive consistent asymptotic variance estimators for using the genealogy of the particle system. Combined with the asymptotic normality, one is able to conduct statistical inference with a single run of the algorithm. Then, Lee and Whiteley [START_REF] Lee | Variance estimation in the particle lter[END_REF] provide a more re ned analysis on the genealogy of the particle system, and an unbiased non-asymptotic variance estimator for is proposed. They also provide new asymptotic variance estimators for both and , for which each term found in the decomposition of the asymptotic variance can be estimated separately. All of these studies are done under the multinomial resampling scheme, namely, for the case = 0. In this thesis, we provide new variance estimators for the case = 1, and it turns out that the information encoded in the survival history has to be taken into consideration. The study on these additional structures of the IPS is still new in the domain and very few theoretical tools are available.

Symmetric and asymmetric resampling schemes

Now, let us return to the static AMS methods introduced in Section 1.1.2. Besides the di erence in the choice of levels, we have mentioned that the mutation mechanism is also di erent: in the static AMS setting, we only apply the mutation procedure to the resampled particles, while the surviving particles stay at the same place. For a given level , let us denote

( ) := 1 { ( ) > } .
It is easy to check that this resampling scheme can be reformulated by the following McKean kernel:

, ( , ) := -1 ( ) ( ) + (1 --1 ( )) ( ) ( -1 )
.

To the best of our knowledge, this setting is not covered in the literature of classic Feynman-Kac particle models. To make it clearer, we consider a slightly generalized setting. We assume that the sequence of potential functions ( ; ≥ 0) is [0, 1]-valued and two sequences of Markov kernels ( ˚ ; ≥ 1) and

( • ; ≥ 1) exist, such that ˚ , = • , = ,
where ˚ , and

•

, denote respectively the Feynman-Kac semigroups de ned by ˚ ( ,

) := -1 ( ) ˚ ( , ),
and

• ( , ) := -1 ( ) • ( , ).
To give an idea, an illustration of the measures ow is presented in Figure 1.4. ( -1 ) .

0 1 +1 • • • ˚ 1 • 1 ˚ +1 • +1 ˚ 1, • 1,
This setting will be referred to as Asymmetric SMC in this thesis, to emphasize that the surviving particles and the resampled particles mutate according to di erent Markov kernels. In particular, when ˚ ≡

• for all ≥ 1, that is, all the particles mutate according to the same mutation kernel at each step, we say that the resampling scheme is symmetric. The setting thus returns to the classic Feynman-Kac particle models for the case = 1 when the potential functions ( ; ≥ 0) are [0, 1]-valued. With a slight abuse of language, this setting is referred to as Symmetric SMC, to emphasize that the surviving particles and the resampled particles mutate according to the same mutation kernel. It is well-known that symmetric SMC admits smaller asymptotic variance than Multinomial SMC (see, e.g., Section 3.6.1). In general, the only additional requirement to implement Symmetric SMC compared to Multinomial SMC is that an upper-bound of the potential function needs to be tractable, in order to de ne "normalized" potential functions that are [0, 1]-valued.

We also want to mention that in the static AMS algorithm, if is a function that only takes nite values, and in addition, the number of particles to kill * is set to be 1, the static AMS algorithm enters the Asymmetric SMC framework. This important and nontrivial observation is inspired by the equivalence between SMC and gAMS revealed by Mathias Rousset in personal communications and the reader is referred to [START_REF] Cérou | On Synchronized Fleming-Viot Particle Systems[END_REF] for discussion on this topic. This is the original motivation for the introduction of Asymmetric SMC, which is one of the main topics discussed in this thesis.

It is well-known that the estimation given by the unnormalized measure is unbiased in the classic Feynman-Kac particle models, namely, when all the particles mutate according to the same Markov kernel at each iteration. However, if the underlying resampling scheme is asymmetric, this lack-of-bias property will be lost in general. At the same time, the consistency and the asymptotic normality are still valid. Considering the diversity of applications that can be reformulated as the estimation of Feynman-Kac measures, we hope that Asymmetric SMC framework may also provide new insights in other disciplines.

Rare-event simulation in Molecular Dynamics

In this section, we present a concrete example in Molecular Dynamics, namely, rare-event estimation in the overdamped Langevin dynamics with two metastable states. This example is the prime motivation for the gAMS framework widely used to sample reactive trajectories in Molecular Dynamics. The interested reader is referred to [BGG + 16] for more details and a very rich list of numerical experiments.

Besides the original gAMS algorithm proposed in [BGG + 16], we also provide a variant with multinomial resampling scheme, which is referred to as Multinomial gAMS. The reference mutation kernel (denoted by in [BGG + 16]) is set to be the most popular one in the AMS context (cf. Figure 1.7). Since the regularity assumptions on the mutation kernel, especially Assumption 2 in [BGG + 16], are fundamental for the unbiasedness of gAMS, and quite abstract in their original form, we also provide some intuitive interpretations. The connection between gAMS and Asymmetric SMC framework is revealed through recent developments on the level-indexed process, which provide novel approaches to study the properties of gAMS thanks to Fleming-Viot particle systems. The reader is referred to [START_REF] Cérou | On the Asymptotic Normality of Adaptive Multilevel Splitting[END_REF] for a more profound and detailed introduction. We also present the gap that exists in theoretical analysis, where the Asymmetric SMC framework failed to cover, and why its continuous-time generalization is an interesting topic to explore. Finally, we discuss some particular generalizations of gAMS algorithm that enter into Asymmetric SMC framework or its continuous-time generalization, and we explain heuristically why they are promising in solving the multi-channel problems in Molecular Dynamics.

Overdamped Langevin dynamics and metastability

The Markov chain X = ( , ∈ N) is de ned by the discretization of the overdamped Langevin dynamics:

∀ ∈ N, +1 -= -∇ ( )ℎ + 2 -1 ( ( +1)ℎ -ℎ ).
(1.5)

Particularly, we let 0 = 0 ∈ := R 3 and ∈ R 3 is a random variable giving the positions of particles in R 3 at time ℎ with ℎ > 0 being the time step size. The function : ↦ → R is the potential energy function of the system. The constant = ( ) -1 denotes the inverse temperature and ( ; ∈ R + ) is a 3 -dimensional Wiener process on . We denote ( , ) the transition kernel of X.

A metastable state for X is an open subset of the state space, such that when the stochastic process X is trapped in it, it takes an extremely long time to escape. In the overdamped Langevin dynamics, a metastable state is typically a well in the energy landscape. In Transition Path Theory (see, e.g. [START_REF] Vanden-Eijnden | Transition Path Theory[END_REF]), it is important to study the reactive trajectories that transit between such metastable states, which is essentially a rare-event simulation problem. Let us denote , two metastable states in R 3 . We assume that 0 ∉ ∪ but 0 is close to . As is illustrated in Figure 1.5, our goal is to estimate the probability of X reaching before , i.e., * := P 0 ( <

) = P ( < | 0 = 0 ) , (1.6)
where the stopping times are de ned respectively by

:= inf { ∈ N : ∈ } and := inf { ∈ N : ∈ }. • 0 rare event of interest R 2
Figure 1.5: Schematic picture of a 2-dimensional toy example with the level sets of (dashed lines).

Reaction coordinate

A reaction coordinate : \( ∪ ) ↦ → R is a function designed to measure the advance of a reactive trajectory towards the metastable state (cf. Figure 1.6). Generally speaking, the introduction of reaction coordinate can be regarded as a model reduction technique that uses a one dimensional summary statistics to collect information on the original Markov process or Markov chain. We suppose that there exists * ∈ R such that

⊂ { ∈ : ( ) ∈] * , ∞[} .
For a given level ∈ R, we denote : N ↦ → N de ned by

∀x = ( ; ∈ N) ∈ N , (x) := inf { ∈ N : ( ) > }.
Similarly, for an open subset ⊂ , we denote

(x) := inf { ∈ N : ∈ }.
We de ne the maximum level of a trajectory x ∈ N by

Ξ(x) := sup ∈N { ( ∧ ∪ (x) )}. (1.7)
Remark that Ξ can be seen as a reaction coordinate in the path space N .

• 0 rare event of interest We know that the choice of the reaction coordinate is crucial for the performance of gAMS, as well as for other rare-event simulation methods in this context. The optimal choice * in terms of asymptotic variance of the gAMS estimator of the probability * is the committor function (see, e.g., [START_REF] Bréhier | Analysis of adaptive multilevel splitting algorithms in an idealized case[END_REF][START_REF] Cérou | Adaptive multilevel splitting: Historical perspective and recent results[END_REF]), which is de ned by * ( )

• • • • • • • * R R 2
= P ( < | 0 = ) . (1.8)
It is known that for the overdamped Langevin dynamics, * is the solution of the following elliptic Partial Di erential Equation:

-∇ • ∇ + -1 Δ = 0 on \( ∪ ),
with the boundary conditions = 0 on ; = 1 on .

A brief introduction to gAMS

Roughly speaking, gAMS is a generalized version of the static AMS algorithm introduced in Section 1.1.2, dedicated to sampling reactive trajectories between metastable states. Before proceeding further, we remark that the resampling in gAMS takes place in the path space E := N , which is indeed a Polish space. Before giving the precise description of the gAMS algorithm, we want to mention that gAMS and static AMS as presented in Section 1.1.2 share the same mechanism when regarded as adaptive Feynman-Kac particle models. More concretely, if we replace the statistics in the static AMS models by Ξ de ned in (1.7), the only di erence between these two algorithms is the construction of the adaptive mutation kernel . In the static AMS models, is built using a Metropolis-Hastings kernel (see Figure 1.3), while in the gAMS framework, the construction of is guided by two regularity assumptions (Assumptions 1 and 2 in [BGG + 16]). A popular choice of that enters into gAMS framework is de ned as follows, and an intuitive illustration is provided in Figure 1.7:

Y ∼ (x, •) ⇔ = if ≤ (X); ∼ ( -1 , •) if > (X). and Y = ∧ ∪ ( ) ∈N
.

(1.9)

Let us consider X := ( ∧ ∪ (X) ; ∈ N) and denote by P Ξ> the law of X given that Ξ(X ) > . It is easy to verify that is indeed P Ξ> -invariant. Notice that this resampling kernel heavily relies on the fact that we are in a dynamical setting, i.e., the random object of interest is a Markov process, whereas the static case discussed in Section 1.1.2 is more general, where the small probability is de ned for any random variable. 0 Now, let us present the mechanism of the gAMS algorithm. This time, we also track the genealogy and the survival history when generating the particle system. Recall that * denotes the minimum number of particles to kill at each resampling step.

• reaction coordinate : ( , ) ↦ → R 2 • • • • • • • • • • • • • • • × × × × × × × × × × × × × × × • • • • • • • • • • • • • • • × × × × × × × × × × × × × × × × × × × × × x : • • • • • • Y : × × • • • ×
(i) Initialization:

Set = = 0;
For = 1, 2, . . . , , sample X ( ,0) independently starting from ( ,0) 0 = 0 ∈ \( ∪ ) with transition kernel described by the time discretization of the overdamped Langevin dynamics (1.5), stopped when reaching or .

(ii) Level calculation:

Order the particles at iteration such that

Ξ(X ( (1), ) ) ≤ Ξ(X ( (2), ) ) ≤ • • • ≤ Ξ(X ( ( -1), ) ) ≤ Ξ(X ( ( ), ) ),
where denotes a permutation on [ ];

Set

= Ξ(X ( ( * ), ) );

Stop the algorithm if > * .

(iii) Absorbing time:

Denote by the set of the indices of the surviving particles at iteration ,

= ∈ [ ] Ξ(X ( , ) ) > .
Stop the algorithm if = and then set = ( -1) ∨ 0.

(iv) Selection:

For ∈ , set X ( , +1) = X ( , ) , = and = 1;

For ∈ [ ]\ , uniformly draw a random index α ∈ and set X ( , ) = X (α , ) , = α and = 0.

(v) Mutation:

Set = + 1 and = + 1; For ∈ [ ]\ , sample X ( , ) ∼ -1 ( X ( , ) , •), where -1
is de ned in (1.9); Go to step (ii).

An unbiased estimator * of the rare-event probability * is de ned by

* := # (X ( , ) ) < (X ( , ) ) -1 =0 # × 1 > * .
In particular, if the selection step (iv) and the mutation step (v) are replaced by the following variants, the corresponding algorithm will be referred to as Multinomial gAMS, for the fact that it can thus be reformulated as an adaptive Feynman-Kac particle model with multinomial resampling scheme. The Multinomial gAMS serves as a benchmark to be compared with the original gAMS algorithm, but is of no interest in practice.

• Selection:

For ∈ [ ], uniformly draw a random index α ∈ and set X ( , ) = X (α , ) , +1 = α and = 0.

• Mutation:

Set = + 1 and = + 1; For ∈ [ ], sample X ( , ) ∼ -1 ( X ( , ) , •); Go to step (ii).
Remark that in Multinomial gAMS, there is no need to track the survival history, as all the particles are resampled at each iteration of the algorithm.

Level-indexed process

In static AMS, there is no unbiasedness property in general. However, by the nontrivial assumptions studied in [BGG + 16], the gAMS turns out to provide unbiased estimations. This in practice allows the implementation of parallel computing. The consistency is therefore ensured by running multiple gAMS algorithms and using the average as the nal estimator. To the best of our knowledge, the consistency and uctuation analysis of gAMS w.r.t. the particle number are still open problems. Recent results can be found in [START_REF] Cérou | On the Asymptotic Normality of Adaptive Multilevel Splitting[END_REF], where the underlying dynamic is a continuous-time Markov process. The key ingredient is a nontrivial construction called levelindexed process. Without loss of generality, we suppose that ( 0 ) = 0. For the Markov chain X = ( ; ∈ N), we de ne the level-indexed process Y = ( ; 0 ≤ ≤ * ) by 0 = 0 and

= ∧ * if (X) < (X); = ∂ if (X) ≥ (X), (1.10)
where ∂ denotes the cemetery point. To simplify the notation, we assume that = { > * }.

In this case, we consider the potential function ( ) := 1 ≠∂ . First, we provide an intuitive interpretation of Assumption 2 of [BGG + 16]. Let be a P Ξ> -invariant kernel. When a reference trajectory x is resampled according to , the part ( 0 , 1 , . . . , (x)-1 ) can not be changed. Otherwise, the mutation kernel does not enter into gAMS framework. Roughly speaking the assumptions of [BGG + 16] can be regarded as the condition such that gAMS w.r.t. X can be reformulated as some continuous-time Symmetrical SMC w.r.t. Y, namely, the Fleming-Viot particle systems. Therefore, the unbiasedness property is provided by the structure of the continuoustime Feynman-Kac particle models. In particular, it is easily checked that in the case where can only take nite values, say 0, 1, 2, . . . , * , the estimation of * can be reformulated by Symmetric SMC framework w.r.t. ( ; 0 ≤ ≤ * ). However, when these assumptions are not veri ed, it is then impossible to construct the level-indexed process in general. Fortunately, we still can reformulate the gAMS algorithm as Asymmetric SMC models, expressed in its continuous-time generalizations on the path space. In such scenario, our work on the discrete-time Asymmetric SMC failed to provide theoretical guarantee. So, it would be interesting to explore the continuous-time Asymmetric SMC framework, which is left for future research.

In our analysis, we will also discuss generalizations of gAMS to new resampling kernels. The motivation for introducing the resampling kernels that do not enter into gAMS framework is the multi-channel problem, which is also presented in [BGG + 16]. Roughly speaking, starting from the neighborhood of , the reactive trajectories may reach from di erent channels that are extremely unbalanced in terms of probability. This induces large variances on the estimators. The multi-channel problem is partially due to the dependence between the trajectories, especially by the fact that a resampled trajectory and its parent share the same piece before the time when they rst pass the reference level of the mutation kernel. To counter this, an elementary approach is to conduct rejection sampling from a lower level, which is not always practical due to the uncontrollable computational costs. A better choice is the Particle Markov Chain Monte Carlo (PMCMC) [START_REF] Andrieu | Particle Markov chain Monte Carlo methods[END_REF] sampler since one can easily construct a P Ξ> -invariant kernel by implementing another gAMS using the freezing technique. The computational costs can therefore be controlled by the size of the IPS. In both cases, the mutation kernels do not enter into gAMS framework, even though the invariant property w.r.t. P Ξ> is veri ed.

Summary of the Chapters

In this section, we summarize the main contributions of the present thesis. We also brie y present some intuition or heuristic knowledge we have collected when studying these subjects, which may help future developments on these topics.

Chapter 2: Variance estimation in Adaptive Sequential Monte Carlo

A generalized version of the Adaptive SMC via summary statistics introduced in Section 2 of [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] is investigated. We provide the conditions under which the asymptotic normality is available, such that the asymptotic variance is identical to a "limiting" non-adaptive SMC model. At the same time, we prove that the variance estimators proposed by Lee and Whiteley in a non-adaptive context [START_REF] Lee | Variance estimation in the particle lter[END_REF] are still consistent in the adaptive context.

Main results

Let ( ; ≥ 0) be a sequence of Polish spaces. For each level ≥ 1, we consider a family of potential functions -1, : -1 ↦ → R + and Markov kernels , : ( -1 , B ( )) ↦ → [0, 1], parametrized by ∈ R . Accordingly, we de ne the family of Feynman-Kac kernels , by , ( , ) := -1, ( ) , ( , ).

We suppose that there exists a sequence of reference parameters ( * ) ≥0 and, for each ≥ 1, we denote ( -1, ( -1 ) ) .

This is an adaptive Feynman-Kac particle model under multinomial resampling scheme. Now, we introduce the regularity assumptions required to establish asymptotic normality and to conduct variance estimation. They are a slight generalization of the ones presented in [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] to obtain consistency and asymptotic normality in the same context.

Assumptions For each ≥ 0, we assume that , is strictly positive and bounded uniformly over ∈ R , i.e.,

,• ∞ := sup ( , ) ∈ ×R , ( ) < +∞.
For any test function +1 ∈ B ( +1 ), there exists a measurable function ℎ :

( × R , B ( ) ⊗ B (R )) → (R , B (R )) such that, for all ( , ) ∈ × R , +1, ( +1 ) ( ) -+1 ( +1 ) ( ) = ℎ ( , ), - * .
The function ℎ is assumed to satisfy the following properties:

• The Euclidean norm |ℎ | is bounded over × R by ℎ ∞ ;

• The application ↦ → ℎ ( , ) is continuous at * uniformly over ∈ . More precisely, for any > 0, there exists > 0, such that - * < implies sup ∈ ℎ ( , )ℎ ( , * ) < ;

• ℎ satis es the equality ℎ (•, * ) = 0.

Moreover, the summary statistics = ( 1 , . . . , ) satis es * = ( ) and is such that, for all ∈ [ ], belongs to B ( ).

Under these assumptions, we have, for any test function ∈ B ( ),

√ ( ) -( ) d -----→ →∞ N 0, 2 ( ) , and
√ ( ) -( ) d -----→ →∞ N 0, 2 ( -( )) .
Moreover, convergent variance estimators of the asymptotic variances are provided respectively by

(1) 2 ( )

P -----→ →∞ 2 ( ), and ( -( ))
P -----→ →∞ 2 ( -( )),
with, if denotes the index of the ancestor of the particle at step 0,

( ) := ( ) 2 - -1 ( -1) +1 ≠ ( ) ( ).
Note that these estimators are the same as the ones proposed by Lee and Whiteley in [START_REF] Lee | Variance estimation in the particle lter[END_REF] in a non-adaptive context.

Other comments

In Chapter 2, we provide a methodology such that the adaptiveness of the Feynman-Kac particle models and the variance estimation problem can be treated separately. The theoretical tools we developed are mainly inspired by the pioneering work [START_REF] Lee | Variance estimation in the particle lter[END_REF] of Lee and Whiteley, and we made some modi cations such that the randomness brought by the adaptiveness is easier to deal with. As a consequence, the framework is also more consistent with previous works such as [START_REF] Cérou | A nonasymptotic theorem for unnormalized Feynman-Kac particle models[END_REF] and [START_REF] Del Moral | Tree based functional expansions for Feynman-Kac particle models[END_REF]. We expect that the adaptive models that are not covered by our regularity assumptions, such as [START_REF] Cérou | Fluctuation analysis of adaptive multilevel splitting[END_REF], can also be studied in a similar way. Heuristically speaking, the regularity assumptions needed to conduct variance estimation are in general weaker than the ones required to establish asymptotic normality. Therefore, the take-home message is simple: it is expected that when an adaptive model has the same asymptotic behaviors as a "limiting" nonadaptive model, the asymptotic variance estimators are also available and are identical to the ones that can be derived in the corresponding non-adaptive context.

Chapter 3: Asymmetric Sequential Monte Carlo

As brie y discussed in Section 1.1.3, Asymmetric SMC is proposed as a generalization of the classic Feynman-Kac particle models, aiming at reducing the computational costs brought by the mutation kernels at each iteration of the algorithm. In a very general setting, we provide consistency and asymptotic normality for the Asymmetric SMC framework, which, in particular, also covers the Symmetric SMC framework. In addition, we construct consistent and e cient asymptotic variance estimators, that are available as a by-product of the simulation of a single run of the particle system. Moreover, we also propose unbiased variance estimators for the unnormalized measure under Symmetric SMC framework. Since the computations of the variance estimators are highly nontrivial, we provide detailed and e cient algorithms w.r.t. time and space complexity in Section 3.5.

Main results

Let ( ; ≤ 0) be a sequence of Polish state spaces. Given a sequence of [0, 1]-valued potential functions ( ; ≥ 0) and a canonical Markov chain ( ; ≥ 0) taking values in ( ; ≥ 0), with initial distribution 0 and transition kernels ( ˚ ; ≥ 1), we de ne

( ) := E ( ) -1 =0 ( ) .
Assuming that (1) > 0 for any ≥ 0, we also de ne ( ) := ( )/ (1). For any test function ∈ B ( ), when the number of particle tends to in nity, the estimators given by Algorithm 2 in Section 3.5, denoted respectively by ( ) and ( ), converge almost surely to ( ) and ( ) if for any ≥ 1, we have

∀ ∈ B ( ), -1 ( -1 × ˚ ( )) = -1 ( -1 × • ( )).
Moreover, we also have

√ ( ) -( ) ˆ ( ) d -----→ →∞ N (0, 1), as well as √ ( ) -( ) ˆ ( -( )) d -----→ →∞ N (0, 1),
where the computations of ˆ ( ) and ˆ ( -( )) are respectively provided in Algorithm 5 and Algorithm 6 in Section 3.5. Moreover, under the condition discussed in Section 3.4.1, which at least contains the case where • ≡ ˚ for any ≥ 1, the estimator ( ) is an unbiased estimator for ( ). Moreover, the estimator provided by Algorithm 7 is an unbiased estimator for the non-asymptotic variance of ( ).

Other comments

The theoretical tools introduced in Chapter 3 can be regarded as a development of the ones introduced in Chapter 2, in order to deal with the challenge brought by the asymmetry in the associated particle system. Although no rigorous mathematical foundation of the adaptive Asymmetric SMC is provided in this chapter, we expect that in the same spirit as for the Adaptive SMC framework studied in Chapter 2, the variance estimators would still be valid if the underlying resampling scheme is also changed accordingly. Many by-products of the analysis on the variance estimators can be found in Section 3.7.8, which also help to understand the di erence between Symmetric and Asymmetric SMC frameworks. We also expect that a similar methodology can be applied to derive variance estimators for Fleming-Viot particle systems [START_REF] Delyon | A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing[END_REF], which enters the continuous-time generalization of Symmetric SMC. In this respect, since we have mentioned many times that gAMS enters into Asymmetric SMC framework when the reaction coordinate only takes nite values and the number of particles to kill is set to be * = 1, it would also be interesting to explore the continuous-time generalization of Asymmetric SMC framework, i.e., when can be chosen as a continuous function.

Chapter 4: Estimating committor function with Mondrian Forests

As brie y mentioned in Section 1.2.2 and illustrated in the rich list of numerical experiments in [BGG + 16], the choice of the reaction coordinate is crucial to the performance of gAMS and many other rare-event estimation algorithms in Molecular Dynamics. In many cases, the optimal choice is the so-called committor function * de ned in (1.8). Interestingly, gAMS can also be used to estimate the values of the committor function. Hence, if we provide a regressor that uses the estimations of gAMS, with some conventional reaction coordinate, as the training data, and we use the trained model as the reaction coordinate to rerun gAMS algorithm to generate new training data, it is expected that such iterative updating strategy may help to improve the accuracy of both gAMS and the regressor. This is the crucial idea of Chapter 4.

In this chapter, we provide some preliminary analysis on the principles of the choice of the regressor, and our choice is a new member in the family of Random Forests called Mondrian Forests, recently proposed in [START_REF] Balaji Lakshminarayanan | Mondrian forests: E cient online random forests[END_REF]. Roughly speaking, Mondrian Forests are a Random Forests-based model such that online learning is available, and implemented in an elegant way. Several interaction strategies between gAMS and Mondrian Forests are presented and we also discuss some possible improvements that can be applied to make Mondrian Forests even more suitable.

Chapter 2

Variance Estimation in Adaptive Sequential Monte Carlo abstract: Sequential Monte Carlo (SMC) methods represent a classical set of techniques to simulate a sequence of probability measures through a simple selection/mutation mechanism. However, the associated selection functions and mutation kernels usually depend on tuning parameters that are of rst importance for the e ciency of the algorithm. A standard way to address this problem is to apply Adaptive Sequential Monte Carlo (ASMC) methods, which consist in exploiting the information given by the history of the sample to tune the parameters. This chapter is concerned with variance estimation in such ASMC methods. Speci cally, we focus on the case where the asymptotic variance coincides with the one of the "limiting" Sequential Monte Carlo algorithm as de ned by Beskos et al. [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF]. We prove that, under natural assumptions, the estimator introduced by Lee and Whiteley [START_REF] Lee | Variance estimation in the particle lter[END_REF] in the nonadaptive case (i.e., SMC) is also a consistent estimator of the asymptotic variance for ASMC methods. To do this, we introduce a new estimator that is expressed in terms of coalescent tree-based measures, and explain its connection with the previous one. Our estimator is constructed by tracing the genealogy of the associated Interacting Particle System. The tools we use connect the study of Particle Markov Chain Monte Carlo methods and the variance estimation problem in SMC methods. As such, they may give some new insights when dealing with complex genealogy-involved problems of Interacting Particle Systems in more general scenarios.

Introduction

Sequential Monte Carlo (SMC) methods are classical Monte Carlo techniques widely used in Bayesian inference, ltering, rare events simulations and many other elds (see for example [START_REF] Doucet | Sequential Monte Carlo methods in practice[END_REF] and references therein). The principle is to approximate a sequence of probability measures ( ) ≥0 by simulating an Interacting Particle System (IPS) via an importance sampling and resampling mechanism. The ow of measures is then approximated by the empirical version ( ) ≥0 . A lot of convergence results when the sample size goes to in nity can be found in the literature (see for example [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF][START_REF] Moral | Mean eld simulation for Monte Carlo integration[END_REF]).

In practice, when applying these SMC methods, it is also very important to have a control on the constructed estimators, such as con dence intervals. For this, if one has a CLT type theorem for the test function such as (see, e.g., [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF][START_REF] Chopin | Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference[END_REF][START_REF] Douc | Limit theorems for weighted samples with applications to sequential Monte Carlo methods[END_REF])

√ ( ) -( ) d -----→ →∞ N (0, ( ) 2 ),
it su ces to provide a consistent estimator ( ) of ( ) since Slutsky's lemma then ensures that √ ( ) -( )

( ) d -----→ →∞ N (0, 1).
A natural way to achieve this aim is by resimulating the IPS independently many times and by estimating ( ) 2 with the crude variance estimator. However, since a single run of the algorithm may take a lot of time, this is usually intractable. In addition, as the estimator ( ) of ( ) provided by SMC is typically biased, it is also nontrivial to implement parallel computing for a large number of IPS with relatively small. As a consequence, a variance estimator available with a single run of the simulation is of crucial interest for applications.

The rst consistent estimator of this type was proposed by Chan and Lai [START_REF] Chan | A general theory of particle lters in hidden Markov models and some applications[END_REF], by using the ancestral information encoded in the genealogy of the associated IPS. Then, Lee and Whiteley [START_REF] Lee | Variance estimation in the particle lter[END_REF] proposed an unbiased variance estimator for the unnormalized measures and a term by term estimator, with insights on the genealogy of the IPS. Both estimators are studied in the classical SMC framework, meaning in a nonadaptive setting where the weight functions and the Markov proposal kernels are xed a priori.

In this chapter, we deal with adaptive SMC methods. At each resampling step, the weight functions and/or Markov proposal kernels depend upon the history of the simulated process. The idea is to approximate an ideal "limiting" SMC algorithm, which is usually out of reach, by exploiting the induced information tracked by some summary statistics. Such approaches are expected to be more e cient and more automated than the nonadaptive ones since they require less user-speci ed tuning parameters.

Speci cally, we are interested in the case where the adaptive SMC algorithm is asymptotically identical to a "limiting" SMC algorithm. More precisely, we expect the asymptotic variance of the adaptive SMC algorithm to be identical to the "ideal" nonadaptive one. This kind of stability property is at the core of the pair of articles [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] and [START_REF] Cérou | Fluctuation analysis of adaptive multilevel splitting[END_REF]. The framework discussed in the present paper is just a slightly generalized version of the one presented in Section 2 in [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] but still ensures the stability property of their Theorem 2.3.

Another remark is about Adaptive Multilevel Splitting (AMS), also known as Subset Simulation, see for example [AB01, AB03, CG07, CDMFG12, CG16]. This is a class of ASMC algorithms dedicated to rare event estimation and simulation. Despite the fact that our assumptions are not veri ed in the AMS framework, we expect that the variance estimator would also work in this context. Nonetheless, we believe that this case requires a speci c analysis as well as di erent assumptions. To account for this, one can notice that the proofs in [START_REF] Cérou | Fluctuation analysis of adaptive multilevel splitting[END_REF] and [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] di er in many points, although the take-home message is the same. In a nutshell, the main di culty in the AMS framework comes from the indicator functions in the potential functions as well as in the transitions kernels, leading to severe regularity issues when dealing with CLT type results and asymptotic variances.

From a theoretical viewpoint, to prove the consistency of the variance estimator proposed in [LW18], we were not able to adapt their technical tools. This is due to the additional randomness brought by the weight functions and Markov kernels in the adaptive case. As a consequence, we propose to develop new techniques in order to estimate the terms Γ that appear in the expansion of the variance given in [START_REF] Cérou | A nonasymptotic theorem for unnormalized Feynman-Kac particle models[END_REF]. The mains ideas are: rst, our term by term estimator is consistent and, second, the di erence between our estimator and the one of Lee and Whiteley goes to 0 in probability when the sample size goes to in nity. However, in practice, one uses the estimator proposed by Lee and Whiteley, which is computationally very simple, while the one we introduce here may be seen as a handy tool to prove the consistency of the former.

The construction of our estimators Γ , uses the idea of many-body Feynman-Kac models, which were designed in [START_REF] Del Moral | On particle Gibbs samplers[END_REF] to study propagation of chaos properties of Conditional Particle Markov Chain Monte Carlo methods [START_REF] Andrieu | Particle Markov chain Monte Carlo methods[END_REF]. Above the speci c context of the present chapter, these connections may give some insights on how to deal with complex genealogyinvolved problems in more general settings.

Notation

Before proceeding, let us provide some notation that will be of constant use in the following.

• For any Polish space , we denote respectively by M ( ), M + ( ) and P ( ) the sets of signed nite measures, nonnegative nite measures, and probability measures on ( , B ( )) while B ( ) denotes the collection of the bounded measurable functions from ( , B ( )) to (R, B (R)) equipped with uniform norm • ∞ .

• For any ∈ M ( ) and any test function ∈ B ( ), we write

( ) := ∫ ( ) ( ).
A nite nonnegative kernel from ( , B ( )) to ( , B ( )) is a function

: × B ( ) ↦ → R +
such that, for all ∈ , ( , •) ∈ M + ( ) and, for all ∈ B ( ), ( , ) is a B ( )measurable function. We say that is a Markov transition kernel if, moreover, for all ∈ , ( , •) is a probability measure in P ( ). For a signed measure ∈ M ( ) and a test function ∈ B ( ), we denote respectively by ∈ M ( ) and ( ) ∈ B ( ) the measure and function respectively de ned by

( ) := ∫ ( ) ( , ) ∀ ∈ B ( ), and
( ) ( ) := ∫ ( , ) ( ) ∀ ∈ .
Given two nite nonnegative kernels 1 and 2 respectively from 0 to 1 and 1 to 2 , 1 2 is the nonnegative kernel from 0 to 2 de ned by 1 2 ( ,

) := ∫ 1 1 ( , ) 2 ( , ) ∀( , ) ∈ 0 × B ( 2 ).
• For two functions , ∈ B ( ), their tensor product is the function

⊗ : 2 ( , ) ↦ → ( ) ( ) ∈ R,
and, in particular, we denote ⊗2 := ⊗ . For two nite nonnegative kernels and from ( , B ( )) to ( , B ( )), we denote

( ⊗ ) (( , ), ( ,
)) := ( , ) × ( , )
for all ( , ) ∈ × and all ( , ) ∈ B ( ) ⊗ B ( ). Accordingly, we write ⊗2 := ⊗ .

• In order to de ne the coalescent tree-based measures of size 2, we introduce the transition operators 0 and 1 as 0 (( , ), ( , )) := ( , ) ( , ), and 1 (( , ), ( , )) := ( , ) ( , ).

In other words, for any measurable function : × ↦ → R, we have 0 ( ) ( , ) = ( , ) and 1 ( ) ( , ) = ( , ).

• For all x = ( 1 , . . . , ) ∈ , we de ne the empirical measure associated to x by

(x) := 1 =1 ∈ P ( ).
We also denote ⊗2 (x) := 1 2 , ( , ) ∈ P ( 2 ),

and 2 (x) := 1 ( -1) ≠ ( , ) ∈ P ( 2 ).
A straightforward computation shows that

⊗2 (x) = -1 2 (x) 0 + 1 2 (x) 1 . (2.1)
With a slight abuse of notation, considering [ ] := {1, 2, . . . , }, we write

( [ ]) := 1 =1 and ⊗2 ( [ ]) := ( [ ]) ⊗ ( [ ]).

Adaptive Sequential Monte Carlo

This section presents the formal de nition and the regularity assumptions of the ASMC framework studied in this chapter. The motivation is mainly from ASMC via summary statistics introduced in Section 2 of [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF]. We refer the reader to the latter for details on motivating examples such as ltering or sequential Bayesian parameter inference.

Framework

The notations that are adopted are essentially those in the pair of books [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF][START_REF] Moral | Mean eld simulation for Monte Carlo integration[END_REF]. Let ( , B ( )) ≥0 be a sequence of Polish spaces. For each level ≥ 1, we consider a family of potential functions -1, :

-1 ↦ → R + and Markov kernels , : ( -1 , B ( )) ↦ → [0, 1] parametrized by ∈ R . Accordingly, we de ne the family of nonnegative Feynman-Kac kernels , by , ( , ) := -1, ( ) , ( , ). We suppose that there exists a sequence of reference parameters ( * ) ≥0 and, for each ≥ 1, we denote

-1 := -1, * -1 , := , * -1
and := , * -1 . Starting with a known probability measure 0 := 0 ∈ P ( 0 ), we de ne the unnormalized Feynman-Kac measures by := 0 1 • • • , along with the normalized measures := 1 (1) .

Assumption 1 below ensures that, for all ≥ 0, is strictly positive so that

(1) = -1 =0 ( ) > 0.
Another formulation of the connection between normalized and unnormalized measures is thus given by

(1) = ( ) -1 =0 ( ). (2.2)
For < , we de ne the Feynman-Kac semigroup

, := +1 • • • ,
and , ( , ) := ( ). In this context, ASMC algorithms aim at approximating the sequences of measures ( ) ≥0 and ( ) ≥0 by exploiting some summary statistics : ↦ → R such that, for all ≥ 0, we have ( ) = * .

ASMC algorithm

In practice, ASMC and SMC algorithms share the same selection/mutation mechanisms. However, since in most situations of interest the parameters ( * ) ≥0 are not analytically tractable, the potential functions ( ) ≥0 and transition kernels ( ) ≥1 are estimated on the y through the design of an adaptive algorithm. Let ∈ N * be the number of particles (or samples). The Interacting Particle System (IPS) associated to the ASMC algorithm is a Markov chain (X n ) ≥0 taking values in ( , B ( ) ⊗ ) ≥0 with genealogy (A n ) ≥0 tracking the indice of the parent of each particle at each level. Specifically, -1 = means that the parent of the particle at layer is -1 at layer -1. The estimation of the normalized measure is given by the empirical measure

:= 1 =1 .
At each level ≥ 0, the estimated parameters are de ned by := ( ). In order to lighten the notation, we denote

-1, := -1, -1 , , := , -1
, and , := , -1 .

Then, considering (2.2), the unnormalized Feynman-Kac measures are estimated by

( ) := ( ) -1 =0 ( , ).
In the following sections, we use the convention

-1 = -1 := 0 .
Let us give the formal de nition of the IPS associated with the ASMC algorithm:

(i) Initial distribution:

At step 0, let X 0 ∼ ⊗ 0 .

(ii) Transition kernels:

For all ≥ 0, set = ( ). The transition +1 is decomposed into two steps:

• Selection: given X p = x , we make an independent multinomial selection of the parent of each particle by

, (x , ) = =1 , ( ) =1 , ( )
(
).

(2.3) Thus, the genealogy of level to level + 1 is tracked by

A p ∼ =1 , (X p , •)
• Mutation: given the parent indices A p = a , each particle at level evolves independently according to the transition kernel +1, , meaning that for ∈ [ ],

+1 ∼ +1, ( , •).

Said di erently, given X p and A p , we have

X p+1 ∼ =1 +1, ( , •)
.

Assumptions

Our assumptions are introduced in a similar way as in [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF], but just slightly weaker. The reason why we can relax their assumptions is because we are only interested in the speci c situation where the asymptotic variance of the ASMC estimator is identical to the "limiting" SMC algorithm which uses ideal potential functions and proposal kernels, namely = , * -1 and =

, * -1 . Considering stability properties, Section 2.7 in [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] explains why this case is particularly interesting in practice. In the following sections, we use A as a shorthand for Assumption.

Assumption 1. For each ≥ 0, we assume that , is strictly positive and bounded uniformly over ∈ R , i.e.,

,• ∞ := sup ( , ) ∈ ×R , ( ) < +∞.
Notice that, under A1, Equation (2.3) above is always well-de ned for the denominator is always strictly positive. In the case where , is only assumed to be nonnegative, as in the AMS framework, one may consider the stopping time de ned by := inf ∈ N : ( , ) = 0 .

We believe that similar techniques can be applied to obtain results of the same taste as in the present paper, but at the cost of considerable technical complications which are out of the scope of this thesis. Let us mention that the strict positivity and boundedness of the potential functions is also required in [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] (see page 1116 and Assumption 1 page 1118). In our second assumption, " •, • " stands for the Euclidean scalar product in R and | • | for the associated norm.

Assumption 2. For any test function +1 ∈ B ( +1 ), there exists a measurable function ℎ :

( × R , B ( ) ⊗ B (R )) → (R , B (R )) such that, for all ( , ) ∈ × R , +1, ( +1 ) ( ) -+1 ( +1 ) ( ) = ℎ ( , ), - * .
The function ℎ is assumed to satisfy the following properties:

• The Euclidean norm |ℎ | is bounded over × R by ℎ ∞ .
• The application ↦ → ℎ ( , ) is continuous at * uniformly over ∈ . More precisely, for any > 0, there exists > 0, such that

- * < implies sup ∈ ℎ ( , ) -ℎ ( , * ) < .
• ℎ satis es the equality ℎ (•, * ) = 0.

Moreover, the summary statistics = ( 1 , . . . , ) satis es * = ( ) and is such that, for all ∈ [ ], belongs to B ( ).

A2 guarantees some regularity properties of the transition kernels , with respect to the parameter and is just a slight generalization of the framework studied in Section 2 of [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF]. Indeed, our function ℎ coincides with the function de ned in (2.17) of [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF], that is

ℎ ( , ) = ∫ 1 0 +1, ( +1 ) ( ) = * + ( - * ) .
As such, the rst two conditions on ℎ are satis ed as soon as Assumption 2 in [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] is veri ed. In this respect, our third condition on ℎ corresponds to their condition (2.19) in Theorem 2.3, which is precisely the "limiting" case mentioned above. Finally, the hypothesis that the summary statistics are bounded is also required in their Assumption 1, while the relation * = ( ) corresponds in their notation to ¯ = -1 ( ).

We also want to mention that the second point is equivalent to

∀ > 0, ∃ ∈ B ( ), ∃ > 0, . . - * < =⇒ ∀ ∈ , ℎ ( , ) -ℎ ( , * ) < ( ) .
We expect that the functions and ℎ can be relaxed to some unbounded functions, belonging for example to L 2 ( ), along with stronger conditions on the test function +1 . We believe that this is one of the main di erences between the ASMC framework studied in [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] and the AMS framework studied in [START_REF] Cérou | Fluctuation analysis of adaptive multilevel splitting[END_REF].

In general, it is not easy to verify the existence of such ℎ . However, we have, at least, a direction to explore in the case where , ( ) is not globally di erentiable with respect to . We also remark that we do not study the consistency of ( ) and ( ) with weaker assumptions, as we are only interested in the CLT type result of Theorem 2.2.1 below and, more speci cally, in the estimation of the asymptotic variance. Nevertheless, let us brie y mention that to establish the consistency of and , one just needs

-1 , ( ) --1 ( ) = p (1)
for any test function ∈ B ( ). This does not require such a strong assumption as A2. However, for CLT type results with the "stable" asymptotic variance, it is necessary that

-1 , ( ) --1 ( ) = p 1 √ .
A stronger regularity assumption like A2 over the parametrization is therefore required.

Central limit theorems

As explained before, the present chapter only deals with the case where the asymptotic variance is identical to the "limiting" one, which is only a special case of the Central Limit Theorem 2.2 given in [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF] under slightly weaker assumptions. This is why, in Section 2.4.2, we propose a di erent strategy for the proof.

Theorem 2.2.1. Assume A1-A2. For any test function ∈ B ( ), we have

√ ( ) -( ) d -----→ →∞ N 0, 2 ( ) , and √ ( ) -( ) d -----→ →∞ N 0, 2 ( -( )) ,
where

2 ( ) := =0 (1) ( , ( ) 2 ) -( ) 2 and 2 ( ) := 2 ( )/ (1) 2 .
One can notice that the CLT for is just a consequence of the CLT for , Slutsky's Lemma and the decomposition

√ ( ) -( ) = 1 (1) √ ( -( )) -( -( )) .
The main goal of this chapter is to estimate the asymptotic variances 2 ( ) and 2 ( -( )) by a single simulation of the particle system, exactly as is done by Lee and Whiteley in [START_REF] Lee | Variance estimation in the particle lter[END_REF] in a nonadaptive context.

Variance estimations

In this section, we recall the coalescent tree-based expansion of the variance rstly introduced in [CDMG11] from which we deduce a new variance estimator. We also recall the variance estimator proposed by Lee and Whiteley in [START_REF] Lee | Variance estimation in the particle lter[END_REF] and explain the connection between both estimators.

Coalescent tree-based variance expansion

We call := ( 0 , . . . , ) ∈ {0, 1} +1 a coalescence indicator where = 1 indicates that there is a coalescence at level . De nition 2.3.1. We associate with any coalescence indicator ∈ {0, 1} +1 the nonnegative measures Γ and Γ ∈ M + ( 2 ) de ned for any ∈ B ( 2 ) by

Γ ( ) := ⊗2 0 0 ⊗2 1 1 • • • ⊗2 ( ), and
Γ ( ) := 1 (1) 2 Γ ( ).
When there is only one coalescence at, say, level , we write Γ ( ) ( ) and Γ( ) ( ) instead of Γ ( ) and Γ ( ) (see Figure 2.1). When there is no coalescence at all, that is = (0, . . . , 0), we have

Γ ( ) ( ) = ⊗2 ( ) and Γ( ) ( ) = ⊗2 ( ). . . . . . . . . . . . . 1 -1 +1 +2 +3 1 -1 +1 +2 +3 0 0 Figure 2.1: A representation of the coalescent tree-based measure Γ ( ) .
By de nition, it is easy to verify that

Γ ( ) ( ⊗2 ) = (1) , ( ) 2 .
As noticed in [START_REF] Cérou | A nonasymptotic theorem for unnormalized Feynman-Kac particle models[END_REF], the latter yields alternative representations for the asymptotic variances of Theorem 2.2.1, namely

2 ( ) = =0 Γ ( ) ( ⊗2 ) -Γ ( ) ( ⊗2 ) , (2.4) and 2 ( ) = =0 Γ( ) ( ⊗2 ) -Γ( ) ( ⊗2 ) . (2.5)
As a consequence, if for any coalescence indicator := ( 0 , . . . , ) ∈ {0, 1} +1 , we can construct a consistent estimator Γ , of Γ , then we automatically deduce consistent estimators for the asymptotic variances of Theorem 2.2.1. This is the idea behind our next de nition.

In this de nition, ˜ [2] = ( ˜ 1 , ˜ 2 ) and [2] = ( 1 , 2 ) denote two couples of indices between 1 and , while an ( + 1)-sequence of couples of indices such that 1 ≠ 2 for all 0 ≤ ≤ is written

[2] 0: = ( [2] 0 , • • • , [2] ) ∈ ( ) 2 ×( +1) .
Additionally, we use the notation

[2] = ( 1 , 2
) to shorten the writings.

De nition 2.3.2. For any test function ∈ B ( 2 ) and any coalescence indicator , we de ne the estimator Γ , of the measure Γ by

Γ , ( ) := -1 ( -1) +1 [2] 0: ∈ ( ( ) 2 ) ×( +1) -1 =0 ( [2] +1 , [2] ) ( ) ( [ 2]
),

where ( ˜ [2]
, [2] ) ∈ {0, 1} is an indicator function de ned by

( ˜ [2] , [2] ) := 1 { =0} 1 { ˜ 1 = 1 ≠ ˜ 2 = 2 } + 1 { =1} 1 { ˜ 1 = 1 = ˜ 2 ≠ 2 } .
The estimator of Γ is de ned by

Γ , ( ) = (1) 2 Γ , ( ).
Since 1 ≠ 2 , we also have

( ˜ [2] , [2] ) = 1 { =0} 1 { ˜ 1 = 1 , ˜ 2 = 2 } + 1 { =1} 1 { ˜ 1 = 1 = ˜ 2 } .
Notice that, for = 0, we get Γ 0, ( )

:= 1 ( -1) [2] 0 ∈ ( ) 2 0 ( ) ( [2] 0 0 ) = 1 ( -1) ≠ 0 ( ) ( 0 , 0 ). (2.6)
We also adopt the convention

Γ -1, ( ) = Γ -1, ( ) := ⊗2 0 0 ( ).
A toy example As the de nition of the estimator Γ , is not completely straightforward, we illustrate the idea on a simple example. For this, we consider the IPS of Figure 2.2. Suppose we want to estimate Γ(3) 6 ( ) by Γ(3) 6,5 ( ). We denote * = (0, 0, 0, 1, 0, 0, 0) the corresponding coalescence indicator. In the associated IPS, we have to nd the choices of [2] 0:6 such that 5 0:6 such that (2.7) is veri ed.

=0 * ( [2] +1 , [2] ) = 1. (2.7) step 0 step 1 step 2 step 3 step 4 step 5 step 6
It turns out that there are 4 possible choices, taking into account that ( , ) is not necessarily symmetric in its variables. Namely, the rst couple of ancestral lines is (see Figure 2.3):

• [2] 0:6 = ((5, 3), (4, 3), (2, 5), (2, 4), (2, 5), (1, 3), (2, 4)) ,

• [2] 0:6 = ((5, 3), (4, 3), (2, 5), (2, 4), (5, 2), (3, 1), (4, 2)) . 0:6 such that (2.7) is veri ed.

The second couple of ancestral lines is (see Figure 2.4):

• [2] 0:6 = ((5, 3), (4, 3), (2, 5), (2, 1), (2, 5), (1, 3), (2, 4)) ,

• [2] 0:6 = ((5, 3), (4, 3), (2, 5), (2, 1), (5, 2), (3, 1), (4, 2)) .

Hence, the number of choices of [2] 0:6 where [2] 6 = (2, 4) is 2, and the number of choices of [2] 0:6

where [2] 6 = (4, 2) is also 2. As a consequence, we have

Γ(3) 6,5 ( ) = 2 × 5 5 4 7 ( 2 6 , 4 6 ) + ( 4 6 , 2 6 ) .
Our next result ensures the convergence of our estimators.

Theorem 2.3.1 (Convergence of Γ , ). Assume A1-A2. For any test functions , ∈ B ( ) and for any coalescence indicator ∈ {0, 1} +1 , we have

Γ , ( ⊗ ) -Γ ( ⊗ ) = O p 1 √ .
The proof is given in Section 2.4.3.

Term by term estimator

Considering (2.4), (2.5), and Theorem 2.3.1, we are now in a position to provide term by term variance estimators for 2 ( ) and 2 ( ).

De nition 2.3.3 (Estimators of the asymptotic variances). Given a test function ∈ B ( ), we let

2 , ( ) := =0 Γ ( ) , ( ⊗2 ) -Γ ( ) , ( ⊗2 ) , and
2 , ( ) := =0 Γ( ) , ( ⊗2 ) -Γ( ) , ( ⊗2 ) .
Theorem 2.3.1 ensures the consistency of both Γ ( ) , ( ⊗2 ) and Γ ( ) , ( ⊗2 ). Returning to the equation (2.4), this amounts to say that 2 , ( )

= =0 Γ ( ) , ( ⊗2 ) -Γ ( ) , ( ⊗2 ) P -----→ →∞ =0 Γ ( ) ( ⊗2 ) -Γ ( ) ( ⊗2 ) = 2 ( ).
Similarly, for the consistency of 2 , ( -( )), since by (2.5) we know that

2 ( ) = =0 Γ( ) ( ⊗2 ) -Γ( ) ( ⊗2 ) ,
it su ces to verify that, for any coalescent indicator ,

Γ , -( ) ⊗2 P -----→ →∞ Γ -( ) ⊗2 . (2.8) Clearly, the linearity of Γ , yields Γ , -( ) ⊗2 = Γ , ( ⊗2 ) -( ) Γ , (1 ⊗ ) + Γ , ( ⊗ 1) + ( ) 2 Γ , 1 ⊗2 .
Mutatis mutandis, the same relation holds for Γ -( ) ⊗2 . Since a by-product of Theorem 2.2.1 is that

( ) -( ) = O p 1 √ ,
the veri cation of (2.8) is just a consequence of Theorem 2.3.1 and Slutsky's Lemma. Hence, we have obtained the following result.

Theorem 2.3.2 (Consistency of 2 , and 2 , ). Assume A1-A2. For ∈ B ( ), we have

2 , ( ) -2 ( ) = O p 1 √ , as well as 2 , ( -( )) -2 ( -( )) = O p 1 √ .
Even if the term by term estimator is very natural in theory, the computational cost is quite heavy in practice since one has to trace the whole genealogy of a particle system and calculate all the corresponding terms one by one. Therefore, we do not provide an e cient algorithm to calculate this estimator. Instead, we show in the next section that this estimator can be connected to the one given by Lee & Whiteley in a nonadaptive context (SMC), which is very simple and fast to calculate. Let us also mention that our term by term estimator is di erent from the one introduced in Section 4.1 of [START_REF] Lee | Variance estimation in the particle lter[END_REF]. The interested reader can nd more details on this point in Appendix 2.5.3.

Disjoint ancestral lines estimator

Let us now recall the variance estimator proposed in [START_REF] Lee | Variance estimation in the particle lter[END_REF], which can be seen as a disjoint ancestral lines estimator. Namely, given a test function ∈ B ( ), consider

( ) := ( ) 2 - -1 ( -1) +1 ≠ ( ) ( ), (2.9)
where is the ancestor index of at level 0. Returning to the toy example of Section 2.3.1, the couples ( , ) such that < and ≠ are: (1, 2), (1, 4), (2, 3), (2, 5), (3, 4), (4, 5). In a nonadaptive context (SMC), this is the variance estimator introduced in [LW18] when the number of particles is the same at each step. The reader is referred to [START_REF] Lee | Variance estimation in the particle lter[END_REF] for an e cient algorithm to compute this estimator.

According to our notation, since ≠ corresponds to the case = (0, . . . , 0) = ( ) of disjoint ancestral lines, we may also write

( ) = ( ) 2 -Γ( ) , ( ⊗2 ).
The following proposition makes a connection between ( ) and our estimators. Notice that this result does not depend on A2, but is provided by the structure of the IPS and the underlying multinomial selection scheme. The proof is housed in Section 2.4.5.

Proposition 2.3.1. Assume A1. For any test function ∈ B ( ), we have

( ) -2 , ( ) = O p 1 , and
( -( )) -2 , ( -( )) = O p 1 .
By combining Theorem 2.3.2 and Proposition 2.3.1, we nally obtain the main result of the present chapter.

Theorem 2.3.3. Assume A1-A2. For any test function ∈ B ( ), we have

(1) 2 ( ) -2 ( ) = O p 1 √ , and
( -( )) -2 ( -( )) = O p 1 √ .
Hence, the main message of the present work is that the computationally very simple estimator proposed by Lee and Whiteley in a nonadaptive framework (SMC) is still consistent in an adaptive one (ASMC). However, since we could not adapt easily their proof in our adaptive context, we propose a new approach to show this consistency result. More details on the connection between both estimators are given in Appendix 2.5.3.

As mentioned before, among other ingredients, the tools we use connect the study of Particle Markov Chain Monte Carlo methods and the variance estimation problem in SMC methods. As such, more generally, they may give some new insights when dealing with complex genealogyinvolved problems of Interacting Particle Systems.

Proofs

Almost sure convergence

In this section, we provide classical almost sure convergence results on SMC framework under our speci c parameterization, namely with adaptive potential functions and transition kernels. We focus on the properties that do not use the additional information given by the genealogy of the associated IPS. Therefore, in order to simplify the story, we give a "rougher" de nition of the associated IPS without considering the genealogy.

• X 0 ∼ ⊗ 0 • For ≥ 1, we let X p ∼ =1 , -1 ( -1 , •)
where, given X p-1 , , -1 is the Markov kernel de ned by

∀( , ) ∈ -1 × B ( ), , -1 ( , ) := -1 , ( , ) -1 -1,
.

It is easy to check that the distributions of the particles are identical to the ones de ned in Section 2.2.2. Let us begin with the consistency of the corresponding adaptive estimators. Recall that, by A2, the summary statistics

= ( 1 , • • • , ) satis es ( ) = * and, for all ∈ [ ], belongs to B ( ).
Theorem 2.4.1. Assume A1-A2. For any ∈ B ( ), we have

( ) . . -----→ →∞ ( ), and ( )
. .

-----→ →∞ ( ).

In particular, we also have = ( )

. .

-----→ →∞ ( ) = * .

Proof. By de nition, it is clear that the convergence of implies the convergence of . Therefore, it is su cient to establish the rst one. We prove by induction that

∀ ∈ B ( ), ( )

. .

-----→ →∞ ( ).

Step 0:

The almost sure convergence of 0 = 0 to 0 = 0 with respect to a test function in B ( 0 ) is given by the strong law of large numbers.

Step ≥ 1:

We assume that

-1 . . -----→ →∞ * -1
and, for any ∈ B ( -1 ), -1 ( )

. .

-----→ →∞ -1 ( ).

For any ∈ B ( ), the triangular inequality yields

( ) -( ) ≤ ( ) --1 , ( ) 1 ( ) + -1 , ( ) --1 ( ) 2 ( ) + -1 ( ) --1 ( ) 3 ( )
.

(2.10)

• For 1 ( ), we denote

, := -1 ( -1, ) ( ) --1 , ( ).
It is readily seen that

1 ( ) = -1 (1) 1 =1 , .
We consider the ltration (E ) 0≤ ≤ de ned by

E := F -1 ∨ ( 1 , . . . , ) with E 0 = F -1 := (X 0 , . . . , X n-1 )
. By de nition, we have

E , E -1 = -1 ( -1, ) -1 , ( ) -1 -1, --1 , ( ) = 0.
Thus, ( , ) 0≤ ≤ is an (E ) 0≤ ≤ -martingale di erence array. Under A1, we have

, ≤ := 2 -1,• ∞ ∞ .
Therefore, for any > 0, Hoe ding-Azuma inequality gives

P =1 , ≥ ≤ 2 exp -2 2 2 . Consequently, Borel-Cantelli Lemma ensures that 1 =1 , . . -----→ →∞ 0.
Combined with the induction hypothesis, we get

1 ( ) = -1 (1) 1 =1 , . . -----→ →∞ 0.
• For 2 ( ), A2 implies that there exists a function ℎ -1 such that

, ( ) ( ) -( ) ( ) = ℎ -1 ( , -1 ), -1 - * -1 .
Hence, since |ℎ -1 | and the potential functions , are bounded, Cauchy-Schwarz inequality gives that

2 ( ) ≤ -1 (1) ℎ -1 ∞ -1 - * -1 ≤ -2 =0 ,• ∞ ℎ -1 ∞ -1 - * -1 .
By induction hypothesis, we conclude that 2 ( )

. .

-----→ →∞ 0.

• For 3 ( ), under A1, we have that ( ) ∈ B ( -1 ). Thus, the induction hypothesis gives 3 ( )

. . -----→ →∞ 0.
Considering (2.10), the veri cation of the convergence

∀ ∈ B ( ), ( ) . . -----→ →∞ ( )
is then complete.

Proof of Theorem 2.2.1

We prove by induction that

√ ( ) -( ) d -----→ →∞ N 0, 2 ( ) .
The veri cation of step 0 comes from the CLT for i.i.d. random variables. For step ≥ 1, we suppose that

∀0 ≤ ≤ -1, √ ( ) -( ) d -----→ →∞ N 0, 2 ( ) .
Notice that, by A2, this implies that

∀0 ≤ ≤ -1, √ -1 - * -1 = O p (1) . (2.11)
For any test function ∈ B ( ), we denote := , ( ) ∈ B ( ). For any ( , ) ∈ 0 ×B ( 0 ) we set 0 ( , ) = 0, ( , ) = ( ). Taking into account the convention -1 = 0 = 0 and the fact that = 0 0, , we have the telescoping decomposition

( ) -( ) = =0 ( ) --1 ( ) = 1 =0 =1 (1) ( ) --1 , ( ) + -1 , ( ) --1 ( ) .
For ∈ [( + 1) ], we denote := and := -× .

We de ne the ltration

∀ ∈ [( + 1) ], E = F ∨ ( 1 , • • • , ).
Then, we set

:= 1 √ (1) ( ) --1 , ( ) , and :
= √ -1 , ( ) --1 ( ) , so that √ ( ) -( ) = ( +1) =1 + 1 = ( +1) =1 + =0 .
(2.12)

From A2, we know that there exists a function ℎ -1 such that

= √ -1 ℎ -1 (•, -1 ) , -1 - * -1 = √ -1 ℎ -1 (•, -1 ) -ℎ -1 (•, * -1 ) , -1 - * -1 + √ -1 ℎ -1 (•, * -1 ) , -1 - * -1 .
• For the rst part, we have by Cauchy-Schwarz inequality

√ -1 ℎ -1 (•, -1 ) -ℎ -1 (•, * -1 ) , -1 - * -1 ≤ √ -1 (1) -1 - * -1 sup ∈ -1 ℎ -1 ( , -1 ) -ℎ -1 ( , * -1 ) .
Then, let us consider

Ω -1 = ∈ Ω : -1 ( ) -----→ →∞ * -1 .
By Theorem 2.4.1, Ω -1 has probability one. Therefore, by A2, for all ∈ Ω -1 and all > 0, there exists ( , ) > 0 such that, for all > ( , ), sup

∈ -1 ℎ -1 ( , -1 ( )) -ℎ -1 ( , * -1 ) < .
This means that sup

∈ -1 ℎ -1 ( , -1 ) -ℎ -1 ( , * -1 )
. .

-----→ →∞ 0.

Thus, we deduce from (2.11) that

√ -1 ℎ -1 (•, -1 ) -ℎ -1 (•, * -1 ) , -1 - * -1 P -----→ →∞ 0.
• For the second part, since Theorem 2.4.1 and A2 imply that

-1 (ℎ -1 (•, * -1 )) . . -----→ →∞ -1 (ℎ -1 (•, * -1 )) = 0, we conclude by (2.11) that √ -1 ℎ -1 (•, * -1 ) , -1 - * -1 P -----→ →∞ 0.
Hence we have proved that

P -----→ →∞ 0, which leads to =0 P -----→ →∞ 0.
Next, it is easy to check that ( ) 1≤ ≤ ( +1) is an (E ) 1≤ ≤ ( +1) -martingale di erence array.

In order to apply Theorem 2.3 in [START_REF] Mcleish | Dependent central limit theorems and invariance principles[END_REF], we just have to check that

• By A1, max 1≤ ≤ ( +1) ≤ 2 √ ∞ max 1≤ ≤ -1 =0 ,• ∞ ≤ 2 √ ∞ =1 -1 =0 ,• ∞ , (2.13) which shows that (max 1≤ ≤ ( +1) | |) is uniformly bounded in 2 norm.
• From (2.13), we also get that

max 1≤ ≤ ( +1) P -----→ →∞ 0.
• Standard calculation gives

( +1) =1 2 = =0 (1) 2 ( 2 ) + ( -1 , ( )) 2 -2 (1) ( ) -1 , ( ) .
As shown above, the convergence of indicates that -1

, ( ) --1 ( )

P -----→ →∞ 0.
Then, by applying Theorem 2.4.1, we obtain

( +1) =1 2 P -----→ →∞ 2 ( ),
Therefore, we have the following central limit theorem

( +1) =1 d -----→ →∞ N 0, 2 ( ) .
Returning to (2.12), the conclusion follows from Slutsky's Lemma.

Proof of Theorem 2.3.1

We want to show that, under A1-A2, for any test functions , ∈ B ( ) and for any coalescence indicator ∈ {0, 1} +1 , we have

Γ , ( ⊗ ) -Γ ( ⊗ ) = O p 1 √ .
Before proceeding, let us introduce some additional notation. With a slight abuse of notation, for a coalescence indicator = ( 0 , . . . , ) ∈ {0, 1} +1 , we denote, for all 0 ≤ ≤ ,

Γ := Γ ( 0 ,..., )
and

Γ , := Γ ( 0 ,..., )
, with the convention

Γ -1, = Γ -1 := ⊗2 0 0 .
Note that, with this convention, we have

Γ = Γ -1 ⊗2 .
We also remark that, for any ∈ {0, 1} and any , ∈ B ( ), there exists and in B ( -1 ) such that

⊗2 ( ⊗ ) = ⊗ .
(2.14) Speci cally, for = 0, it su ces to consider = ( ) and = ( ), while for = 1 one can take = ( ) and = (1) = -1 . As usual, the proof is done by induction.

-Step 0:

• If 0 = 1, (2.6) and De nition 2.3.1 give

E Γ 0, ( ⊗ ) = E 1 =1 ( 0 ) ( 0 ) = 0 ( ) = Γ 0 ( ⊗ ).
Hence, the central limit theorem yields

√ Γ 0, ( ⊗ ) -Γ 0 ( ⊗ ) d -----→ →∞ N 0, 0 ( 2 2 ) -0 ( ) 2 , so that Γ 0, ( ⊗ ) -Γ 0 ( ⊗ ) = O p 1 √ . • If 0 = 0, the central limit theorem ensures that 0 ( ) -0 ( ) = O p 1 √ and 0 ( ) -0 ( ) = O p 1 √ .
Therefore, we have

0 ( ) 0 ( ) -0 ( ) 0 ( ) = 0 ( ) -0 ( ) 0 ( ) + 0 ( ) 0 ( ) -0 ( ) = O p 1 √ .
Thanks to (2.1), one has

Γ 0, ( ⊗ ) = -1 0 ( ) 0 ( ) - 1 2 =1 ( 0 ) ( 0 ) .
Combined with De nition 2.3.1 and the law of large numbers, one deduces that

Γ 0, ( ⊗ ) -Γ 0 ( ⊗ ) = Γ 0, ( ⊗ ) -0 ( ) 0 ( ) = O p 1 √ .
-Step ≥ 1:

We suppose that for any test functions , ∈ B ( -1 ) and coalescence indicator , we have

Γ -1, ( ⊗ ) -Γ -1 ( ⊗ ) = O p 1 √ .
Next, we consider the following decomposition

Γ , ( ⊗ ) -Γ ( ⊗ ) = Γ , ( ⊗ ) -Γ -1, ⊗2 , ( ⊗ ) 1 ( ) + Γ -1, ⊗2 , ( ⊗ ) -Γ -1, ⊗2 ( ⊗ ) 2 ( ) + Γ -1, ⊗2 ( ⊗ ) -Γ -1 ⊗2 ( ⊗ ). 3 ( ) (2.15)
The tools to terminate the proof are the following ones:

• Lemma 2.4.1 shows that 1 ( ) = O p 1 √ .
• Lemma 2.4.2 and the fact that one may write ( ⊗ ) as ⊗ for any ensure that

2 ( ) = O p 1 √ .
• Finally, the convergence rate

3 ( ) = O p 1 √ .
is a direct consequence of (2.14) and the induction hypothesis.

Technical results

This section presents some useful technical results. Before going further, remind that

Γ , (1) := (1) 2 -1 ( -1) +1 [2] 0: ∈ ( ( ) 2 ) ×( +1) -1 =0 ( [2] +1
, [2] ) .

If we set

Λ [2] := [2] 0: -1 ∈ ( ( ) 2 ) × -1 =0 ( [2] +1 , [2] ) (2.16)
together with the convention Λ

[2] 0 0 := 1, we may write

Γ , (1) := (1) 2 -1 ( -1) +1 [2] ∈ ( ) 2 Λ [2]
,

(2.17) so that

Γ , (1) 2 = (1) 4 -1 ( -1) +1 2 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 Λ [2] Λ [2]
.

(2.18)

Note that (2.17) is still true when = 0. Then, for ≥ 1, we have by de nition

Λ [2] = [2] -1 ∈ ( ) 2 Λ [2] -1 -1 -1 ( [2] -1 , [2] -1 ). (2.19)
This decomposition will appear several times in the sequel for it is a keystone to study the behavior of the coalescent tree-based measures.

Proposition 2.4.1. Assume A1. For any coalescence indicator , we have

sup >1 E Γ , (1) 2 < +∞.
In particular, the sequence (Γ , (1); ≥ 1) is uniformly tight.

Proof. We give a proof by induction. The veri cation for step 0 is trivial as Γ 0, (1) = 1. For ≥ 1, we suppose that sup

>1 E Γ -1, (1) 2 < +∞.
As de ned in Section 2.2.2, the IPS associated with ASMC is a Markov chain (X n ) ≥0 with genealogy (A n ) ≥0 tracking the indice of the parent of each particle at each level. More precisely,

-1 = means that the parent of the particle is -1 . Accordingly, the ltration (G ) ≥0 with the genealogy of the IPS is de ned by

G 0 := (X 0 ) and, for ≥ 1, G := (A 0 , . . . , A n-1 , X 0 , . . . , X n ).
By combining (2.18) and (2.19), and taking into account that

(1) = -1 =0 ( , ) = -1 (1) -1 ( -1, ) = -1 (1) (X n-1 ) ( -1, ) is G -1 -measurable, we have E Γ , (1) 2 G -1 = -1 (1) 4 -1 ( -1) +1 2 ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ [2] -1 -1 Λ [2] -1 -1 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 (X n-1 ) ( -1, ) 4 E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 . (2.20)
For the notation concerning the indices in the IPS, we use

[ ] := ( 1 , . . . , ) ∈ [ ] : Card{ 1 , . . . , } = .
In particular, we denote ( ) := [ ] . We also write

( ) 2 × := ( ) 2 × ( ) 2 × • • • × ( ) 2 times .
With a slight abuse of notation, we admit that

(( , ), ( , )) = ( , , , ).
With this notation, for ≥ 4, we have the decomposition

(( ) 2 ) ×2 = (( ) 2 ) ×2 ∩ [ ] 4 2 ∪ (( ) 2 ) ×2 ∩ [ ] 4 3 ∪ ( ) 4 .
The idea of the proof consists in analyzing (2.20) with respect to the three terms that appear in the right-hand side of the latter. Recall from (2.3) that, given X n-1 , we make an independent multinomial selection of the parent of each particle at step according to the discrete probability measure

-1, (X n-1 , •) = =1 -1, ( -1 ) =1 -1, ( -1 ) = =1 -1, ( -1 ) (X n-1 ) ( -1, ) , with, for all ∈ [ ], 0 < -1, ( -1 ) (X n-1 ) ( -1, ) ≤ -1,• ∞ (X n-1 ) ( -1, ) .
We also recall that

-1 ( [2] -1 , [2] -1 ) = 1 { -1 =0} 1 { 1 -1 = 1 -1 ≠ 2 -1 = 2 -1 } + 1 { -1 =1} 1 { 1 -1 = 1 -1 = 2 -1 ≠ 2 -1 } . • Case 1: ( [2] , [2] ) ∈ (( ) 2 ) ×2 ∩ [ ] 4 2 .
In this case, there are only two distinct random variables among

1 -1 , 2 -1 , 1 -1 , 2 -1 . Recall that 1 ≠ 2 by construction. Let us rst suppose that 1 = 1 and 2 = 2 .
Thus, we deduce that

E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ E -1 ( [2] -1 , [2] -1 ) G -1 = 1 { -1 =0} P 1 -1 = 1 -1 , 2 -1 = 2 -1 G -1 + 1 { -1 =1} P 1 -1 = 1 -1 = 2 -1 G -1 ≤ -1,• ∞ (X n-1 ) ( -1, ) 2 .
The analysis for the case where 1 = 2 and 2 = 1 is analogue. Hence, we conclude that

E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ -1,• ∞ (X n-1 ) ( -1, ) 2 .
(2.21)

Meanwhile, we notice that

# (( ) 2 ) ×2 ∩ [ ] 4 2 = 2 ( -1).
Putting all things together yields

( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 2 (X n-1 ) ( -1, ) 4 E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ 2( -1) -1,• 4 ∞ .
• Case 2:

( [2] , [2] ) ∈ (( ) 2 ) ×2 ∩ [ ] 4 3 .
As noticed in the previous case, the number of di erent indices within ( [2] , [2] ) is the only thing that matters for the upper-bound in (2.21). Accordingly, the same reasoning gives this time

E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ -1,• ∞ (X n-1 ) ( -1, ) 3 .
Since the total number of choices is

# (( ) 2 ) ×2 ∩ [ ] 4 3 = 4 ( -1) ( -2), it comes ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 3 (X n-1 ) ( -1, ) 4 E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ 4( -1) ( -2) 2 -1,• 4
∞ .

• Case 3:

( [2]
, [2] ) ∈ ( ) 4 . This time, we get

E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ -1,• ∞ (X n-1 ) ( -1, ) 4 , and # ( ) 4 = ( -1) ( -2) ( -3), so that ( [2] , [2] ) ∈ ( ) 4 (X n-1 ) ( -1, ) 4 E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ ( -1) ( -2) ( -3) 3 -1,• 4 ∞ .
As a consequence, since

2 + 4( -2) + ( -2) ( -3) 2 ≤ 7, an upper-bound for (2.20) is E Γ , (1) 2 G -1 ≤7 -1 ( -1) +1 2 -1 -1,• 4 ∞ -1 (1) 4 ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ [2] -1 -1 Λ [2] -1 -1 .
Replacing with ( -1) in (2.18) allows us to conclude that

E Γ , (1) 2 G -1 ≤ 7 -1 -1,• 4 ∞ Γ -1, (1) 2 .
Finally, by applying the induction hypothesis, we have

sup ≥4 E Γ , (1) 2 = sup ≥4 E E Γ , (1) 2 G -1 ≤ 28 3 -1,• 4 ∞ sup ≥4 E Γ -1, (1) 2 < +∞,
which ends the proof of Proposition 2.4.1.

Lemma 2.4.1. Under A1, for any test functions , ∈ B ( ), we have, for all ≥ 1,

E Γ , ( ⊗ ) G -1 = Γ -1, ⊗2 , ( ⊗ ), (2.22)
as well as

Γ , ( ⊗ ) -Γ -1, ⊗2 , ( ⊗ ) = O p 1 √ .
Proof. First, by exploiting the notation de ned in (2.16), we have

Γ , ( ⊗ ) := (1) 2 -1 ( -1) +1 [2] ∈ ( ) 2 Λ [2] ( ⊗ ) ( [2]
), and (2.22) is then a direct consequence of Proposition 2.5.1 since for any [2] ∈ ( ) 2

E (1) 2 -1 ( -1) +1 Λ [2] ( ⊗ ) ( [ 2]
)

G -1 = 1 ( -1) Γ -1, ⊗2 , ( ⊗ ),
where the right-hand side does not depend on [2] . Second, thanks to Chebyshev's inequality, it su ces to verify that

Var Γ , ( ⊗ ) -Γ -1, ⊗2 , ( ⊗ ) = O 1 .
For this, by (2.22), we just have to show that

E Γ , ( ⊗ ) 2 -(Γ -1, ⊗2 , ( ⊗ )) 2 = O 1 .
Then, recall that, by de nition,

Γ , ( ⊗ ) 2 = (1) 4 -1 ( -1) +1 2 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 Λ [2] Λ [2] ( ⊗ ) ⊗2 ( [2] , [ 2]
)

= (1) 4 -1 ( -1) +1 2 ( [2] , [2] ) ∈ ( ) 4 Λ [2] Λ [2] ( ⊗ ) ⊗2 ( [2] , [ 2]
)

1 ( ) + (1) 4 -1 ( -1) +1 2 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ [2] Λ [2] ( ⊗ ) ⊗2 ( [2] , [ 2]
) 2 ( ) .

• For 1 ( ), our goal is to establish that

E 1 ( ) -(Γ -1, ⊗2 , ( ⊗ )) 2 = O 1 .
In fact, for any

( [2] , [2] ) ∈ ( ) 4 , ( 1 , 1 , 2 , 2 ) and ( 1 , 1 , 2 , 2 )
are conditionally independent given G -1 by construction of the IPS. Hence, by applying Proposition 2.5.1 respectively for [2] and for [2] , we have

E (1) 4 -1 ( -1) +1 2 Λ [2] Λ [2] ( ⊗ ) ( [2] ) ( ⊗ ) ( [2] ) G -1 = 1 2 ( -1) 2 (Γ -1 ⊗2 , ( ⊗ )) 2 .

PROOFS

51

Then, since

# ( ) 4 = ( -1) ( -2) ( -3),
we deduce that

E 1 ( ) -(Γ -1, ⊗2 , ( ⊗ )) 2 G -1 = ( -1) ( -2) ( -3) 2 ( -1) 2 -1 (Γ -1 ⊗2 , ( ⊗ )) 2 ,
and

E 1 ( ) -(Γ -1, ⊗2 , ( ⊗ )) 2 = ( -1) ( -2) ( -3) 2 ( -1) 2 -1 E (Γ -1 ⊗2 , ( ⊗ )) 2 = O 1 ,
where the nal equality is due to Proposition 2.4.1, taking into account that and are bounded, and so is -1, uniformly with respect to by A1.

• For 2 ( ), the nonnegativity of

Λ [2] implies E[ 2 ( )] ≤ E       (1) 4 -1 ( -1) +1 2 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ [2] Λ [2]       2 ∞ 2 ∞ .
So the proof will be nished once we have shown that

E       (1) 4 -1 ( -1) +1 2 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ [2] Λ [2]       = O 1 .
Once again, we proceed by induction. At step 0, we have

1 2 ( -1) 2 ( [2] 0 , [2] 0 ) ∈ ( ( ) 2 ) ×2 \( ) 4 1 = 1 - ( -1) ( -2) ( -3) 2 ( -1) 2 = O 1 .
For step ≥ 1, we suppose that

E        -1 (1) 4 -2 ( -1) 2 ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ [2] -1 -1 Λ [2] -1 -1        = O 1 .
The adaptation of (2.20) to the present context gives

E       (1) 4 -1 ( -1) +1 2 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ [2] Λ [2] G -1       = -1 (1) 4 -1 ( -1) +1 2 ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ [2] -1 -1 Λ [2] -1 -1 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 (X n-1 ) ( -1, ) 4 E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 . Now, for ≥ 4, it is clear that (( ) 2 ) ×2 \( ) 4 = (( ) 2 ) ×2 ∩ [ ] 4 2 ∪ (( ) 2 ) ×2 ∩ [ ] 4 3 .
-Case 1:

( [2] , [2] ) ∈ (( ) 2 ) ×2 ∩ [ ] 4 2 . By de nition of -1, in (2.3), E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ -1,• ∞ (X n-1 ) ( -1, ) 2 1 { -1 =1, 1 -1 = 1 -1 } + 1 { -1 =0, 1 -1 = 1 -1 , 2 -1 = 2 -1 } ≤ -1,• ∞ (X n-1 ) ( -1, ) 2 1 {#{ 1 -1 , 1 -1 , 2 -1 , 2 -1 }<4} ,
and since

# (( ) 2 ) ×2 ∩ [ ] 4 2 = 2 ( -1), it comes ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 2 (X n-1 ) ( -1, ) 4 E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ 2 ( -1) 2 -1,• 4 ∞ 1 {#{ 1 -1 , 1 -1 , 2 -1 , 2 -1 }<4} .
-Case 2:

( [2] , [2] ) ∈ (( ) 2 ) ×2 ∩ [ ] 4 3 .
First, we suppose that 1 = 1 . As for the previous case, we have

E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ -1,• ∞ (X n-1 ) ( -1, ) 3 1 { -1 =1, 1 -1 = 1 -1 } + 1 { -1 =0, 1 -1 = 1 -1 } ≤ -1,• ∞ (X n-1 ) ( -1, ) 3 1 {#{ 1 -1 , 1 -1 , 2 -1 , 2 -1 }<4} .
By the same reasoning, for 1 = 2 , 2 = 1 and 2 = 2 , we also have

E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ -1,• ∞ (X n-1 ) ( -1, ) 3 1 {#{ 1 -1 , 1 -1 , 2 -1 , 2 -1 }<4} .
In addition, since

# (( ) 2 ) ×2 ∩ [ ] 4 3 = 4 ( -1) ( -2),
we get this time

( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 3 (X n-1 ) ( -1, ) 4 E -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1 ≤ 4 ( -1) ( -2) 3 -1,• 4 ∞ 1 {#{ 1 -1 , 1 -1 , 2 -1 , 2 -1 }<4} .
By gathering both cases, we have

E       (1) 4 -1 ( -1) +1 2 ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ [2] Λ [2] G -1       ≤ -1 (1) 4 -2 ( -1) 2 2 ( -1) 2 2 ( -1) 2 + 4 ( -1) ( -2) 3 -1,• 4 ∞ ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ [2] -1 -1 Λ [2] -1 -1 1 {#{ 1 -1 , 1 -1 , 2 -1 , 2 -1 }<4} ≤6 2 ( -1) 2 -1,• 4 ∞ -1 (1) 4 -2 ( -1) 2 ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ [2] -1 -1 Λ [2] -1 -1 .
The conclusion follows from the induction hypothesis by taking the expectation on both sides.

This ends the proof of Lemma 2.4.1.

Lemma 2.4.2. Assume A1-A2, then for any test functions , ∈ B ( ) and for all ≥ 1,

Γ -1, ⊗2 , ( ⊗ ) -Γ -1, ⊗2 ( ⊗ ) = O p 1 √ .
Proof. The veri cation shares some resemblance with the convergence of 2 ( ) in the proof of Theorem 2.4.1. Speci cally, we start with the following decomposition:

Γ -1, ⊗2 , ( ⊗ ) -Γ -1, ⊗2 ( ⊗ ) ≤ Γ -1, ⊗2 , ( ⊗ ) -Γ -1, ( , ⊗ ) ( ⊗ ) 1 ( ) + Γ -1, ( , ⊗ ) ( ⊗ ) -Γ -1, ⊗2 ( ⊗ ) 2 ( )
.

For 1 ( ), we may write

( ⊗2 , ( ⊗ ) -( , ⊗ ) ( ⊗ )) ( , ) = , ( ) ( ) ( , ( ) ( ) -( ) ( )).
By A2, for any ∈ B ( ), there exists a bounded function ℎ -1 such that

, ( ) ( ) -( ) ( ) = ℎ -1 ( , -1 ), -1 - * -1 ≤ ℎ -1 ∞ -1 - * -1 .
Since, in addition,

| , ( ) ( )| ≤ -1,• ∞ ∞ , it comes 1 ( ) ≤ Γ -1, (1) -1,• ∞ ∞ ℎ -1 ∞ -1 - * -1 .
By Proposition 2.4.1, one has

Γ -1, (1) = O p (1) .
In addition, a by-product (2.11) of Theorem 2.2.1 is that

-1 - * -1 = O p 1 √ .
Hence, one concludes that

1 ( ) = O p 1 √ .
The reasoning for 2 ( ) is the same.

Connection between the estimators

In this section, we give some combinatorial results on the coalescent tree-based measures Γ , . In particular, these results allow us to connect the variance estimator (2.9) of Lee & Whiteley and our term by term estimators. As mentioned before, these results do not depend on A2: they are provided by the structure of the IPS and the underlying multinomial selection scheme.

Proposition 2.4.2. Under A1, for any test function ∈ B ( 2 ), we have the decompositions:

( ) ⊗2 ( ) = ∈ {0,1} +1 =0 ( -1) 1- Γ , ( ), and
( ) ⊗2 ( ) = ∈ {0,1} +1 =0 ( -1) 1- Γ , ( ). Proof. Since Γ , ( ) := -1 ( -1) +1 [2] 0: ∈ ( ( ) 2 ) ×( +1) -1 =0 ( [2] +1 , [2] ) ( ) ( [2]
),

we have

∈ {0,1} +1 =0 ( -1) 1- Γ , ( ) = -1 ( -1) +1 ∈ {0,1} +1 [2] 0: ∈ ( ( ) 2 ) ×( +1) =0 ( -1) 1- -1 =0 ( [2] +1 , [2] ) ( ) ( [ 2]
).

Enumerating all the possibilities for the coalescence indicator ∈ {0, 1} +1 leads to

∈ {0,1} +1 =0 ( -1) 1- Γ , ( ) = [2] 0 ∈ ( ) 2 • • • [2] -1 ∈ ( ) 2 -1 =0 1 1 { 1 +1 = 2 +1 = 1 ≠ 2 } + -1 1 { 1 +1 = 1 ≠ 2 +1 = 2 } -1 -1 2 (X n ) 0 ( ) + 1 2 (X n ) 1 ( ) .
To conclude, one just has to observe that

[2] 0 ∈ ( ) 2 • • • [2] -1 ∈ ( ) 2 -1 =0 1 1 { 1 +1 = 2 +1 = 1 ≠ 2 } + -1 1 { 1 +1 = 1 ≠ 2 +1 = 2 } = -1 , while, by (2.1), -1 2 (X n ) 0 ( ) + 1 2 (X n ) 1 ( ) = ⊗2 (X n ) ( ) = ( ) ⊗2 ( ).
Multiplying both sides by (1) 2 gives the corresponding relation for ( ) ⊗2 ( ).

We can now proceed with the proof of Proposition 2.3.1. Recall that the goal is to show that

( ) -2 , ( ) = ( ) - =0 Γ( ) , ( ⊗2 ) -Γ( ) , ( ⊗2 ) = O p 1 , and
( -( )) - =0 Γ( ) , -( ) ⊗2 -Γ( ) , -( ) ⊗2 = O p 1 .
By construction, we have

( ) = ( ) 2 -Γ( ) , ( ⊗2 ) = ( ) ⊗2 ( ⊗2 ) -Γ( ) , ( ⊗2 ).
An implication of Proposition 2.4.1 is that, for any test function and any coalescence indicator , Γ , ( ⊗2 ) = O p (1) .

Thus, a consequence of Proposition 2.4.2 is

( ) ⊗2 ( ⊗2 ) = -1 +1 Γ( ) , ( ⊗2 ) + 1 -1 =0 Γ( ) , ( ⊗2 ) + O p 1 2 .
The desired formula is then obtained by remarking that

-1 = 1 -O 1 and -1 +1 -1 = - + 1 + O 1 2 . Similarly, since Γ , -( ) ⊗2 = O p (1) ,
the same algebraic manipulation yields

( -( )) - =0 Γ( ) , -( ) ⊗2 -Γ( ) , -( ) ⊗2 = O p 1 .
This closes the proof of Proposition 2.3.1.

Many-body Feynman-Kac models

The many-body Feynman-Kac model was proposed in [START_REF] Del Moral | On particle Gibbs samplers[END_REF] to study the propagation of chaos property of the Conditional Particle Markov Chain Monte Carlo introduced in [START_REF] Andrieu | Particle Markov chain Monte Carlo methods[END_REF]. The basic idea is to trace the information of all particles in the IPS along with its genealogy, and to construct an instrumental particle block which is heavily dependent (identical) to some speci c particles. We call these instrumental particles the coupled particle block of the IPS.

Duality formula

At each layer, the particles in the original IPS are denoted by X p , with its genealogy A p-1 . The coupled particle block of particles is denoted by ˜ [ ] , with its genealogy denoted by ˜ [ ] -1 . The corresponding variables in the integral operators will be denoted by

x , a -1 , ˜ [ ] and ˜ [ ] -1
respectively. Before giving speci c de nitions, we want to mention that the mathematical object we would like to look into is the whole particle system, namely the original IPS and the coupled particle block with genealogy. At each layer , we are interested by the tuple:

(X p , A p-1 , ˜ [ ] , ˜ [ ] -1 ).
As for the basic idea of Particle Markov Chain Monte Carlo method [ADH10], we study respectively the distributions of

X p , A p-1 ˜ [ ] , ˜ [ ] -1 and ˜ [ ] , ˜ [ ] -1 X p , A p-1 .
Thanks to the speci c construction, as well as the relatively simple multinomial resampling scheme of Feynman-Kac IPS, Lemma 2.5.1 provides a duality formula to connect both distributions and leads in particular to Proposition 2.5.1. This latter result is crucial to prove the consistency of our term by term estimator in Theorem 2.3.1.

In this section, a transition kernel denoted by the letter is a Feynman-Kac kernel, meaning that its total mass is not necessarily 1, and it can be expressed by the product of a positive potential function and a Markov kernel. All transition kernels denoted by the letter are Markov kernels.

Notice that the transition from level -1 to level of the IPS with its genealogy de ned in Section 2.2.2 can be expressed as

(A p-1 , X p ) ∼ =1 Φ , (X p-1 , ( -1 , ))
with Φ , de ned by

Φ , (x -1 , ( -1 , )) = -1, (x -1 , -1 ) × , ( -1 -1 , ).
We de ne the transition of the original IPS with its genealogy by

M (x -1 , (a -1 , x p )) := =1 Φ , (x -1 , ( -1 , ))
and the potential function of the particle block of size by

G ( ) -1 (x -1 ) := (x -1 ) ( -1, ) .
We denote the associated Feynman-Kac transition kernel

Q ( ) (x -1 , (a -1 , x p )) := G ( ) -1 (x -1 ) × M (x -1 , (a -1 , x p )). Given [ ] ∈ ( ) , ˜ [ ] -1 ∈ [ ] and ˜ [ ] ∈ , we de ne M ˜ [ ] -1 , [ ] , ˜ [ ] (x -1 , (a -1 , x p )) := ∈ [ ]\{ 1 ,..., } Φ , (x -1 , ( -1 , )) × ˜ [ ] ( [ ] ) × ˜ [ ] -1 ( [ ] -1 )
the conditional transition for the original particle system given the coupled particle block

˜ [ ] = ˜ [ ] at position [ ] with frozen genealogy ˜ [ ] -1 = ˜ [ ] -1 .
In particular, we denote

M [ ] 0 , ˜ [ ] 0 0 ( x 0 ) :=        ∈ [ ]\{ 1 0 ,..., 0 } 0 ( 0 )        × ˜ [ ] 0 ( [ ] 0 0 ).
We also de ne

Q ( ) (x -1 , ( ˜ [ ] -1 , ˜ [ ] )) := ( [ ]) ⊗ ( ˜ [ ] -1 ) ⊗ , ( ˜ [ ] -1 -1 , ˜ [ ] ),
and

M [ ] (a -1 , x ), ( ˜ [ ] -1 , ˜ [ ] ) := [ ] -1 ( ˜ [ ] -1 ) [ ] ( ˜ [ ] ).
Then we have the following pivotal duality formula, which is simply a generalization of Lemma 4.1 in [START_REF] Del Moral | On particle Gibbs samplers[END_REF]. We will apply it in the proof of Proposition 2.5.1 with = 2. Lemma 2.5.1. For ≥ 1, ∈ [ ] and [ ] ∈ ( ) , we have the following duality formula between integral operators

Q ( ) (x -1 , (a -1 , x p )) M [ ] (a -1 , x ), ( ˜ [ ] -1 , ˜ [ ] ) = Q ( ) (x -1 , ( ˜ [ ] -1 , ˜ [ ] )) M ˜ [ ] -1 , [ ] , ˜ [ ] (x -1 , (a -1 , x p )),
and

⊗ 0 ( x 0 ) [ ] 0 0 ( ˜ [ ] 0 ) = ⊗ 0 ( ˜ [ ] 0 ) M [ ] 0 , ˜ [ ] 0 0 ( x 0 ).

Proof.

Step 0 is clear. For ≥ 1, it su ces to check that the nonidentical parts are equal, namely

G ( ) -1 (x -1 ) =1 -1, ( -1 ) (x -1 ) ( -1, ) ⊗ ( [ ] -1 ) ⊗ , ( [ ] -1 -1 , [ ] ) M [ ] (a -1 , x ), ( ˜ [ ] -1 , ˜ [ ] ) = Q ( ) (x -1 , ( ˜ [ ] -1 , ˜ [ ] )) ˜ [ ] ( [ ] ) ˜ [ ] -1 ( [ ]
-1 ).

Fixing [ ] ∈ ( ) and x -1 ∈ -1 , consider a function F ∈ B ( [ ] ×[ ] × × ). Moreover, let [ ] = ( 1 , . . . , ) and [ ] = ( 1 , . . .

, ) denote generic variables belonging respectively to

[ ] and . Then, we may write

∫ (x -1 ) ( -1, ) =1 -1, ( -1 ) (x -1 ) ( -1, ) ⊗ ( [ ] -1 ) ⊗ , ( [ ] -1 -1 , [ ] ) [ ] ( ˜ [ ] ) [ ] -1 ( ˜ [ ] -1 ) F( ˜ [ ] -1 , [ ] -1 , [ ] , ˜ [ ] ) = ∫ ( [ ]) ⊗ ( [ ] ) ⊗ -1, ( [ ] -1 ) ⊗ , ( [ ] -1 , [ ] ) F( [ ] , [ ] , [ ] , [ ] ) = ∫ ( [ ]) ⊗ ( [ ] ) ⊗ , ( [ ] -1 , [ ] ) F( [ ] , [ ] , [ ] , [ ] ) = ∫ ( [ ]) ⊗ ( ˜ [ ] -1 ) ⊗ , ( ˜ [ ] -1 -1 , ˜ [ ] ) ˜ [ ] ( [ ] ) ˜ [ ] -1 ( [ ] -1 ) F( ˜ [ ] -1 , [ ] -1 , [ ] , ˜ [ ] ) = ∫ Q ( ) (x -1 , ( ˜ [ ] -1 , ˜ [ ] )) ˜ [ ] ( [ ] ) ˜ [ ] -1 ( [ ] -1 ) F( ˜ [ ] -1 , [ ] -1 , [ ] , ˜ [ ] ).
This ends the proof of the duality formula.

Let us recall (2.16) and (2.19):

Λ [2] = [2] 0: -1 ∈ ( ( ) 2 ) × -1 =0 ( [2] +1 , [2] ) = [2] -1 ∈ ( ) 2 Λ [2] -1 -1 -1 ( [2] -1 , [2] -1 ).
(2.23) with the convention Λ

[2] 0 0 = 1. In fact, this gives another representation of the approximation of the coalescent tree-based measures:

Γ , ( ⊗ ) = (1) 2 -1 ( -1) +1 [2] ∈ ( ) 2 Λ [2] ( ⊗ ) ( [2]
).

Recall that

G -1 := (X 0 , . . . , X n-1 , A 0 , . . . , A n-2 ).

The upcoming result is useful in the proof of Lemma 2.4.1.

Proposition 2.5.1. Under A1, for any [2] ∈ ( ) 2 , any coalescence indicator , and any test functions and in B ( ), we have, for all ≥ 1, that

E (1) 2 -1 ( -1) +1 Λ [2] ( ⊗ ) ( [ 2]
)

G -1 = 1 ( -1) Γ -1, ⊗2 , ( ⊗ ).
Proof. By applying (2.23), we obtain

(1) 2 -1 ( -1) +1 Λ [2] ( ⊗ ) ( [2] ) = (1) 2 -1 ( -1) +1 [2] -1 ∈ ( ) 2 Λ [2] -1 -1 -1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [ 2]
).

Since Λ

[2] -1

-1 is G -1 -measurable, it is su cient to show that for each [2] -1 ∈ ( ) 2 , we have

E (X n-1 ) ( -1, ) 2 -1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [2] ) G -1 = 1 2 -1 ⊗2 , ( ⊗ ) ( [2] -1
-1 ).

(2.24) Before starting our reasoning, for the sake of simpli cation, we remark that

-1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [ 2]
) can be seen as a bounded measurable function of (A n-1 , X n ), rather than a measurable function of

( [2]
,

[2]

-1 ). With this in mind, for any test function

∈ B ( [ ] × • • • × [ ] ( -1) times × 0 × • • • × -1 ),
we have, by de nition of

Q (2) (x -1 , (a -1 , x p )), E (X n-1 ) ( -1, ) 2 -1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [2] ) (A 0:n-2 , X 0:n-1 ) = ∫ Q (2) (x -1 , (a -1 , x n )) -1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [ 2]
)

(a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ), (2.25)
where -1 denotes the measure corresponding to the underlying joint distribution of the IPS from step 0 to step -1 with genealogy. Taking into account that

M [2] (a -1 , x ), ( ˜ [2] -1 , ˜ [2] ) = [2] -1 ( ˜ [2] -1 ) [2] ( ˜ [2] )
is a Markov kernel, we can introduce it in the right-hand side of (2.25) to obtain

E (X n-1 ) ( -1, ) 2 -1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [2]
) (a 0: -2 , x 0: -1 )

= ∫ Q (2) (x -1 , (a -1 , x n ))M [2] (a -1 , x n ), ( ˜ [2] -1 , ˜ [2] ) -1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [2]
) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ).

The design of many-body Feynman-Kac models allows us to replace

( [2] -1 , [ 2]
) with (

˜ [2] -1 , ˜ [2]
) in the observation functions, as they are equal by de nition. Hence, one has the following equality:

∫ Q (2) (x -1 , (a -1 , x n ))M [2] (a -1 , x n ), ( ˜ [2] -1 , ˜ [2] ) -1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [2]
) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 )

= ∫ Q (2) (x -1 , (a -1 , x n ))M [2] (a -1 , x n ), ( ˜ [2] -1 , ˜ [2] ) -1 ( ˜ [2] -1 , [2] -1 ) ( ⊗ ) ( ˜ [2]
) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ).

Now, the duality formula given in Lemma 2.5.1 yields

∫ Q (2) (x -1 , (a -1 , x n ))M [2] (a -1 , x n ), ( ˜ [2] -1 , ˜ [2] ) -1 ( ˜ [2] -1 , [2] -1 ) ( ⊗ ) ( ˜ [2] ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ) = ∫ Q (2) (x -1 , ( ˜ [2] -1 , ˜ [2] )) M ˜ [2] -1 , [2] , ˜ [2] (x -1 , (a -1 , x n )) -1 ( ˜ [2] -1 , [2] -1 ) ( ⊗ ) ( ˜ [2]
) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ).

In addition, since

M ˜ [2] -1 , [2] , ˜ [2] (x -1 , (a -1 , x n ))
is a Markov kernel for any choice of (

˜ [2] -1 , [2] , ˜ [2] ), we deduce that ∫ Q (2) (x -1 , ( ˜ [2] -1 , ˜ [2] )) M ˜ [2] -1 , [2] , ˜ [2] (x -1 , (a -1 , x n )) -1 ( ˜ [2] -1 , [2] -1 ) ( ⊗ ) ( ˜ [2] ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ) = ∫ Q (2) (x -1 , ( ˜ [2] -1 , ˜ [2] )) -1 ( ˜ [2] -1 , [2] -1 ) ( ⊗ ) ( ˜ [2] ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ).
Next, let us recall that

Q (2) (x -1 , ( ˜ [2] -1 , ˜ [2] )) := ( [ ]) ⊗2 ( ˜ [2] -1 ) ⊗2 , ( ˜ [2] -1 -1 , ˜ [2]
) and

-1 ( ˜ [2] -1 , [2] -1 ) := 1 { -1 =1, ˜ 1 -1 = ˜ 2 -1 = 1 -1 ≠ 2 -1 } + 1 { -1 =0, ˜ 1 -1 = 1 -1 ≠ ˜ 2 -1 = 2
-1 } , whence we get the equality concerning the operator -1 . More precisely, if -1 = 0, we have

∫ Q (2) (x -1 , ( ˜ [2] -1 , ˜ [2] )) -1 ( ˜ [2] -1 , [2] -1 ) ( ⊗ ) ( ˜ [2] ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ) = ∫ 1 2 ⊗2 , ( ⊗ ) ( [2] -1
-1 ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ).

(2.26)

Otherwise, if -1 = 1, we get, with the convention

(1,1) -1 -1 = ( 1 -1 -1 , 1 -1 -1 ), ∫ Q (2) (x -1 , ( ˜ [2] -1 , ˜ [2] )) -1 ( ˜ [2] -1 , [2] -1 ) ( ⊗ ) ( ˜ [2] ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ) = ∫ 1 2 ⊗2 , ( ⊗ ) ( (1,1) -1
-1 ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ).

(2.27)

Combining (2.26) and (2.27), we safely deduce that

∫ Q (2) (x -1 , ( ˜ [2] -1 , ˜ [2] )) -1 ( ˜ [2] -1 , [2] -1 ) ( ⊗ ) ( ˜ [2] ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ) = ∫ 1 2 -1 ⊗2 , ( ⊗ ) ( [2] -1 -1 ) (a 0: -2 , x 0: -1 ) -1 ( a 0: -2 , x 0: -1 ) =E 1 2 -1 ⊗2 , ( ⊗ ) ( [2] -1 -1 ) (A 0:n-2 , X 0:n-1 ) .
In conclusion, we have established that

E (X n-1 ) ( -1, ) 2 -1 ( [2] -1 , [2] -1 ) ( ⊗ ) ( [2] ) (A 0:n-2 , X 0:n-1 ) =E 1 2 -1 ⊗2 , ( ⊗ ) ( [2] -1 -1 ) (A 0:n-2 , X 0:n-1 ) ,
which terminates the veri cation of (2.24) and the proof of Proposition 2.5.1.

Some intuition

In general, the coupled particle block does not necessarily have the parents-children relations.

Let us see a representation of the duality formula given in Lemma 2.5.1 recursively applied in a mini IPS from level 0 to level 5 to some randomly chosen indices [2] 0:5 (see Figure 2.5). , and the black twisted ones pointing to the particles in the coupled particle block represent the Feynman-Kac transition kernels , . The red dotted bending arrows are identities. The indices of the original particles in the coupled particle block are

[2] 0 = (4, 5), [2] 1 = (2, 4), [2] 2 = (2, 3), [2] 3 = (3, 5), [2] 4 = (1, 5) and [2] 5 = (1, 2).
However, we can get any ancestral relations or coalescent tree-based form by manipulating the genealogical information encoded in the coupled genealogy. This is the essential idea we used by introducing many-body Feynman-Kac models. To make it clearer, we consider an event de ned by

[2] -2 = ˜ [2] -2 , 1 -1 = ˜ 1 -1 = ˜ 2 -1 ≠ 2 -1 , [2] = ˜ [2] .
(2.28)

On this event, we are able to track the coalescent tree-based form as in Figure 2.6.

˜ 1 -2 ˜ 1 -1 • • • ˜ 1 ˜ 1 +1 • • • ˜ 2 ˜ 2 +1 • • • ˜ 2 -2 ˜ 2 -1 • • • Figure 2.6:
The coupled particle block tracked by the event de ned by (2.28).

The coupled particle block and its genealogy are de ned as the copies of certain particles and parents indices in the associated original IPS. On one hand, we select certain events such that the desired structure is trapped in the coupled particle block. On the other hand, we de ne the estimator based on the information re ected in the original IPS as no additional randomness are added by introducing the coupled particle block. Since their distributions are connected by the duality formula, we can use the information coded in the original IPS to estimate the measures corresponding to these coalescent tree-based particle blocks (see Figure 2.7).

[2] 0 = (3, 5), [2] 1 = (3, 4), [2] 2 = (2, 5), [2] 3 = (1, 2), [2] 4 = (2, 5) and [2] 5 = (1, 3).
The duality formula provides a way to touch the adaptive versions of the coalescent treebased measures Γ , i.e., all the Feynman-Kac transition kernels in the de nition are replaced with the adaptive version , . This is the idea underlying the construction of the estimators Γ , .

Connection with SMC

To conclude, let us say a few words about the behavior of Γ , . One remark is that, in general, this estimator is not unbiased in the ASMC framework. This is a consequence of the adaptive parametrization, as witnessed by Lemma 2.4.2. On the opposite, in a nonadaptive case (SMC), the estimation is unbiased, exactly as is an unbiased estimation of (see for example [DM04] Section 3.5.1). It turns out that the classical SMC framework corresponds to the case where the function ℎ in A2 is equal to zero, meaning that , = for all . Thus, Lemma 2.4.1 and (2.15) give the following proposition.

Proposition 2.5.2. Assume A1-A2 and suppose that ℎ ≡ 0 for all ≥ 0. Then, for all test functions , ∈ B ( ), E Γ , ( ⊗ ) = Γ ( ⊗ ).

In particular, we also have

E (1) 2 ( ) = Var ( ) .
In fact, the essential technical results in Section 2.4.4 and Section 2.4.5 only require A1. In other words, A2 can be studied separately in order to adapt to applications not covered in this chapter.

Another remark is about the di erence between Γ , and as de ned in Section 3.2. of [START_REF] Lee | Variance estimation in the particle lter[END_REF] in the nonadaptive context. However, since it is not straightforward to compare these estimators that are extremely notation-heavy, we would just like to brie y and heuristically mention that the main di erence comes from the step where there is a coalescence, namely = 1. If we consider Figure 2.3 in Section 2.3.1, our estimator is not the most "precise" that one could propose. Let us look at the case where

[2] 0:6 = ((5, 3), (4, 3), (2, 5), (2, 4), (2, 5), (1, 3), (2, 4)) .

For the terminal point 4 3 , the conditional distribution of 4 2 is simply the categorical distribution since 4 3 is a terminal point. Roughly speaking, once all the genealogy of the terminal points is calculated, one can deduce . Hence, the take-home message is simple: if one is interested in estimating Γ numerically, then the estimator proposed in [START_REF] Lee | Variance estimation in the particle lter[END_REF] is expected to be more accurate, meaning that the variance should be smaller in general.

Nevertheless, as a theoretical tool, our estimator is easier to deal with in the adaptive framework. Indeed, induction is highly involved in our proof of consistency, so estimators that are stepwise easy to manipulate are required. Another di erence is that we do not use instrumental random variables such as 1 and 2 in the de nition of . This also simpli es the analysis in an adaptive context where there is already more randomness than in a nonadaptive context.

Chapter 3

Asymmetric Sequential Monte Carlo abstract: Sequential Monte Carlo is a general framework aiming at sampling a sequence of measures ( ; ≥ 0) connected by some nonlinear operators. In the classical setting, the simulation consists in a multinomial resampling selection step and a Markov mutation step at each iteration of the algorithm. When the potential functions are [0, 1]-valued, a well-known variant is to conduct a Bernoulli survival test before the multinomial resampling step: the surviving particles will not be resampled, whilst the non-survived particles perform a multinomial resampling. We go one step further, that is, we suppose that the surviving particles and non-surviving particles will mutate according to di erent Markov kernels. This setting is referred to as "asymmetric resampling" in this chapter. The idea is natural in rare-event simulation, where the Markov kernel at step is -invariant. In this scenario, since the surviving particles have already "well placed", there is no need to implement another mutation. The idea is to reduce the computational burden induced by the mutation kernels. We provide a CLT-type result as well as consistent variance estimators, which allows to conduct statistical inference with a single run of the simulation. We also provide an unbiased variance estimator for the unnormalized measures under certain conditions. To do this, we introduce generalized coalescent tree-based measures and their particle approximations as a complement of the ones introduced respectively in [CDMG11] and [DG19](Chapter 2). We rmly believe that they represent an important and natural family of mathematical objects in the general framework of SMC. We expect the same methodology may also inspire further analysis for the models in a continuous-time setting, such as Fleming-Viot particle systems (see, e.g., [START_REF] Delyon | A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing[END_REF]).

Introduction

Sequential Monte Carlo (SMC) methods are powerful numeric algorithms widely used in many elds in computational statistics, such as Bayesian inference, ltering, rare-events simulations, etc. The reader is referred to [START_REF] Doucet | Sequential Monte Carlo methods in practice[END_REF] for a larger list of available applications. The basic idea is to simulate an Interacting Particle System (IPS) in order the approximate a sequence of probability measures ( ) ≥0 or positive nite measures ( ) ≥0 connected by some non-linear operators. The estimators are naturally designed as the associated empirical measures at each level of the IPS. The proper mathematical foundation and more theoretical aspects such as convergence results and bias analysis can be found for example in the pair of books [DM04, DM13] and references therein.

Classical SMC methods consist in a multinomial selection step and a Markov mutation step at each step of the algorithm. This resampling strategy is well-understood both in theory and in practice. It corresponds to a natural interpretation of the Boltzmann-Gibbs transformation w.r.t. the potential functions ( ; ≥ 0) on the empirical measures ( ; ≥ 0). There are a lot of variants on this resampling strategy, such as residual resampling, strati ed resampling and systematic resampling, etc. The reader is referred to [START_REF] Hol | On resampling algorithms for particle lters[END_REF] for a quick survey. Some theoretical analysis can be found in the recent work [START_REF] Gerber | Negative association, ordering and convergence of resampling methods[END_REF], which includes the most important variants of the resampling schemes mentioned above.

In contrast to the resampling strategy discussed above, we are interested in the case where there are two di erent Markov kernels, denoted respectively by ˚ and • , at each iteration of the algorithm. In the rest of this chapter, they are referred to as mutation kernels. Roughly speaking, at each iteration, say, from level -1 to level , each particle performs a Bernoulli survival test w.r.t. the [0, 1]-valued potential function -1 : the survived particle mutates according to • while the non-surviving ones execute a multinomial resampling w.r.t. the potential function -1 , after which a mutation according to the kernel ˚ will be executed. The precise mathematical de nition will be given in Section 3.2.4.

The main motivating example is the generalized Adaptive Multilevel Splitting (gAMS) methods introduced in [BGG + 16], where the kernel • is set to be the identity ( ). It is a natural choice in the applications such as Particle Tempering and other Subset Simulation methods, where ˚ is designed to be an -invariant kernel. The primal motivation of this implementation is to reduce the unnecessary computational costs brought by the mutation kernel ˚ . Moreover, since the invention of Particle Markov Chain Monte Carlo methods (PMCMC, cf. [START_REF] Andrieu | Particle Markov chain Monte Carlo methods[END_REF]), the design of mutation kernels ˚ becomes much easier and more computationally demanding at the same time. One typical example is SMC 2 methods (see, e.g., [START_REF] Chopin | SMC 2 : an e cient algorithm for sequential analysis of state space models[END_REF][START_REF] Chopin | Towards automatic calibration of the number of state particles within the SMC 2 algorithm[END_REF]). The basic idea is to use another SMC-based IPS and freezing techniques to construct an -invariant kernel at each level. One can imagine that the computational costs are mainly from the implementation of the mutation kernels ( ˚ ; ≥ 1), which can be dramatically reduced by applying asymmetric resampling scheme if we choose • to be the identity or some other "cheaper" kernels. Another motivation is the variance estimation in the context of symmetric sampling, namely, the case where all the particles mutate according to the same Markov kernel at each step. More precisely, it means that • ≡ ˚ for each ≥ 1 and this setting enters into the classic Feynman-Kac particle models (e.g. [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF]). It is well-known that the asymptotic variance is smaller than the one under classical multinomial resampling scheme. The exact di erence at each step is presented in (3.20). As the symmetric resampling can also slightly reduce the computational costs required by the multinomial resampling, there is no practical reason to implement multinomial resampling scheme if an upper bound for is available.

In order to conduct statistical inference, it is important to study the asymptotic behaviors of the empirical measures associated to the IPS (see, e.g., [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF][START_REF] Chopin | Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference[END_REF][START_REF] Douc | Limit theorems for weighted samples with applications to sequential Monte Carlo methods[END_REF]). For example, if one has a CLT-type theorem for some test function such as

√ ( ) -( ) d -----→ →∞ N (0, ( ) 2 ),
it is su cient to provide a consistent estimator ( ) of ( ) since Slutsky's lemma guarantees that √ ( ) -( )

( ) d -----→ →∞ N (0, 1).
An asymptotic con dential interval can therefore be derived once the IPS is simulated. Although SMC methods are intensely studied for over 20 years, the classical way to achieve this is still by resimulating the IPS independently many times and by estimating ( ) 2 with the crude variance estimator. This is not always practical: a single run of an IPS may take a lot of time, and one also expects that all the computing budget is used to improve precision, rather than to estimate the variance. In addition, as the estimator ( ) of ( ) provided by SMC methods is typically biased, it is also nontrivial to implement parallel computing for a large number of independent IPS with relatively small. As a consequence, a variance estimator available with a single run of the simulation is of crucial interest for applications.

The breakthrough is made by Chan and Lai in [START_REF] Chan | A general theory of particle lters in hidden Markov models and some applications[END_REF]. The rst consistent variance estimators are proposed by exploiting the ancestral information encoded in the genealogy of the associated IPS. Then, Lee and Whiteley [START_REF] Lee | Variance estimation in the particle lter[END_REF] provided an unbiased variance estimator for the unnormalized measures and a term by term estimator, which provide a deeper understanding on the role of the genealogy in variance related problems. In the sequel, a more numerically stable variance estimator is provided in [START_REF] Olsson | Numerically stable online estimation of variance in particle lters[END_REF], as a natural xed-lag version of the original one proposed in [START_REF] Chan | A general theory of particle lters in hidden Markov models and some applications[END_REF], when more stability properties of the IPS are available. Another recent result is given in [DG19](Chapter 2), by extending the estimator of Lee & Whiteley to the adaptive SMC context (cf. [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF]). All these estimators are studied in the classical SMC framework, meaning under multinomial resampling scheme.

From a theoretical viewpoint, the current setting can be regarded as a "toy model" for more sophisticated algorithms in the adaptive context and/or in a continuous-time setting: there is no additional attention required to deal with complicated regularity assumptions, and we can thus focus on the structural properties of the IPS. Similar to the case where the variance estimators provided by Lee & Whiteley in [START_REF] Lee | Variance estimation in the particle lter[END_REF] are still valid in the adaptive SMC framework with some additional assumptions (cf. Assumption 2, [DG19](Chapter 2)), we expect that the variance estimators provided in this chapter are still valid in more general settings, and our methodology can also be extended in such scenarios. The rigorous mathematical formulation of the current setting can be seen as a generalization of the discrete-time Feynman-Kac particle models presented in the literature such as [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF][START_REF] Moral | Mean eld simulation for Monte Carlo integration[END_REF]. Our technical tools consist in a new family of mathematical objects, i.e., the so-called coalescent Feynman-Kac measures and coalescent tree occupation measures. They are introduced as an extension to the methodology developed in [DG19](Chapter 2), which can potentially be a universal strategy to conduct variance estimation in the one-parent IPS context. We hope these theoretical tools will be of bene t to the analysis of more complex and advanced models in the IPS context.

Main results

On one hand, in a very general setting, we provide consistent estimations for the target measures in the SMC context with controllable asymptotic uncertainty under our speci c asymmetric resampling scheme. Since the computations of the variance estimators are highly nontrivial, we provide detailed and e cient algorithms with time and space complexity analysis in Section 3.5. If there is any ambiguity w.r.t. the notation, the reader is referred to Section 3.1.2. Theorem 3.1.1. Let ( ; ≤ 0) be a sequence of Polish state spaces. Given a sequence of [0, 1]valued potential functions ( ; ≥ 0) and a canonical Markov chain ( ; ≥ 0) taking values in ( ; ≥ 0), with initial distribution 0 and transition kernels ( ˚ ; ≥ 1), we de ne the family of measures ( ; ≥ 0) by

( ) := E ( ) -1 =0 ( ) .
Assuming that (1) > 0 for any ≥ 0, we also de ne ( ) := ( )/ (1). For any test function ∈ B ( ), when the number of particle tends to in nity, the estimators given by Algorithm 2 in Section 3.5, denoted respectively by ( ) and ( ), converge almost surely to ( ) and ( ) if for any ≥ 1, we have

∀ ∈ B ( ), -1 ( -1 × ˚ ( )) = -1 ( -1 × • ( )).
Moreover, we also have

√ ( ) -( ) ˆ ( ) d -----→ →∞ N (0, 1), as well as √ ( ) -( ) ˆ ( -( )) d -----→ →∞ N (0, 1),
where the computation of ˆ 2 ( ) and ˆ 2 ( -( )) are respectively provided in Algorithm 5 and Algorithm 6 in Section 3.5.

On the other hand, under mild assumptions, we provide an unbiased non-asymptotic variance estimator of ( ), which, again, represents the output of Algorithm 2.

Theorem 3.1.2. Assume the same setting as in Theorem 3.1.1. Under the condition discussed in Section 3.4.1, which at least contains the case where • ≡ ˚ for any ≥ 1, the estimator ( ) is an unbiased estimator for ( ). Moreover, the estimator provided by Algorithm 7 in Section 3.5 is an unbiased estimator for the non-asymptotic variance of ( ).

Notation

Before getting into details, let us provide a few notations which are useful in the following.

• The underlying probability space is denoted by (Ω, F, P). For -elds E, G ⊂ F, E ∨ G denotes the smallest -eld on Ω containing E and G. For any , ∈ R, we denote ∧ := min{ , } and ∨ := max{ , }. We also adopt the standard convention inf = ∞.

• Let X be a number, a function or a random variable. We adopt the following convention:

1 X 1 X≠0 := 1 X if X ≠ 0, 0 otherwise. (3.1)
Therefore, we admit the calculation

X × 1 X 1 X≠0 = 1 X≠0 .
• Random variables take values in Polish spaces, i.e., a topological space which is metrizable, separable and complete for some distance . It is endowed with the Borel -algebra generated by , denoted by B ( ).

• We denote respectively by M ( ), M + ( ) and P ( ) the set of all signed nite measures, the subset of all nonnegative nite measures and the subset of all probability measures on ( , B ( )). The set P ( ) is endowed with the Prohorov-Lévy metric, i.e., the weak convergence " d -→" is tested with continuous bounded functions.

• B ( ) denotes the collection of all the bounded measurable functions from ( , B ( )) to (R, B (R)) equipped with uniform norm • ∞ , among which the constant function will be denoted by 1 with a slight abuse of notation. Given a probability measure in P ( ) and for all test functions in B ( ), we denote -ess sup( ) the essential supremum of . It is de ned by -ess sup(

) := inf { ∈ R : ( ∈ : ( ) > ) = 0} .
• For all ∈ M ( ) and for all test functions ∈ B ( ), ( ) denotes the integration ∫ ( ) ( ).

A nite transition kernel from ( , B ( )) to ( , B ( )) is a function

: × B ( ) ↦ → R + .
More precisely, for all ∈ , ( , •) is a nite nonnegative measure in M + ( ) and for all ∈ B ( ), ↦ → ( , ) is a B ( )-measurable function. We say that is a Markov transition kernel if is a nite transition kernel and for all ∈ , ( , •) is a probability measure in P ( ). For a signed measure ∈ M ( ) and a test function ∈ B ( ), we denote respectively ∈ M ( ) and ∈ B ( ) are respectively de ned as follows:

∀ ∈ B ( ), ( ) := ∫ ( ) ( , ), and
∀ ∈ , ( ) := ∫ ( , ) ( ).
Let 1 and 2 be two nite transition kernels respectively from 0 to 1 and from 1 to 2 . When well-de ned, we denote 1 • 2 or simply 1 2 , the transition kernel from 0 to 2 de ned by

∀( , ) ∈ 0 × B ( 2 ), 1 2 ( , ) := ∫ 1 1 ( , ) 2 ( , ).
Note that, there is no reason that 1 2 is still a nite transition kernel in general. We say that 1 is a uniformly nite transition kernel from space 0 to 1 if sup

∈ 0 ∫ 1 ( , ) < +∞.
For example, a Markov transition kernel is a uniformly nite transition kernel. Let 2 be a uniformly nite transition kernel from 1 to 2 , we have that 1 2 is also a uniformly nite transition kernel from 0 to 2 .

• For two test functions , ∈ B ( ), we denote

⊗ : 2 ( , ) ↦ → ( ) ( ) ∈ R.
In particular, we denote ⊗2 := ⊗ .

Accordingly, we denote

B ( ) ⊗2 := { ⊗ : , ∈ B ( )} .
For two nite transition kernels and from ( , B ( )) to ( , B ( )), we denote, for all ( , ) ∈ × and for all ( , ) ∈ B ( ) ⊗ B ( ), ⊗ (( , ), ( , )) := ( , ) × ( , ).

Similarly, we also denote ⊗2 := ⊗ .

• In order to de ne the coalescent tree-based measures of size 2, we introduce the transition operators 0 and 1 as 0 (( , ), ( , )) := ( , ) ( , ), and 1 (( , ), ( , )) := ( , ) ( , ). In other words, for any measurable function : × ↦ → R, we have 0 ( ) ( , ) = ( , ) and 1 ( ) ( , ) = ( , ).

• For all x = ( 1 , . . . , ) ∈ , we de ne the empirical measure associated to x by : x ↦ → (x) := 1

=1 ∈ P ( ).
We denote ⊗2 : x ↦ → ⊗2 (x) := 1 2 , ( , ) ∈ P ( ),

and 2 : x ↦ → 2 (x) := 1 ( -1) ≠ ( , ) ∈ P ( ).
A straightforward computation shows that

⊗2 (x) = -1 2 (x) 0 + 1 2 (x) 1 . (3.2)
With a slight abuse of notation, considering [ ] := {1, 2, . . . , }, we write

( [ ]) := 1 =1 and ⊗2 ( [ ]) := ( [ ]) ⊗ ( [ ]), as well as 2 ( [ ]) := 1 ( -1) ≠ ( , ) .

SMC framework

In this section, we de ne the SMC framework studied in this chapter. We present some standard convergence results on the consistency and asymptotic normality of the associated Interactive Particle System (IPS) in the discrete time setting. We mainly use the language of Feynman-Kac particle models, and the reader is referred to the pair of books [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF] and [START_REF] Moral | Mean eld simulation for Monte Carlo integration[END_REF] for more details. The main goal is to establish central limit theorems and to specify the asymptotic variance in our speci c asymmetric setting.

Setting

Let ( , B ( )) ≥0 be a sequence of Polish spaces and let us x a probability measure 0 ∈ P ( 0 ). We consider a sequence of [0, 1]-valued measurable potential functions ( ) ≥0 and a sequence of Markov transition kernels ( ˚ ) ≥1 s.t. ˚ : ( -1 , B ( )) ↦ → [0, 1]. We de ne the Feynman-Kac kernels as follows

∀( , ) ∈ ( -1 , B ( )), ˚ ( , ) := -1 ( ) ˚ ( , ).
It is readily checked that is a uniformly nite transition kernel. Therefore, we de ne the unnormalized Feynman-Kac measure by

∀ ≥ 1, := 0 ˚ 1 ˚ 2 • • • ˚ ,
with 0 := 0 . By de nition, for all ≥ 1, is a sub-probability measure. For all ≥ 0, we suppose that we have a meaningful sampling problem at each step, i.e., we assume that (1) > 0. Therefore, one can de ne the normalized Feynman-Kac measures ∀ ≥ 1, := (1) .

We adopt the convention

-1 = -1 = 0 .
By standard convention for the product symbol " ", it is readily checked that

∀ ≥ 0, = -1 =0 ( ) . (3.3)
Di erent from the classical framework of SMC methods, we suppose that there exists an additional sequence of Markov transition kernels (

•

) ≥0 , such that for the Feynman-Kac kernel de ned by ∀( , ) ∈ -1 × B ( ),

• ( ,

) := -1 ( ) × • ( , ),
we have, for all ≥ 0,

• +1 = ˚ +1 = +1 .
Using the Feynman-Kac kernels ˚ and

•

, we de ne the Feynman-Kac kernel by := -1 ( -1 )

•

+ [1 --1 ( -1 )] ˚ .
More rigorously, for any ∈ M ( -1 ) and for any ∈ B ( ), we have

( ) := -1 ( -1 ) • ( ) + [1 --1 ( -1 )] ˚ ( ).
Hence, for all 0 ≤ < < +∞, the associated Feynman-Kac partial semigroup is de ned as follows:

, := +1 • • • .
The term "partial" comes from the fact that the state spaces may vary w.r.t. the time horizon . Hence, it is not a semigroup. In particular, the partial unit elements at each step is de ned by , ( , ) := ( ) on the space . Remark. Technically speaking, such

0 = 0 1 +1 • • • ˚ 1 • 1 ˚ +1 • +1 ˚ 2 • • • ˚ • 2 • • • • 1 1, +1 Figure
• always exists. For example, one may consider the choice

• ≡ ˚ .
Even in this simple symmetric setting, the variance related problems are already very challenging. The CLT-type results are well-known (see, e.g., Chapter 7 of [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF]). However, to the best of our knowledge, there is no consistent asymptotic variance estimators available with a single simulation of the particle system. Meanwhile, it is natural to implement the asymmetric resampling in the case where ˚ is an -invariant kernel: we are interested by the choice • ( , ) := ( ) since it requires the least computational cost, which is widely-used by the practitioners in tempering and rare-event simulation. Therefore, we combine these two examples and go one step further: we consider the asymmetric resampling scheme and we provide some theoretical analysis. When ˚ is not an -invariant kernel, it is also always possible to construct a nontrivial • using ˚ . In fact,

• can still be thought as some "cheaper" version of the latter: ˚ can be designed as the composition of • and an -invariant kernel, for which one may consider the PMCMC-type kernel, which is always available with ( ; ≥ 0) and ( ˚ ; ≥ 1) under the current setting. Intuitively speaking, this SMC 2 -type design can help to reduce the dependence due to the multinomial resampling step.

De nition 3.2.1. We introduce the asymmetric McKean kernel , from -1 to , parameterized by some positive nite measure ∈ M + ( -1 ) such that ( -1 ) > 0, de ned as follows 1 ) .

∀ ∈ B ( ), , (
, ) := -1 ( ) • ( , ) + (1 --1 ( )) ( -1 × ˚ ( )) ( -
Accordingly, we also de ne the McKean-type Feynman-Kac kernel , by

, := ( -1 ) , ( , ),

with the convention ∀ ∈ 0 , 0, ( , ) := 0 ( ).

(3.4)

Remark. Standard calculations show that the McKean-type kernels , -1 and , -1 also connect the Feynman-Kac measures ow:

-1 , -1 = and -1

, -1 = .

(3.5)

Assuming that is upper bounded by 1 rather than a nite positive number ∞ is purely for technical reasons, in order to simplify the relatively heavy notation. There is no loss of generality for the case where ∞ is known: we could always consider the "normalized" version of potential function

¯ := ∞ ,
in order to construct a potential function varying on the interval [0,1]. However, when ∞ is not explicitly tractable, it is not possible to design the asymmetric version of SMC sampler with xed normalizer. When the normalizer is set to be +∞, we return to the classical multinomial resampling scheme. This is a crucial problem in applications such as tempering, when determining a reasonable upper bound of the potential function is not always trivial. One possible solution is to consider the adaptive normalizer, depending upon the entry measure , rather than a pre xed one. An interesting example of the adaptive normalizer is de ned by -ess sup( -1 ), where denotes the entry measure of the McKean kernel. This is the "laziest" resampling scheme we could ever design, which gives potentially the smallest asymptotic variance, and no upper bound of the potential function is required. However, we failed to provide the general analysis for this case since the calculation of the asymptotic variance in the CLT-type results will become more challenging and it is possible that stronger mixing properties for ˚ and • have to be assumed. Heuristically speaking, in order to establish the CLT-type results and to conduct the asymptotic variance estimation, one needs a convergence of the following type:

∃ ∈ R * + , 1/ max 1≤ ≤ ( ) P -----→ →∞ .
This requires much stronger convergence than the well-known almost sure convergence of the empirical measures. However, if the convergence above holds, we expect that the methodology in this chapter would still be valid, with only minor notational complications. In a nutshell, one needs to discuss the property above in concrete applications, such as the mixing property of the Markov kernels, etc. Meanwhile, the goal of the present work is to obtain some general structural results without further assumptions. As a consequence, we decide to leave this important case for future research.

Interacting particle system

The Interacting Particle System (IPS) in this chapter refers to a Markov chain (X n ; ≥ 0) with absorption in the product spaces ( , B ( ) ⊗ ; ≥ 0). As we have seen in the previous section (3.5), the normalized Feynman-Kac measures and +1 are connected by +1, , which depends on the measure of the previous step . Hence, it is not possible to simulate directly according to the kernel +1, . The idea of the IPS is to simulate particles X n = ( 1 , 2 , . . . , ) step by step. Therefore, by exploiting the empirical measure (X n ) to approximate its "limiting" measure , we are able to simulate the next layer of particles X n+1 with the approximated kernel +1, (X n ) . In this section, we deal with the version without the genealogy (i.e., the indices of the parent of each particle) and the survival history of IPS. The mechanism of the IPS is de ned as follows:

(i) X 0 ∼ ⊗ 0 ; (ii) Stop the algorithm at step ≥ 0 if (X n ) ( ) = 0; (iii) If not stopped at step ≥ 0, X n+1 ∼ =1 +1, (X n ) ( , •).
A more detailed explanation on the algorithm can be found in Section 3.2.4. The particle approximation of the normalized measure is de ned by

:= (X n ) = 1 =1 .
According to (3.3), the unnormalized version is de ned by :=

-1 =0 ( ) .
The absorbing time of the Feynman-Kac IPS is de ned by

:= inf { ∈ N : (X n ) ( ) = 0} .

Asymptotic results

In this section, we establish some basic convergence results such as law of large numbers and central limit theorem for the empirical Feynman-Kac measures. These results are standard in the case where

• ≡ ˚ (see, e.g., Chapter 7 of [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF]) and the proofs are housed respectively in Section 3.7.3 and Section 3.7.4. The goal is to understand the consequences of the introduction of • , especially on the form of the asymptotic variances.

Theorem 3.2.1. For any test function ∈ B ( ), we have

( )1 ≥ . . -----→ →∞ ( ).
The almost sure convergence also holds for 1 ≥ . In particular, by taking the test function 1 for , we get 1 ≥

. .

-----→ →∞ 1.

Moreover, if we assume symmetric resampling, that is

• ≡ ˚ for any ≥ 1, we also have

∀ ≥ 0, E ( )1 ≥ = ( ).
Theorem 3.2.2. For any test function ∈ B ( ), we have

∀ ≥ 0, √ ( )1 ≥ -( ) d -----→ →∞ N 0, 2 ( ) ,
with the asymptotic variance de ned as follows:

2 ( ) := =0 ⊗2 1 ⊗2 , ( ⊗2 ) -⊗2 -1 1 ⊗2 , -1 ⊗2 , ( ⊗2 ) . (3.6)
Similarly, we also have

∀ ≥ 0, √ ( )1 ≥ -( ) d -----→ →∞ N 0, 2 ( -( )) ,
with 2 de ned by

∀ ∈ B ( ), 2 ( ) := 2 ( )/ (1) 2 . (3.7)
Let us emphasize that one of the main goals of this chapter is to provide consistent estimators w.r.t. the particle numbers for the asymptotic variances 2 and 2 de ned above. In practice, thanks to Slutsky's lemma, the consistent variance estimators allow us to deduce con dential intervals with one single simulation of IPS.

Genealogy and survival history

In this section, we give a more detailed version of the IPS de ned in Section 3.2.2, namely, the actual simulation algorithm we execute in practice. Speci cally, we trace two kinds of information: the genealogy A n = ( 1 , . . . ,

) ∈ [ ] and the so-called survival history B n = ( 1 , . . . , ) ∈ {0, 1} . They are both intermediate random variables introduced in the real-world algorithm, so that one can simulate according to an approximated kernel +1, (X n ) . Note that = means that the parent of +1 at level is . Besides, = 1 indicates that the particle has survived at step , i.e., the parent of +1 is ( = ) and

+1 ∼ • ( , •).
Note that this does not mean that a non-survived particle at step , i.e., a particle such that = 0, is disappeared in the IPS after step : it can still be selected as a parent by the multinomial resampling step. Unlike the multinomial selection scheme, the information encoded in IPS and its genealogy is not enough to conduct the variance estimation. This is the reason why survival history has to be taken into consideration. Another remark is for rare-event simulation, or more generally, the case where ( ; ≥ 0) are all indicator functions: the survival history is already encoded in ( ( ), ≥ 0, ∈ [ ]). Hence, there is no need to track them separately. Now, let us give the proper de nition of the IPS with its genealogy and survival history: (i) Initial distribution: At step 0, we let X 0 ∼ ⊗ 0 .

(ii) Stopping criterion:

Stop the algorithm at step ≥ 0 if (X n ) ( ) = 0.

(iii) Transition kernels:

If not stopped at ≥ 0, we execute the elementary transition +1 for all 1 ≤ ≤ conditionally independently, following the three steps:

• Survival test: Let be a Bernoulli random variable with probability ( ), that is

∼ ( ) 1 + (1 -( )) 0 .
• Selection: If = 1, we let = . Otherwise, the parent index is selected by the following multinomial selection

∼ =1 ( ) =1 ( ) .

Therefore, given

= β , we have

∼ β + (1 -β ) =1 ( ) =1 ( ) .
• Mutation: Given = β and = , each particle evolves independently from level to level + 1 according to the following transition kernel:

+1 ∼ β • +1 ( , •) + (1 -β ) ˚ +1 ( , •).

Variance estimations

In this section, we provide estimators for the asymptotic variances 2 ( ) and 2 ( ): we provide a term by term asymptotic variance estimator, an unbiased variance estimator under symmetric resampling scheme, and nally, an e cient asymptotic variance estimator. The strategy is almost identical as in [DG19](Chapter 2). First, we give an alternative representation of the asymptotic variance 2 ( ) using some generalized coalescent tree-based measures. Next, we provide convergence results of the particle approximations of these generalized coalescent tree-based measures, which gives naturally a term by term variance estimator. Finally, we connect this term by term estimator to the non-asymptotic variance using a nontrivial combinatorial property of the IPS given in Theorem 3.6.1, from which we derive an e cient variance estimator that can be computed with the optimal O( ) time complexity.

Asymptotic variance expansion

In this section, we revisit the asymptotic variance 2 ( ) of Theorem 3.2.2 using some novel coalescent tree-based measures. More precisely, unlike the multinomial case, the form of the asymptotic variance 2 ( ) is relatively complex under asymmetric resampling and there is no free coalescent tree-based expansion as in [DG19](Chapter 2). Hence, we need to introduce some generalized coalescent tree-based measures as a supplement of the one introduced in [START_REF] Cérou | A nonasymptotic theorem for unnormalized Feynman-Kac particle models[END_REF].

The goal is plain and simple: we want to establish an alternative representation of the asymptotic variance based on some coalescent tree-based measures. To begin, let us de ne the so-called coalescent Feynman-Kac kernels:

(i) Q †,0 := ⊗2 ; Q †,1 := 1 ⊗2 --1 ( -1 ) 2 1 • ⊗2 . (ii)            Q †,0 := ⊗2 ; Q †,1 := -1 ( -1 ) ( -1 × • ) ⊗ ˚ + ˚ ⊗ ( -1 × • ) + -1 ( 2 -1 ) • ⊗2 - • ⊗ ˚ -˚ ⊗ • .
The full description of this new family of kernels can be found in Section 3.6.2. Using the partial semigroup property of the coalescent Feynman-Kac kernels de ned above, we introduce some generalized coalescent tree-based measures. They will be referred to as coalescent Feynman-Kac measures in this chapter. In the next de nition, we call := ( 0 , . . . , ) ∈ {0, 1} +1 a coalescence indicator where = 1 indicates that there is a coalescence at level .

De nition 3.3.1. For any ≥ 1 and for any coalescence indicator ∈ {0, 1} +1 , we de ne the signed nite measures Γ †, and Γ †, by

∀ ∈ B ( 2 ), Γ †, ( ) := ⊗2 0 Q †, 0 1 Q †, 1 2 • • • Q †, -1 ( ), and
∀ ∈ B ( 2 ), Γ †, ( ) := ⊗2 0 Q †, 0 1 Q †, 1 2 • • • Q †, -1 ( ),
with the convention Γ †, 0 ( ) := ⊗2 0 0 and Γ †, 0 ( ) := ⊗2 0 .

When there is only one coalescence at level , we write respectively Γ †,( ) ( ) and Γ †,( ) ( ) instead.

When there is no coalescence, we denote respectively Γ †,( ) ( ) and Γ †,( ) ( ).

The connection of the original coalescent tree-based measures proposed in [START_REF] Cérou | A nonasymptotic theorem for unnormalized Feynman-Kac particle models[END_REF] and the generalized version de ned above will be discussed in Section 3.6.1 and Section 3.6.2. By exploiting this novel pair of coalescent Feynman-Kac measures, we have the following alternative representation of the asymptotic variance 2 ( ). The rigorous veri cation is housed in Section 3.7.2.

2 ( ) := =0 Γ †,( ) ( ⊗2 ) -Γ †,( ) ( ⊗2 ) + -1 =0 Γ †,( ) ( ⊗2 ).
(3.8)

Term by term asymptotic variance estimators

Thanks to the alternative representation (3.8) given in the last section, the variance estimation problem is reformulated as how we can estimate the corresponding coalescent Feynman-Kac measures. Using the same idea as in [DG19](Chapter 2), we construct the particle approximation of Γ †, and Γ †, . They will be referred to as coalescent tree occupation measures in this chapter.

In the following, ˜ [2] = ( ˜ 1 , ˜ 2 ) and [2] = ( 1 , 2 ) denote two couples of indices between 1 and , while an ( + 1)-sequence of couples of indices such that 1 ≠ 2 for all 0 ≤ ≤ is written

[2] 0: = ( [2] 0 , • • • , [2] ) ∈ ( ) 2 ×( +1) ,
where ( ) 2 := {( , ) ∈ [ ] 2 : ≠ }. Especially, we denote

[2]

: +1 := ( [2] , [2] +1 ).
Additionally, we use the notation

[2] = ( 1 , 2
) to shorten the writings. One can also nd a toy example in [DG19](Chapter 2) in order to get more intuitions for the following de nition.

De nition 3.3.2. For any ≥ 0 and for any coalescence indicator ∈ {0, 1} +1 , the estimator Γ ‡, , of Γ †, is de ned by

∀ ∈ B ( 2 ), Γ ‡, , ( ) = 1 ( -1) [2] ∈ ( ) 2          -1 =0 [2] ∈ ( ) 2 G ‡ p (X p ) ( [2] +1
, [2] )

         ( ) ( [ 2]
),

with G ‡ p de ned by ∀x ∈ , G ‡ p (x ) := -1 ⊗2 (x ) ( ⊗2 ),
and

( ˜ [2]
, [2] ) ∈ {0, 1} is the indicator function de ned by

( ˜ [2] , [2] ) := 1 { =0} 1 { ˜ 1 = 1 ≠ ˜ 2 = 2 } + 1 { =1} 1 { ˜ 1 = 1 = ˜ 2 ≠ 2 } .
Notice that, by standard convention, we get

Γ ‡, 0, := 1 ( -1) [2] 0 ∈ ( ) 2 0 ( ) ( [2] 0 0 ) = 1 ( -1) ≠ 0 ( ) ( 0 , 0 ).
(3.9)

Remark. In fact, Γ ‡, , de ned above is exactly the same estimator as Γ , de ned in De nition 3.2 of [DG19](Chapter 2). The change of notation is due to the change of resampling scheme, and the exact reason lies in a technical result (cf. Proposition 3.7.9). The inhomogeneity of the notation w.r.t. " ‡" and " †" is due to some nontrivial combinatorial structure of the asymmetric SMC framework. The detailed explanation can be found in Proposition 3.6.3 and other remarks in Section 3.6.2. De nition 3.3.3. For any test function ∈ B ( 2 ) and any coalescence indicator , the estimator Γ †, , of Γ †, is de ned by

Γ †, , ( ) := 1 ( -1) [2] ∈ ( ) 2          -1 =0 [2] ∈ ( ) 2 G †, p ( [2] : +1 , B p , X p ) ( ) ( [2] +1
, [2] )

         ( [ 2]
) with G †, de ned as follows:

∀( [2] : +1 , , x ) ∈ ( ) 2 ×2 × {0, 1} × , we let G †,0 p ( [2] : +1 , , x ) := G ‡ p (X p ) - 1 -1 G †,1 p ( [2]
: +1 , , x ), and

G †,1 p ( [2] : +1 , , x ) :=β 1 +1 β 2 +1 (x ) ( 2 ) + β 1 +1 (1 -β 2 +1 ) (x ) ( ) ( 1 ) (x ) ( ) -(x ) ( 2 ) ≠ 1 1 -( ) / + β 2 +1 (1 -β 1 +1 ) (x ) ( ) ( 2 ) (x ) ( ) -(x ) ( 2 ) ≠ 2 1 -( ) /
.

We also de ne

∀ ∈ B ( 2 ), Γ ( ) , ( ) := Γ †,( ) , ( ).
Remark. In particular, if is an indicator function for all ≥ 0, we consider

G †,1 p ( [2] : +1 , , x ) := β 1 +1 β 2 +1 (X p ) ( 2 ),
which leads to a simpler form of G †,0 p , i.e.,

G †,0 p ( [2] : +1 , , x ) := β 1 +1 β 2 +1 (x ) 2 ( ) + -1 (1 -β 1 +1 β 2 +1 ) ⊗2 (x ) ( ⊗2 ). (3.10)
Returning to the coalescent tree-based expansion given in (3.8), it is natural to de ne the term by term estimators 2 ( ) as follows:

2 ( ) := =0 Γ ‡,( ) , ( ⊗2 ) -Γ ‡,( ) , ( ⊗2 ) + -1 =0 Γ †,( ) , ( ⊗2 ) 1 ≥ .
Then, by (3.7), it is natural to consider 2 ( ) := 2 ( ) / (1) 2 .

Therefore, thanks to Theorem 3.6.3 and Corollary 3.6.3.1 , we have the consistency of these term by term variance estimators.

Theorem 3.3.1 (Consistency of 2 and 2 ). For any test function ∈ B ( ), we have

sup ≥0 √ E 2 ( ) -2 ( ) < +∞, as well as 2 -( ) -2 ( -( )) = O p 1 √ .
Remark. We do not provide the algorithm to compute these estimators, since, to the best of our knowledge, they can only be computed with time complexity O( 2 ). Therefore, they mainly serve as theoretical handy tools prove the consistency of the e cient estimator given in Algorithm 5 and Algorithm 6. However, with the same techniques as in these two Algorithms, one should be able to design an algorithm such that each term in the asymptotic variance can be evaluated separately, with time complexity O( ). The details are given in Section 3.6.6.

Unbiased non-asymptotic variance estimator

In this section, we provide an unbiased non-asymptotic variance estimator which is only valid under symmetric resampling scheme, i.e.,

• ≡ ˚ for all ≥ 0. However, we prove that in the general case, this estimator also yields a consistent asymptotic variance estimator. In fact, in order to provide an unbiased non-asymptotic variance estimator, the idea is much more straightforward: thanks to Theorem 3.2.1, we know that under symmetric sampling scheme, the estimation ( )1 ≥ is unbiased. As a consequence, one has

Var ( )1 ≥ = E ( ) 2 1 ≥ -E ( )1 ≥ 2 = ( ) 2 = ⊗2 ( ⊗2 ).
.

It is then clear that constructing an unbiased non-asymptotic variance estimator is equivalent to constructing an unbiased estimator for the measure ⊗2 . Therefore, the following proposition is a direct consequence of Proposition 3.7.3. The detailed computation is provided in Algorithm 7 in Section 3.5.

Theorem 3.3.2. Assume symmetric resampling, that is,

• ≡ ˚ for all ≥ 0. For any test function ∈ B ( ), the estimator ( ) de ned below is an unbiased variance estimator of ( )1 ≥ :

( ) := ( ) 2 -Γ ( ) , ( ⊗2 ) 1 ≥ . (3.11)
Remark. In fact, this unbiased estimator can also be used by AMS methods if the image of the reaction coordinate is a nite set, and under some regularity assumption on the resampling kernel (e.g. Assumption 1, 2 of [BGG + 16]) is satis ed. Although the associated IPS is not simulated by symmetric resampling, recent results (cf. [START_REF] Cérou | On the asymptotic normality of adaptive multilevel splitting[END_REF]) show that one can construct an IPS with particles de ned by some level-indexed processes that are "mathematically symmetrically resampled". It can be regarded as an almost sure equivalence between an arti cial asymmetric IPS, i.e. the real-world algorithm, and a symmetric IPS constructed by some abstract mathematical objects. More generally, when the reaction coordinate is nite-valued, the AMS method enters the asymmetric SMC framework. More discussions on this topic can be found in Section 3.4.1. As a consequence, for all the unbiasedness results in this chapter, we only use the condition under symmetric resampling or assume symmetric resampling in order to simplify the writings.

Connection between the estimators

The connection between the term by term estimators and the non-asymptotic variance estimator is based on Theorem 3.6.1. Unfortunately, to the best of our knowledge, both of these estimators can only be computed with O( 2 ) time complexity. However, this connection inspired the construction of the e cient consistent estimator provided in the next section. The proof is provided in Section 3.7.5.

E cient asymptotic variance estimators

The e cient variance estimator can be regarded as a "mix" of the term by term estimator and the non-asymptotic estimator provided respectively in Section 3.3.2 and Section 3.3.3. Inspired by a by-product (3.41) in the proof of Proposition 3.3.1, an intermediate estimator can be proposed as follows

( ) 2 -Γ ‡,( ) , ( ⊗2 ) + -1 =0 Γ †,( ) , ( ⊗2 ) 1 ≥ .
By the same technique as the one proposed in [START_REF] Lee | Variance estimation in the particle lter[END_REF], the former term can be approximated by

( ) 2 -Γ ‡,( ) , ( ⊗2 ) ,
which can be computed with O( ) time complexity. This term corresponds to V ‡ founded in Algorithm 5 in Section 3.5. Hence, the design of an e cient variance estimator amounts to constructing an e cient estimator for Γ †,( ) . Since we failed to provide an e cient algorithm to compute Γ †,( ) , , we consider some particle approximation Γ ‡,( ) , of Γ †,( ) , such that Γ ‡,( ) , is "close" enough to the original one Γ †,( ) ,

, and which, at the same time, can be computed with O( ) time complexity.

De nition 3.3.4. For any test function ∈ B ( 2 ) and any coalescence indicator , the e cient estimator Γ ‡, , of Γ †, is de ned by

Γ ‡, , ( ) := 1 ( -1) [2] ∈ ( ) 2          -1 =0 [2] ∈ ( ) 2 G ‡, p ( [2] : +1 , B p , X p ) ( ) ( [2] +1
, [2] )

         ( [ 2]
) with G ‡, de ned by, ∀( [2] : +1 , , x ) ∈ ( )

2 ×2 × {0, 1} × , G ‡,0 p ( [2] : +1 , , x ) := G ‡ p (x ); G ‡,1 p ( [2] : +1 , , x ) := G †,1 p ( [2] : +1 , , x ).
Proposition 3.3.2. For any test function ∈ B ( 2 ) and any coalescence indicator ∈ {0, 1} +1 , we have sup

>1 E Γ ‡, , ( ) -Γ †, , ( ) < +∞.
Finally, we provide the e cient asymptotic variance estimators, which are respectively the output of Algorithm 5 and Algorithm 6. For any test function ∈ B ( ), we de ne

ˆ 2 ( ) := ( ) 2 -Γ ‡,( ) , + -1 =0 Γ ‡,( ) , ( ⊗2 ), (3.14) and ˆ 2 ( ) = ˆ 2 ( )/ (1) 2 .
Thanks to Proposition 3.6.4, Proposition 3.3.2, Proposition 3.6.4, Theorem 3.3.1 and Theorem 3.2.1, we have the following consistency result.

Theorem 3.3.3 (Consistency of ˆ 2 and ˆ 2 ). For any test function ∈ B ( ), we have

sup >1 √ E ˆ 2 ( ) -2 ( ) < +∞,
as well as

ˆ 2 -( ) -2 ( -( )) = O p 1 √ .

Discussions

Unbiasedness condition

We discuss the condition such that the estimations for the unnormalized measures are unbiased. More precisely,

E ( )1 ≥ = ( ) and E Γ ( ) , ( )1 ≥ = ( ) 2 .
The major motivation is to give an intuitive interpretation of Assumption 2 of [BGG + 16] in our xed level setting. Recent works (cf. [START_REF] Delyon | A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing[END_REF] and [START_REF] Cérou | On the asymptotic normality of adaptive multilevel splitting[END_REF]) show that Assumption 2 of [BGG + 16] can eventually be regarded as a requirement to reformulate AMS methods as Fleming-Viot particle systems, which enters the continuous-time generalization of the current framework, with symmetric resampling, namely, the case where

• ≡ ˚ .
More concretely, if the realworld simulation is under asymmetric resampling scheme, but we are somehow able to nd out an underlying mathematical structure that is symmetrically resampled, the unbiasedness will be recovered. In addition, as discussed in Section 3.6.1, it also gives a smaller asymptotic variance than the multinomial resampling. Unfortunately, the only setting we could provide that allows this property is the family of AMS methods in the dynamical setting. The readers are referred to [START_REF] Delyon | A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing[END_REF], [START_REF] Cérou | On the asymptotic normality of adaptive multilevel splitting[END_REF] and [BGG + 16] for more details. In this respect, since the real mathematical object is simply a symmetric SMC model, we did not emphasize the more complex condition discussed above, which is also due to the fact that the variance estimators are the same in the symmetric and asymmetric resampling schemes. Moreover, it would be interesting to see if we can implement the same technique in Particle Tempering methods, at least in some speci c situations.

Other comments

On the adaptive SMC models First, we want to mention that the current setting covers the Adaptive Multilevel Splitting methods when the image of the reaction coordinate is a nite set. This is due to the fact that under asymmetric resampling scheme, when the potential of a particle is 1, there is no additional computational cost required to evolve the IPS. In practice, it corresponds to the case where the reaction coordinate function is calculated by some pre xed grid. Next, since the ingredients we need to implement asymmetric SMC are nearly the same as for the classical multinomial SMC, we can therefore consider the corresponding adaptive methods as in our previous work in [DG19](Chapter 2). Since all the technical results are done in a similar style, we expect the adaptive version with Assumption 2 of [DG19](Chapter 2) to be a simple generalization from a mathematical point of view, at the price of some notational complications. Therefore, the variance estimators may be used as a reference if the underlying resampling scheme is changed to the asymmetric one. We also expect the asymptotic variance estimators to apply to another family of adaptive SMC models, which contain an online adaptive resampling strategy, such as [START_REF] Giraud | Nonasymptotic analysis of adaptive and annealed feynman-kac particle models[END_REF][START_REF] Del Moral | On adaptive resampling strategies for sequential monte carlo methods[END_REF]. Roughly speaking, the resampling is executed when some summary statistics, such as the popular E ective Sample Size, attains some pre xed threshold. In this scenario, as the adaptive model and the xed reference model are connected by a coupling argument, one is also encouraged to use our estimators in the real-world applications as a reference. However, there are situations where we are sure that the estimators provided in this chapter will fail. The rst example is in Section 2 of [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF]. If the stability property given in Theorem 2.3 is not veri ed: namely, if the "limit" model has di erent asymptotic variance, then, it is not possible to conduct variance estimations with our estimators. The same argument also applies to the Adaptive Tempering introduced in section 3 of [START_REF] Beskos | On the convergence of adaptive sequential Monte Carlo methods[END_REF]: it is not possible to use our variance estimator as a reference even if we change the underlying resampling scheme. In all cases, the rigorous analysis of the adaptive context requires more attention on the regularity of the adaptive parametrizations.

On the non-asymptotic variance expansion and long-term behaviors

There is an angle that we did not cover in the present work: using the ner analysis given in Lemma 3.7.10 and the decomposition in Theorem 3.6.1, a very sharp upper bound of the non-asymptotic variance can be obtained, w.r.t. both and . Since it is well understood that the non-asymptotic variance may also contribute to the bound encountered in the propagation of chaos property of the particle system (cf. [START_REF] Del Moral | Tree based functional expansions for Feynman-Kac particle models[END_REF]), we expect that one can also derive the sharp propagation of chaos bound for the non-asymptotic empirical measures. In particular, this kind of analysis can provide information on the bias associated to the estimation ( )1 ≥ . The same idea can also be applied to obtain sharp L p -bound estimates. This is a relatively large topic, the rigorous analysis is thus left for future investigations. Returning to the variance estimation problems, we remark that all the consistent variance estimators provided in this chapter are essentially for "short-term" models, namely, is set to be nite and tends to in nity. When more stability properties of the Feynman-Kac kernels are available, it would be interesting to investigate the xed-lag variance estimator such as the one introduced in [START_REF] Olsson | Numerically stable online estimation of variance in particle lters[END_REF]. The same kind of estimators, i.e., by considering only part of the genealogy, as well as the survival history, is expected to be more numerically stable in the long term. We expect that the regularity requirements will be the same as in the multinomial case.

On the PMCMC-type kernels Another remark is on the PMCMC-type kernels: starting from a trajectory of the particle system, the new sample is constructed by simulating an IPS with this frozen trajectory, and we pick randomly and uniformly an ancestral lineage in the novel IPS, using its terminal point as the new sample. This kind of kernel does not enter the framework of gAMS since Assumption 2 of [BGG + 16] is not veri ed. Therefore, no level-indexed process can be derived. However, it is a widely used kernel in the Particle Filters and Particle Tempering contexts. In rare-event simulation context, it is also promising in resolving the high dimensional multimodal metastable problems. As a complement, the present work can implement this type of kernels: in fact, one may pre x a very ne grid of levels: since the resampling scheme we use does not require additional computation when all the particles have survived, the implementation is very close to the last-particle AMS methods in practice. Moreover, one can also construct the PMCMC-type kernel by using the standard transition kernel that satis es Assumption 2 of [BGG + 16]. It is possible to study the performance of this Markov kernel with the theoretical tools we provided in this chapter. This kind of connection is also well illustrated in [START_REF] Andrieu | Uniform ergodicity of the iterated conditional smc and geometric ergodicity of particle gibbs samplers[END_REF]. Nevertheless, the rigorous analysis is also left for future research.

Algorithms to compute variance estimators

We provide all the supporting algorithms in this section. The matrix-type data structures will be denoted by bold abbreviations, such as IPS, GENE and SH. They stand respectively for Interacting Particle System, genealogy and survival history. For any set , we denote M × ( ) the collection of all the × matrices with elements taking values in . For example, the notation SH[ , ] stands for the element at -th row and -th column of the matrix SH. In particular, since the state space may vary w.r.t. time horizon, IPS is not necessarily a matrix, however, we still use the notation IPS[ , ] to denote the -th particle at level of IPS.

Algorithm 1: Simulation of an IPS with genealogy and survival history.

Require: particle number , time horizon , potentials ( ; 0 ≤ ≤ -1), Markov kernels ( ˚ ; 1 ≤ ≤ ) and ( Above, we also provide Algorithm 7 to compute unbiased non-asymptotic variance estimator for ( ). A tremendous amount of e ort has been spent in order to nd an O( ) time complexity algorithm, which, unfortunately, does not pay back. The Algorithm 7 is of O( 2 ) time complexity, which means that even with a very little , the computation will be intractable when > 10 6 . In fact, the construction for the consistent asymptotic variance estimators, i.e., Algorithm 5 and Algorithm 6, of O( ) time complexity is highly nontrivial, and we failed to apply the same technique to reduce the time complexity for the unbiased non-asymptotic variance estimator. Still, this estimator may be useful for rare-event simulation problems or other applications whose target measure is . One can thus take advantage of the parallel computing for relatively small . Though, the crude estimator is a by-product in this case, the average of this estimator may still be typically more accurate and the statistical inference for the variance of the variance estimator is also available if an unbiased variance estimator is provided. Note that the lack-of-bias is not free in general, and we provide a relatively general condition in Section 3.4.1. In fact, even without unbiased condition, this estimator multiplied by is also a consistent asymptotic variance estimator for ( ). Another remark is that V ‡ found in Algorithm 5 and Algorithm 6 represent respectively the estimators ( ) (1) 2 and ( -( )) provided in [START_REF] Lee | Variance estimation in the particle lter[END_REF]. Due to the change of resampling scheme, some modi cations, namely, V † have to be taken into consideration. Now, we provide the time complexity and space complexity of the algorithms in the SMC context. The multinomial resampling scheme will be set as benchmark, with the variance estimators provided in [START_REF] Lee | Variance estimation in the particle lter[END_REF].

• ; 1 ≤ ≤ ), initial distribution 0 . Result: absorbing time T ∈ {0, 1, . . . , }, particle system IPS of size ( + 1) × , genealogy GENE ∈ M × ([ ]), survival history SH ∈ M × ({0, 1}), ancestor indices EVE ∈ M ( +1)× ( [ ]).

Estimation

Time complexity Space complexity

( ) or ( ) O( ) O( ) non-asymptotic variance of ( ) O( ) O( ) asymptotic variance of ( ) O( ) O( ) asymptotic variance of ( ) O( ) O( )
( ) or ( ) O( ) O( ) non-asymptotic variance of ( ) O( 2 ) O( ) asymptotic variance of ( ) O( ) O( ) asymptotic variance of ( ) O( ) O( )
Table 3.2: Time and space complexity under asymmetric resampling scheme.

We remark that we did not provide the algorithm to compute ( ) and ( ) with O( ) space complexity, which is readily obtained with some modi cation of Algorithm 1, since we mainly focus on the variance estimation problems. We can see from Table 3.1 and Table 3.2 that the main drawback of the presented setting is the space complexity and the unbiased variance estimator for the non-asymptotic variance of ( ). However, the main computational consumption, in general, is brought by the resampling kernels ˚ at each iteration of the algorithm. If a cheap kernel • is available, it is expected that the asymmetric resampling scheme would dramatically reduce the computational cost of the simulation of IPS. In fact, in real-world applications, the computational consumption of the asymptotic variance estimators is negligible compared to the simulation of IPS. In addition, the asymmetric setting also gives smaller variance in some speci c situations, such as AMS methods.

Coalescent tree-based expansions

As we have seen in Theorem 3.2.2, the asymptotic variance becomes sophisticated when the asymmetric resampling is implemented. Therefore, we need to develop a novel mathematical language in order to conduct calculations and eventually, to understand the structures behind. In this section, we give a detailed development of the coalescent tree-based expansions encountered in the asymmetric SMC framework. Before going further, let us list some de nitions and properties associated to McKean-type Feynman-Kac kernel , within reach by some straightforward algebraic calculations in the following proposition. For the sake of simpli cation, these properties will be of constant use in the following sections and may be applied without reference.

Proposition 3.6.1. For any probability measure ∈ P ( -1 ) and test function ∈ B ( ), we have the following properties:

(i) sup ∈ -1 , ( ) ( ) ≤ 2 ∞ .
(ii) We de ne , by

∀( , ) ∈ -1 × B ( ), , (
, ) := ( -1 ) • ( , ) --1 ( ) ˚ ( ), (3.15)
with the convention ∀ ∈ 0 , 0, ( , ) := 0 ( ).

(3.16)

Then, we have , ( ) ( ) = ˚ ( ) + , ( ) ( ), and

-1 , -1 ( ) = 0. (iii) We have , ( ) ( ) 2 = ˚ ( ) 2 + , ( ) ( ) 2 + 2 ˚ ( ) , ( ) ( ), as well as -1 , -1 ( ) 2 = -1 ˚ ( ) 2 + -1 , -1 ( ) 2 .
(3.17)

Original coalescent tree-based measures

First, let us recall the original coalescent tree-based expansion introduced in [START_REF] Cérou | A nonasymptotic theorem for unnormalized Feynman-Kac particle models[END_REF]. The following de nition is adopted from the De nition 3.1 of [DG19](Chapter 2), which is essentially the same as the one introduced in [START_REF] Cérou | A nonasymptotic theorem for unnormalized Feynman-Kac particle models[END_REF]. A more general version for the particle block of size greater than 2 can be found in [START_REF] Del Moral | Tree based functional expansions for Feynman-Kac particle models[END_REF].

De nition 3.6.1. For any ≥ , we associate with any coalescence indicator ∈ {0, 1} +1 the nonnegative measures Γ ∈ M + ( 2 ) de ned for any ∈ B ( 2 ) by

Γ ( ) := ⊗2 0 0 ˚ ⊗2 1 1 • • • ˚ ⊗2 ( ).
When there is only one coalescence at, say, level , we write Γ ( ) ( ) instead of Γ ( ) (see Figure 3.2). When there is no coalescence at all, that is = ( ), we have

Γ ( ) ( ) = ⊗2 ( ). . . . . . . . . . . . . ˚ 1 ˚ -1 ˚ ˚ +1 ˚ +2 ˚ +3 ˚ ˚ 1 ˚ -1 ˚ ˚ +1 ˚ +2 ˚ +3 ˚ Figure 3.2: A representation of the original coalescent tree-based measure Γ ( ) .
Comparison of asymptotic variance. Now, we suppose that

• ≡ ˚ for all ≥ 1. Let us go back to the form of the asymptotic variance 2 ( ) de ned in (3.6). It is easy to verify that

Γ ( ) ( ⊗2 ) = (1) , ( ) 2 . (3.18)
By applying (3.17) with some standard algebraic manipulations, we have

2 ( ) = =0 Γ ( ) ( ⊗2 ) -Γ ( ) ( ⊗2 ) - =1 -1 (1) -1 , -1 , ( ) 2 . (3.19)
One may notice that the rst term corresponds to the asymptotic variance of the multinomial resampling scheme (see., e.g Theorem 2.1 of [DG19](Chapter 2)). Since the term

-1 (1) -1 , -1 , ( ) 2 (3.20)
is nonnegative, we deduce that the choice • ≡ ˚ is always better than the multinomial resampling scheme in terms of asymptotic variance. Moreover, we notice that the original coalescent tree-based measures introduced in [CDMG11] and [DG19](Chapter 2) failed to provide a full description of the asymptotic variance, even in this simple symmetric case. This is the main difculty compared to the multinomial resampling scheme, where the alternative representation is free. Therefore, we need to develop some new tools to understand the term given in (3.20).

Coalescent Feynman-Kac kernels

As we have seen in the last section, the original coalescent tree-based measures fail to provide insights on the asymptotic variance 2 ( ). In order to go one step further, let us go back to De nition 3.6.1. We consider the following alternative writing

Γ ( ) := ⊗2 0 0 ⊗2 1 Q 0 1 1 ⊗2 2 Q 1 2 • • • -1 ⊗2 Q -1 ( ),
which gives a similar de nition as based on the partial semigroup structure of the Feynman-Kac kernels:

Γ ( ) := ⊗2 0 • Q 0 1 • Q 1 2 • • • Q -1 ( ).
We say Q -1 conserves the structure of coalescence if

∀ -1 ∈ {0, 1}, -1 Q -1 ≡ Q -1 .
This simple observation gives an interesting idea on how we could possibly overcome the difculties encountered in the asymptotic variance representation: we change the construction of the partial semigroup according to our asymmetric resampling scheme, in order to establish the coalescent tree-based expansion of the asymptotic variance. The rst pair of replacement is for Q 0 and Q 1 . We de ne

       Q 0 := ˚ + ⊗2 -1 (1 ⊗ -1 ) ( • -˚ ) ⊗2 ; Q 1 := 1 Q 0 . (3.21)
This replacement is compatible with the notation above if

• ≡ ˚ . Note that ⊗2 -1 (1 ⊗ -1 ) = -1 ( -1 ).
Next, we need to introduce several important coalescent Feynman-Kac kernels, which, at the moment, is not as intuitive as the one introduced above. Their introduction is motivated by an observation in the proof of a technical result (cf. Proposition 3.7.9):

(i) Q †,0 := Q 0 ; Q †,1 := Q 1 -⊗2 -1 ( ⊗2 -1 ) 1 • ⊗2 . (ii)            Q †,0 := Q 0 ; Q †,1 := ⊗2 -1 (1 ⊗ -1 ) ( -1 × • ) ⊗ ˚ + ˚ ⊗ ( -1 × • ) + ⊗2 -1 ( 1 ⊗2 -1 ) • ⊗2 - • ⊗ ˚ -˚ ⊗ • . (iii) Q †,0 := Q 0 ; Q †,1 := 1 Q †,1 -⊗2 -1 ( 1 ⊗2 -1 ) 1 • ⊗2 . (iv) ∀ -1 ∈ {0, 1}, Q †, -1 ,( ) := 1 -1 Q †, -1 . (v) ∀ -1 ∈ {0, 1}, Q †, -1 ,( ) := 1 -1 Q †, -1 . (vi) Q ‡,0 ,( ) := Q †,0 + Q †,1 ,( ) ; Q ‡,1 ,( ) := Q †,1 + Q †,1 ,( ) .
It is readily checked that they are all uniformly nite transition kernels and that, except the kernels with " ", namely,

Q †, -1 and Q †, -1
, all of the other kernels conserve the coalescence structure. This observation may be the intrinsic reason why they play a particularly important role in variance related problems. No matter how anecdotal it seems, we claim that these kernels are at the core of the analysis of the variance related problems. Although we are not able to clarify the exact purpose of the construction of these coalescent Feynman-Kac kernels at the moment, we can explain, however, the logic of our notation: the number of daggers " †" indicates the number of kernels between -1 = 0 and -1 = 1, that is changed from the original de nition (3.21). At the same time, the kernel for -1 = 1 is always replaced before the kernel for -1 = 0. This is why all the kernels that have only one dagger share the same Q 0 for the case -1 = 0. Since the number of particles is also involved in the de nition, we add parenthesis "( )" to specify the number of particles , in order to di erentiate from the coalescent tree occupation measures. With a slight abuse of notation, when there is no ambiguity, we omit the part "( )" for simplicity. For example, we may use Q ‡, to denote Q ‡, ,( ) . All the kernels de ned from point (i) to point (v) are introduced to describe the binary decomposition w.r.t. "+" in the de nition of point (vi). We say that the kernels de ned above are in the class Q (2) , or the kernels are of Q (2) -class.

Next, we de ne the generalized coalescent tree-based measures. In this chapter, they will be referred to as the coalescent Feynman-Kac measures, by using the partial semigroup properties of these coalescent Feynman-Kac kernels. For example, we denote

Q , := Q +1, Q +1 +2, • • • Q -1 , ,
with the convention Q -1 , := ⊗2 , . De nition 3.6.2. For any ≥ 1, ≥ 2 and for any coalescence indicator ∈ {0, 1} +1 , we de ne the signed nite measures Γ ‡, ,( ) by

∀ ∈ B ( 2 ), Γ ‡, ,( ) ( ) := ⊗2 0 Q ‡, 0 1,( ) Q ‡, 1 2,( ) • • • Q ‡, -1 ,( ) ( ),
with the convention

Γ ‡, 0,( ) ( ) = ⊗2 0 0 .
Similar as in De nition 3.6.1, when there is only one coalescence at level , we write Γ ‡,( ) ,( ) ( ) instead. When there is no coalescence , we denote Γ ‡,( ) ,( ) ( ).

Meanwhile, we de ne the Q (2) ˆ -class kernels by replacing all the ⊗2 -1 in the de nition above with the empirical sub-probability measure

( -1 ) 2 1 ≥ -1 := 2 (X n-1 )1 ≥ -1 .
In regard to the notation, all the " " in the de nition will be replaced by " ˆ " correspondingly:

(i)        Q 0 ˆ := ˚ + ( -1 ) 2 (1 ⊗ -1 )1 ≥ -1 ( • -˚ ) ⊗2 ; Q 1 ˆ := 1 Q 0 ˆ . (ii) Q †,0 ˆ := Q 0 ˆ ; Q †,1 ˆ := Q 1 ˆ -( -1 ) 2 ( ⊗2 -1 )1 ≥ -1 1 • ⊗2 . (iii)            Q †,0 ˆ := Q 0 ˆ ; Q †,1 ˆ := ( -1 ) 2 (1 ⊗ -1 )1 ≥ -1 ( -1 × • ) ⊗ ˚ + ˚ ⊗ ( -1 × • ) +( -1 ) 2 ( 1 ⊗2 -1 )1 ≥ -1 • ⊗2 - • ⊗ ˚ -˚ ⊗ • . (iv) Q †,0 ˆ := Q 0 ˆ ; Q †,1 ˆ := 1 Q †,1 ˆ -( -1 ) 2 ( 1 ⊗2 -1 )1 ≥ -1 1 • ⊗2 . (v) ∀ -1 ∈ {0, 1}, Q †, -1 ˆ ,( ) := 1 -1 Q †, -1 ˆ . (vi) ∀ -1 ∈ {0, 1}, Q †, -1 ˆ ,( ) := 1 -1 Q †, -1 ˆ . (vii) Q ‡,0 ˆ ,( ) := Q †,0 ˆ + Q †,1 ˆ ,( ) ; Q ‡,1 ˆ ,( ) := Q †,1 ˆ + Q †,1 ˆ ,( ) .
We remark that the kernels of Q (2) ˆ -class will not be used to de ne the coalescent tree-based measures, the introduction is purely for technical reasons (cf. Proposition 3.7.9, Proposition 3.7.10, Lemma 3.7.10 and Lemma 3.7.11). They are eventually proved to be very "close" to the Q -class kernels (cf. Lemma 3.7.12) by the propagation of chaos property of the IPS (cf. Proposition 3.7.5).

Binary decompositions

Next, we de ne some auxiliary coalescent Feynman-Kac kernels using the same idea. Before that, we need some new notation to describe the coalescence structure that is a little bit more complicated than the basic binary structure illustrated in Figure 3.2. For one coalescence indicator , we use | | to denote the number of 1 in , namely

| | := =0 .
Using the same de nition as above, for two coalescence indicators and in {0, 1} +1 , the notation | -| denotes the number of di erent elements between and . More precisely,

| -| = =0 - = # ∈ {0, 1, . . . , } : ≠ .
In particular, when | | = 0, we denote ( ) := (0, . . . , 0). For two coalescence indicator and in {0, 1} +1 , we say ≤ if for any 0 ≤ ≤ , we have ≤ . More over, if ≤ , and ≠ , we say < . We also consider the set of coalescence indicators S ( ), S > ( ) and S ( ) de ned as follows:

• S ( ) := ∈ {0, 1} +1 = ;

• S > ( ) := ∈ {0, 1} +1 > and = ;

• S ( ) := S ( )\{ }.

Let us consider the following auxiliary coalescent Feynman-Kac kernels:

               Q 0|0 := Q †,0 Q 1|0 := Q †,1 Q 0|1 := Q †,1 Q 1|1 := Q †,1
and

               Q 0|0 ,( ) := Q †,0 Q 1|0 ,( ) := Q †,1 ,( ) Q 0|1 ,( ) := Q †,1 ,( ) Q 1|1 ,( ) := Q †,1 (3.22)
We remark that these kernels are constructed in respect of the decomposition of the partial group structure w.r.t. the composition "+", namely, at level , we have

0 : Q ‡,0 ,( )
0|0:Q †,0 + Q †,1 ,( ) :1|0 and 1 : Q ‡,1 ,( ) , 1|1:Q †,1 + Q †,1 ,( ) :0|1 (3.23)
whence the four cases which correspond to the four possible choices when the partial semigroup structure is passing through the coalescent Feynman-Kac kernel Q ‡, -1 ,( ) . With this in mind, we de ne some auxiliary coalescent tree-based measures that are useful for the decomposition mentioned above.

De nition 3.6.3. For any ≥ ≥ 1, ≥ 2 and for any coalescence indicators , ∈ {0, 1} +1 , we de ne the signed nite measures Γ | and Γ | , respectively by

∀ ∈ B ( 2 ), Γ | ( ) := ⊗2 0 Q 0 | 0 1 Q 1 | 1 2 • • • Q -1 | -1 ( ),
and

∀ ∈ B ( 2 ), Γ | ,( ) ( ) := ⊗2 0 Q 0 | 0 1,( ) Q 1 | 1 2,( ) • • • Q -1 | -1 ,( ) ( ),
with the convention

Γ | 0 ( ) = Γ | 0,( ) ( ) := ⊗2 0 0 .
We nally de ne all the coalescent tree-based measures as a generalization of the work in [START_REF] Cérou | A nonasymptotic theorem for unnormalized Feynman-Kac particle models[END_REF]. The following proposition is a direct consequence of the introduction of

Q †, -1 ,( ) = 1 -1 Q †, -1 and Q †, -1 ,( ) = 1 -1 Q †, -1 .
Proposition 3.6.2. For any ≥ ≥ 1, ≥ 2, for any coalescence indicators , ∈ {0, 1} +1 and for any test function ∈ B ( 2 ), we have the following equalities:

(i) Γ †, ,( ) ( ) = 1 -1 | | Γ †, ( ); (ii) ∀ ∈ S > ( ), Γ | ,( ) ( ) = 1 -1 | |-| | Γ | ( ); (iii) Γ | ,( ) ( ) = 1 -1 | -| Γ | ( ).
Since we have all the necessary ingredients at hand, we provide the most important result of this section. For all test function ∈ B ( 2 ), we have

Γ ‡, ,( ) ( ) = ∈S ( ) Γ | ,( ) ( ) =Γ | ,( ) ( ) + ∈ S ( ) Γ | ,( ) ( ) =Γ | ( ) + ∈ S ( ) 1 -1 | -| Γ | ( ) =Γ †, ( ) + ∈ S ( ) 1 -1 | -| Γ | ( ).
(3.24)

In particular, for the case where = ( ), we have

Γ ‡,( ) ,( ) ( ) = ∈S ( ) Γ | ,( ) ( ) =Γ ( ) | ( ) ,( ) ( ) + ∈ S ( ( )) Γ | ( ) ,( ) ( ) =Γ †,( ) ( ) + ∈S > ( ( )) 1 -1 | | Γ †, ( ) =Γ ( ) ( ) + 1 -1 -1 =0 Γ †,( ) ( ) + ∈S > ( ), | | ≥2 1 -1 | | Γ †, ( ).
(3.25)

Taking into account that all the coalescent tree-based meaures are nite signed measures, the calculations above give the following proposition.

Proposition 3.6.3. For any ≥ ≥ 1 and for any coalescent indicator ∈ {0, 1} +1 , we have

∀ ∈ B ( 2 ), Γ ‡, ,( ) ( ) -Γ †, ( ) = O 1 .
In particular, we have

∀ ∈ B ( 2 ), Γ ‡,( ) ,( ) ( ) -Γ ( ) ( ) - 1 -1 -1 =0 Γ †,( ) ( ) = O 1 2 .
Remark. Above lies part of the reason why we have inhomogeneity in the notation w.r.t. " ‡" and " †" in De nition 3.3.2. In fact, the strategy to prove the consistency given in Theorem 3.6.3 is divided into two steps, and the latter is done by Proposition 3.6.3 above:

• Γ ‡, , ( )1 ≥ -Γ ‡, ,( ) ( ) = O L 1 1 √ ; • Γ ‡, ,( ) ( ) -Γ †, ( ) = O 1 √ .

Coalescent tree occupation measures

In this section, we introduce the particle approximations of the coalescent tree-based measures discussed in the last section. To do this, we need to adopt some notation from [DG19](Chapter 2). The readers are also referred to Appendix A.2 of [DG19](Chapter 2) to nd more intuitions and the connection between the construction of these particle approximations and the Particle Markov Chain Monte Carlo methods. Due to the fact that the underlying resampling scheme is changed, the analysis also becomes more challenging. Fortunately, the basic idea remains the same: we exploit the information encoded in the genealogy, and in addition, the information encoded in the survival history, to approximate the coalescent Feynman-Kac measures. The key idea is to collect all the corresponding coalescent tree-type forms illustrated in Figure 3.2, and the coalescent tree occupation measures are constructed as weighted empirical terminal measures of these particle blocks. The intuition of this procedure remains identical to the previous work in [DG19](Chapter 2). More precisely, the major di erence is about the weights mentioned above, which correspond to the potential function of the original IPS in many-body Feynman-Kac models (cf. G ( ) (x ) de ned in A.1 of [DG19](Section 2.5.1)). Under asymmetric resampling scheme, it is necessary to consider the in uence of the survival history. Thus, it is expected that the constructions become more sophisticated. Another remark is on the measure Γ †, : since the related coalescent Feynman-Kac kernels do not conserve the coalescence structure, its particle approximation also turns out to be a little bit di erent. Recall that ( ˜ [2] , [2] ) ∈ {0, 1} is an indicator function de ned by

( ˜ [2]
, [2] )

:= 1 { =0} 1 { ˜ 1 = 1 ≠ ˜ 2 = 2 } + 1 { =1} 1 { ˜ 1 = 1 = ˜ 2 ≠ 2 } .
De nition 3.6.4 (Coalescent tree occupation measures). For any ≥ ≥ 0 and for any coalescence indicator ∈ {0, 1} +1 , the random measure Γ , is de ned by

∀ ∈ B ( 2 ), Γ , ( ) := -1 ( -1) +1 [2] 0: ∈ ( ( ) 2 ) ×( +1) -1 =0 ( [2] +1 , [2] ) ( ) ( [ 2]
).

The next theorem is brought from Proposition 4.2 of [DG19](Chapter 2), and the proof is a slightly modi ed version of the one given in Section 4.5 of [DG19] (Section 2.4.5). It is one of the most important ingredients that connect the coalescent tree occupation measures and nonasymptotic variance of Feynman-Kac IPS. It provides information on the combinatorial structure of the IPS in regard to the coalescent tree occupation measures Γ , , which does not depend on the resampling scheme and regularity assumptions whilst the IPS is well-de ned. Namely, it reveals the essential combinatorial properties that apply to all genealogy tree-based particle systems when each particle has only one parent. This combinatorial property, in particular, is also valid in continuous-time settings and/or in the frameworks with even more complex resampling schemes. In this chapter, it is the bridge between the asymptotic variance, non-asymptotic variance, and eventually, the construction of our variance estimators. The proof is given in Section 3.7.6. Theorem 3.6.1. For any test function ∈ B ( 2 ), we have the following decompositions.

( ) ⊗2 ( )1 ≥ = ∈ {0,1} +1 -1 +1-| | 1 | | Γ , ( )1 ≥ , . . and
( ) ⊗2 ( )1 ≥ = ∈ {0,1} +1 -1 +1-| | 1 | | Γ ‡, , ( )1 ≥ .
. .

Below we list the most important lack-of-bias and convergence results of the coalecent tree occupation measures, serving as the particle approximations of the coalescent Feynman-Kac measures. The proofs are divided into several technical results: they are direct consequences of the combination of Lemma 3.7.1, Lemma 3.7.2, Proposition 3.7.2, Proposition 3.7.3 and Proposition 3.6.3, with some standard manipulations of bounded i.i.d. random variables for the case = 0, which is provided in Lemma 3.7.4. The only remark is that for any coalescent indicator and for any coalescent Feynman-Kac kernel, e.g., Q †, , we have, by de nition,

∀ , ∈ B ( ), ∃ , ∈ B ( -1 ) . . Q †, ( ⊗ ) ≡ ⊗ .
The property above also holds for Q ‡, .

Feynman-Kac measures ow in a random environment.

In this section, we gives some intuition on the construction of the coalescent tree occupation measures from another point of view. One of the main message of [DG19](Chapter 2) is to provide some intuition on the construction of the coalescent tree occupation measures using the manybody Feynman-Kac models introduced in [START_REF] Del Moral | On particle Gibbs samplers[END_REF], by considering a Gibbs sampler w.r.t. the original IPS and coupled particle block on a sophisticated path space. Then, we de ne the event that traps the desired coalescent particle block and eventually, we construct the estimator given in De nition 3.6.4. This methodology gives the foundation of the present work: De nition 3.3.3 is also obtained by this procedure, though it is not discussed in detail as in [START_REF] Du | Variance estimation in adaptive sequential monte carlo[END_REF](Chapter 2). Now, let us look at this family of random measures from a di erent angle. We begin with some basic observations. To facilitate the writings, let us x a time horizon ∈ N * . The following discussion is valid on the event { ≥ } and 0 ≤ ≤ . Given W (cf. Section 3.7.1) and xing

[2] -1 ∈ ( ) 2 , we have ∀ ∈ {0, 1} +1 , ∀ ∈ [ ], [2] ∈ ( ) 2 -1 ( [2] -1 , [2] -1 ) = 1.
Therefore, let us consider the state space ( ) 2 , the matrix of size ( -1) × ( -1), with some pre xed ordering rule on the set ( ) 2 , is denoted by

-1 ( [2] -1 , [2] -1 ) [2] -1 , [2] ∈ ( ( ) 2 ) ×2 , (3.26)
which can then be regarded as a random transition matrix, with

-1 ( [2] -1 , [2] -1 )
denoting the probability of transition from the site [2] -1 to the site [2] . For the general theory regarding to the Markov chain in a random environment, the readers are referred to [START_REF] Cogburn | Markov chains in random environments: The case of markovian environments[END_REF]. Returning to the de nition of coalescent tree occupation measures Γ ‡, , , we can nd a similar semigroup structure: the initial distribution on the state space ( ) 2 is 2 ( [ ]) and the potential function on the site [2] -1 is G ‡ n (X n-1 ), which is a constant function given W . Therefore, by denoting

H ‡, [ [2] -1 , [2] ] := G ‡, n (X n-1 ) -1 ( [2]
-1 , [2] -1 ), and by respecting the composition law of the matrix multiplication, we de ne

H ‡, • H ‡, +1 [ [2] -1 , [2] +1 ] := [2] ∈ ( ) 2 H ‡, [ [2] -1 , [2] ] × H ‡, +1 [ [2] , [2] +1 ].
In addition, for any random measure Λ -1 on the state space ( ) 2 , we de ne

Λ -1 • H ‡, [ [2] ] := [2] ∈ ( ) 2 Λ -1 [ [2] -1 ] × H ‡, [ [2] -1 , [2] ],
and for any random test function F on the state space ( ) 2 , we de ne

H ‡, (F) [ [2] -1 ] := [2] ∈ ( ) 2 H ‡, [ [2] -1 , [2] ] × F[ [2] ].
In particular, we denote

Λ (F) := [2] ∈ ( ) 2 Λ [ [2] ] × F[ [2] ].
Obviously, these composition law does not depend on the pre xed ordering rule on the set ( ) 2 since the sum "+" of the random variables is commutative. Now, we are able to give an alternative representation of Γ ‡, , . We de ne

Λ ‡, [ [2] ] := 2 ( [ ]) • H ‡, 1 • H ‡, 2 • • • H ‡, [ [2] ], (3.27)
with the convention Λ ‡, 0 := 2 ( [ ]). Accordingly, the random test function F is de ned by

F [ [2] ] := ( ) ( [ 2]
).

Consequently, we have

Λ ‡, (F ) = Γ ‡, , ( ).
(3.28)

The random measure notation above is frequently used in the proof of technical results, an explicit form can be found later in (3.56). Similarly, we denote

H †, [ [2] -1 , [2] ] := G †, n-1 ( [2] -1: , B n-1 , X n-1 ) -1 ( [2] -1 , [2] -1 ), and Λ †, [ [2] ] := 2 ( [ ]) • H †, 1 • H †, 2 • • • H †, [ [2] ], with the convention Λ †, 0 := 2 ( [ ]).
Apart from the fact that this writing guided and simpli ed some of the proofs of the technical results, the main motivation is to provide a decomposition result similart to the one in (3.25), which is essentially due to the partial R-algebra homomorphism. We de ne H 0 := H †,0 . By de nition, we have

H 0 [ [2] -1 , [2] ] + 1 -1 H †,1 [ [2] -1 , [2] ] = H ‡, [ [2] -1 , [2] ],
which yields, for the associated random matrix,

H 0 + 1 -1 H †,1 = H ‡, .
Therefore, we have the following decomposition:

Λ ‡,( ) (F n ) =Λ ( ) (F n ) + ∈S > ( ( )) 1 -1 | | Λ †, (F n ) =Λ ( ) (F n ) + 1 -1 -1 =0 Λ †,( ) (F n ) + ∈S > ( ), | | ≥2 1 -1 | | Λ †, (F n ),
which is equivalent to

Γ ‡,( ) , ( ) =Γ ( ) , ( ) + ∈S > ( ( )) 1 -1 | | Γ †, , ( ) =Γ ( ) , ( ) + 1 -1 -1 =0 Γ †,( ) , ( ) + ∈S > ( ), | | ≥2 1 -1 | | Γ †, , ( ).
(3.29)

Thanks to Proposition 3.7.8 and the decomposition (3.29) above, we have the following proposition.

Proposition 3.6.4. For any ≥ ≥ 1 and for any coalescent indicator ∈ {0, 1} +1 , we have

∀ ∈ B ( 2 ), Γ ‡,( ) , ( ) -Γ ( ) , ( ) - 1 -1 -1 =0 Γ †,( ) , ( ) 1 ≥ = O L 2 1 2 .
3.6.6 E cient estimator of Γ †,( )

As is mentioned before, we failed to provide an O( ) time complexity algorithm to compute the term by term variance estimator and the non-asymptotic variance estimator provided in the previous sections. Therefore, we give a new asymptotic variance estimators that can be computed with O( ) time complexity. The idea is to construct some new coalescent tree occupation measures that are very "close" to Γ †,( ) ,

, which is easier to obtain by some numerical techniques to reduce the computational costs. First, let us de ne a new sequence of random matrix on the event { ≥ }. For any > 1, we consider

H ‡,0 := H ‡,0 = H †,0 + 1 -1 H †,1 ; H ‡,1 := H †,1
.

Next, using the same semigroup property of these random matrix and random test function as in (3.27) and (3.28), by consider the initial distribution 2 ( [ ]), we de ne the new coalescent tree occupation measure Γ ‡, , for each ∈ {0, 1} +1 by

∀ ∈ B ( 2 ), Γ ‡, , ( ) := Λ ‡, (F ) = 2 ( [ ]) • H ‡, 1 • H ‡, 2 • • • H ‡, (F ), with F [ [2] ] := ( [ 2]
).

The theoretical property of the approximation discussed above is provided in the following proposition. The proof is housed in Section 3.7.7.

Proposition 3.6.5. For any test function ∈ B ( 2 ) and for any coalescence indicator , we have

Γ ‡, , ( ) -Γ †, , ( ) 1 ≥ = O L 2 1 .
In particular, for any ∈ {0, 1, . . . , -1}, we have

Γ ‡,( ) , ( ) -Γ †,( ) ,
( ) 1 ≥ = O L 2 1 .
Finally, without loss of generality, we explain why the estimator given in (3.14) can be computed with O( ) time complexity. In fact, the rst part of the estimator

=0 Γ ‡,( ) , ( ⊗2 ) -Γ ‡,( ) , ( ⊗2 )
can indeed be approximated by the variance estimator ( ) (1) 2 proposed by Lee & Whiteley [START_REF] Lee | Variance estimation in the particle lter[END_REF]. Hence, an O( ) algorithm is therefore available. It is then su cient to provide an O( ) algorithm to compute Γ ‡,( ) , ( ⊗2 ). This is possible due to the homogeneity of the potential function G ‡ n (X n ) w.r.t. the di erent indices [2] and the following technical lemma.

Lemma 3.6.1. Let (R, +, ★) be a ring, and (E ) ∈ [ ] be a disjoint partition of [ ] for some ≥ 1, then for any sequence ( ) ∈ [ ] composed by elements of R, we have the following equality:

∈ E , ∈ E 1≤ ≠ ≤ ★ = =1 ★ =1 - =1 ∈ E ★ .
In our case, the partition (E ) ∈ [ ] is the divided by the ancestor indices of the particles of level . The ring R is R 3 and the "★" product refers to the operation

R 3 × R 3 ( , , ) ★ ( , , ) ↦ → ( 1 × , × , × ) ∈ R 3 ,
which represents an intermediate step in the Algorithm 4, whose nal output is ( , , ), ( , ,

) ★ := 1 × + × + × .
This is useful in calculating the term Λ ‡,( ) [ [2] ]F ( [2] ).

In fact, for any test function ∈ B ( ) ⊗2 , the term above can be a.s. reformulated as

∃( ) ∈ [ ] ∈ R 3 , . . ∀ [2] ∈ ( ) 2 , Λ ‡,( ) [ [2] ]F ( [2] ) = 1 , 2 ★ .
Hence, by applying Lemma 3.6.1, one can therefore compute Γ ‡,( ) , ( ⊗2 ) with O( ) time complexity. The details can be found in Algorithm 5, and the design of the Algorithm 6 is similar. We remark that we are not able to construct the ring homomorphism discussed above for the measure Γ †, , . This is the intrinsic reason why we failed to apply the same technique to reduce the time complexity of the non-asymptotic variance estimator ( ). In the Algorithm 7 of Section 3.5, the corresponding term 2 ( [ ])H ( ) 0, [ [2] ]F ( [2] ) is therefore calculated by violently searching all the possible pairs of [2] . This is why the computation is of time complexity O( 2 ).

Proofs

In this section, we list all the proofs in the present work. Some notation are gathered in Section 3.7.1, such as the formal de nitions of the ltrations frequently used in the proofs, along with the most important martingale decompositions. A little plan on the organization of the technical results is also provided. In order to facilitate the writing, the stochastic bounds introduced in Thanks to Cauchy-Schwarz inequality and Markov's inequality, we have

= O . . ( ) =⇒ = O L 2 ( ) =⇒ = O L 1 ( ) =⇒ = O p ( ).
We also remark that for all these 4 types of stochastic bounds, they are weaker than the corresponding convergence. For example, if

/ L 1 /P -----→ →∞ Const. < +∞, one also has = O L 1 /p ( ).

Martingales

We present some important martingales encountered in the analysis of SMC framework. They are crucial to some of the technical results in this chapter. We also hope that the similar construction may inspire the future work in di erent context. Before proceeding further, let us de ne some ltrations associated to the Feynman-Kac IPS.

Filtrations. (F ) ≥0 denotes the ltration that consists the information of the values of particles. More precisely,

F -1 := { , Ω} and ∀ ≥ 0, F := (X 0 , . . . , X n ).
If we only add one particle at each step, a more re ned ltration (E ) ≥0 can be de ned by

∀ ∈ [( + 1) ], E = F -1 ∨ ( 1 , • • • , ),
where for any ∈ [( + 1) ], we adopt the notaition := and := -× .

Next, (G ) ≥0 denotes the ltration that contains the genealogy of IPS, which is de ned by

∀ ∈ {-1, 0}, G := F and ∀ ≥ 1, G := F ∨ (A 0 , . . . , A n-1 ).
Finally, the ltration that contains all the information including survival history of the particle system are denoted by (W ) ≥0 , namely,

∀ ∈ {-1, 0}, W := F and ∀ ≥ 1, W := G ∨ (B 0 , . . . , B n-1 ).
Moreover, as is used several times in some technical results, we also consider an updatated ltration (W ) ≥0 de ned by

W := W ∨ (B ).
taking into account the convention

-1 = 0 = 0 and 0, -1 ( 0, ) ( ) ≡ 0 ( 0, ) = ( ).
Note that, for the case • ≡ ˚ , we have almost surely -1, ( ) ≡ 0. In this case, , ( )

0≤ ≤ is a (F ; 0 ≤ ≤ )-martingale.
Now, to facilitate the writing, we x a nite time horizon ∈ N * , and a test function ∈ B ( ). As a natural extension, we discuss a similar family of bias-martingales decomposition brought by the partial semigroup structure of coalescent Feynman-Kac kernels. Let us consider the term de ned as follows:

X ‡, ( )

:= Γ ‡, , Q ‡, , ( )1 ≥
. First, the integrability is guaranteed by Proposition 3.7.7. Then, thanks to Proposition 3.7.9, we get a almost sure equality which is very "close" to a martingale structure:

E Γ ‡, Q ‡, , ( )1 ≥ G -1 = Γ ‡, -1, Q ‡, -1 ˆ Q ‡, , ( )1 ≥ -1 .
Note that, since Q ‡, +1 conserves the coalescence structure, the term disappears. We denote

♯ D ‡, ( ) := X ‡, ( ) -E X ‡, ( ) G -1 ,
as well as ♭ D ‡, ( ) := E X ‡, ( ) G -1 -X ‡, -1 ( ). Thanks to Lemma 3.7.10, Lemma 3.7.12, the Minkowski's inequality and conservation of coalescence structure, we deduce the following Lemma 3.7.1. Then, the Proposition 3.7.2 is a direct application of Doob decomposition theorem.

Lemma 3.7.1. For any test function ∈ B ( 2 ) and any coalescence indicator ∈ {0, 1} +1 , we have

=1 ♯ D ‡, ( ) = O L 1 1 √ , and
=1 ♭ D ‡, ( ) = O L 1 1 √ .
Proposition 3.7.2. For any test function ∈ B ( 2 ) and any coalescence indicator ∈ {0, 1} +1 , the integrable process (X ‡, ( ); 0 ≤ ≤ ) can be decomposed to a (G ; 0 ≤ ≤ )-martingale (M ‡, ( ); 0 ≤ ≤ ) , and a integrable predictable process (A ‡, ( ); 0 ≤ ≤ ), respectively de ned by

M ‡, ( ) := X ‡, 0 ( ) + =1 ♯ D ‡, ( ), and
A ‡, ( ) := =1 ♭ D ‡, ( ).
Similarly, we also discuss the martingale decomposition associated to the measure Γ †, , . We de ne X †, ( )

:= Γ †, , Q †, , ( )1 ≥ , as well as ♯ D †, ( ) := X †, ( ) -E X †, ( ) W -1 , and ♭ D †, ( ) := E X †, ( ) W -1 -X †, -1 ( ).
Thanks to Proposition 3.7.10, Lemma 3.7.11 and Lemma 3.7.12, we have the following results.

The unbiasedness given in Proposition 3.7.3 is a direct consequence of the de nition of Q †,( ) ˆ and Q †,( ) .

Lemma 3.7.2. For any test function ∈ B ( 2 ) and any coalescence indicator ∈ {0, 1} +1 , we have

=1 ♯ D †, ( ) = O L 1 1 √ , and =1 ♭ D †, ( ) = O L 1 1 √ .
Proposition 3.7.3. For any test function ∈ B ( 2 ) and any coalescence indicator ∈ {0, 1} +1 , the integrable process ( X †, ( ); 0 ≤ ≤ ) can be decomposed to a (W ; 0 ≤ ≤ )-martingale ( M †, ( ); 0 ≤ ≤ ) , and a integrable predictable process ( A †, ( ); 0 ≤ ≤ ), respectively de ned by

M †, ( ) := X †, 0 ( ) + =1 ♯ D †, ( ), and
A †, ( ) := =1 ♭ D †, ( ).
In particular, under symmetric resampling scheme, that is

• ≡ ˚ for all ∈ [ ],
we also have

♭ D †,( ) ( ) ≡ 0, . . which yields X †,( ) ( ) 0≤ ≤ is a (W ; 0 ≤ ≤ )-martingale.

Veri cation of asymptotic variance expansion

In this section, we verify the asymptotic variance expansion (3.8) given in Section 3.3.1. Recall that, with the introduction of coalescent Feynman-Kac kernels, we have

2 ( ) = =0 ⊗2 1 Q ( ) , ( ⊗2 ) -⊗2 -1 1 ⊗2 , -1 Q ( ) , ( ⊗2 ) .
By de nition, since

⊗2 -1 Q 0 = -1 ˚ + -1 ( -1 ) -1 ( • -˚ ) ≡0. ⊗2 = ⊗2 , we have ⊗2 1 Q ( ) , ( ⊗2 ) = Γ ( ) ( ⊗2 ).
Next, for the latter term, since

Q ( ) , ( ⊗2 ) = , ( ) ⊗2
, by applying (3.17), we deduce that

∀ ∈ [ ], ⊗2 -1 1 ⊗2 , -1 Q ( ) , ( ⊗2 ) = ⊗2 -1 ( 1 (1)) ⊗2 -1 ˚ ⊗2 Q ( ) , ( ⊗2 ) Γ ( ) ( ⊗2 ) + ⊗2 -1 1 ⊗2 , -1 Q ( ) , ( ⊗2 ) Note that ∀ ∈ B ( ), , -1 ( ) ( ) 2 = -1 ( -1 ) 2 • ( ) ( ) 2 + -1 ( ) 2 -1 ˚ ( ) 2 -2 -1 ( -1 ) -1 ˚ ( ) ( -1 × • ) ( ) ( ), whence ⊗2 -1 1 ⊗2 , -1 ( ⊗2 ) = -1 ( -1 ) 2 ⊗2 -1 1 • ⊗2 ( ⊗2 ) - --1 ( 2 -1 ) ⊗2 -1 ˚ ⊗2 -1 ( 2 -1 ) ⊗2 -1 [ ˚ ⊗2 - • ⊗ ˚ -˚ ⊗ • ] +2 -1 ( -1 ) ⊗2 -1 [ ˚ ⊗ ( -1 × • )] ( ⊗2 ) ⊗2 -1 Q †,1 ( ⊗2 )= Γ †,( ) -1 Q †,1 ( ⊗2 )
.

Replacing ⊗2 above by Q ( ) , ( ⊗2 ) = Q †,( ) , ( ⊗2 ), we get ∀ ∈ [ ], ⊗2 -1 1 ⊗2 , -1 Q ( ) , ( ⊗2 ) =Γ ( ) ( ⊗2 ) -Γ †,( ) -1 Q †,1 Q †,( ) , ( ⊗2 ) + -1 ( -1 ) 2 ⊗2 -1 1 • ⊗2 Q ( ) , ( ⊗2 ) =Γ ( ) ( ⊗2 ) -Γ †,( -1) ( ⊗2 ) + -1 ( -1 ) 2 Γ †,( ) -1 1 • ⊗2 Q ( ) , ( ⊗2 ).
Taking into account that

∀ ∈ [ ], Γ ( -1) ( ⊗2 ) --1 ( -1 ) 2 Γ †,( ) -1 1 • ⊗2 Q ( ) , ( ⊗2 ) = Γ †,( -1) ( ⊗2 ), and
Γ ( ) ( ⊗2 ) = Γ †,( ) ( ⊗2 ),
as well as the convention (3.4) for the case = 0, that writes

⊗2 -1 1 ⊗2 0, -1 Q ( ) 0, ( ⊗2 ) = Γ ( ) ( ⊗2 ),
we nally obtain the coalescent tree-based asymptotic variance expansion:

2 ( ) := =0 Γ †,( ) ( ⊗2 ) -Γ ( ) ( ⊗2 ) + -1 =0 Γ †,( ) ( ⊗2 ).

Proof of Theorem 3.2.1

For any test function ∈ B ( ), the unbiased property

E ( )1 ≥ = ( )
for the case

•
≡ ˚ is a direct consequence of the martingale decomposition (3.31). For the almost sure convergence, the proof is done by induction. For the step 0, the almost sure convergence of 0 is a direct consequence of law of large numbers for i.i.d. random variables. For step ≥ 1, we suppose that for each 0 ≤ ≤ -1, we have

∀ ∈ B ( ), ( ) . .
-----→ →∞ ( ).

We rst check that 1

( +1) =1 ( )
. .

-----→ →∞ 0.

(3.32)

Taking into account that ( ) ≤ 3 ∞ , . .

we have, thanks to Azuma-Hoe ding inequality, for any α > 0,

P ( +1) =1 ( ) > α ≤ 2 exp -2 α 2 9( + 1) 2 ∞ .
Hence, the almost sure convergence (3.32) is then ensured by Borel-Cantelli lemma. On the other hand, the induction hypothesis gives

∀ ∈ [ ], , ( ) . .
-----→ →∞ 0, which yields =1 , ( )

. .

-----→ →∞ 0.

The veri cation of the almost sure convergence for ( )1 ≥ is then complete. The almost sure convergence of is then trivial since for any test function ∈ B ( ), the convention (3.1) allows the writing

( )1 ≥ = ( )1 ≥ (1)1 ≥ .

Proof of Theorem 3.2.2

Lemma 3.7.3. Let be an empirical on -1 , we suppose that there exists a probability measure on -1 , such that for any test function ∈ B ( -1 ), one has

( ) . . -----→ →∞ ( ).
Then, for any test function ∈ B ( ), we have the following almost sure convergence:

(i) , ( ) 2 . . -----→ →∞ , ( ) 2 ; (ii) , ( ) 2 . . -----→ →∞ , ( ) 2 .
Proof. Before starting the proof, let us recall that for any probability measure ∈ P ( -1 ) and for any test function ∈ B ( ), we have

, ( ) ( ) = ˚ ( ) + , ( ) ( )
with , de ned in (3.15). Basic algebraic mulipulation gives

, ( ) ( ) 2 = ˚ ( ) 2 + , ( ) ( ) 2 + 2 ˚ ( ) , ( ) ( ).
Recall that, by de nition, we have

, ( ) ( ) = ( -1 ) • ( ) ( ) --1 ( ) ˚ ( ) and , ( ) ( ) 2
= ( -1 ) 2 • ( ) ( ) 2 + -1 ( ) 2 ˚ ( ) 2 -2 ( -1 ) ˚ ( ) -1 ( ) • ( ) ( ),
whence we deduce that

, ( ) 2 = ( -1 ) 2 • ( ) 2 + ( 2 -1 ) ˚ ( ) 2 -2 ( -1 ) ˚ ( ) -1 • ( ) .
Since -1 , ˚ ( ) ∈ B ( -1 ), Theorem 3.2.1 gives that

( -1 ) 2 • ( ) 2 + ( 2 -1 ) ˚ ( ) 2 -2 ( -1 ) ˚ ( ) -1 • ( ) . . . -----→ →∞ ( -1 ) 2 • ( ) 2 + ( 2 -1 ) ˚ ( ) 2 -2 ( -1 ) ˚ ( ) -1 • ( ) .
On the other hand, as

, ( ) 2 = ( -1 ) 2 • ( ) 2 + ( 2 -1 ) ˚ ( ) 2 -2 ( -1 ) ˚ ( ) -1 • ( ) .
we safely deduce that , ( ) 2

. .

-----→ →∞ , ( ) 2 , which terminates the veri cation for the point (i). Next, by standard calculation, we obtain

, ( ) ( ) 2 = ˚ ( ) 2 + , ( ) ( ) 2 + 2 ˚ ( ) , ( ) ( ), whence , ( ) 2 =
(1) ˚ ( ) 2 + , ( ) 2 + 2 ˚ ( ) ( -1 ) ( • -˚ ) ( ).
Finally, as ˚ ( ),

• ( ) ∈ B ( -1 ), point (i) and Theorem 3.2.1 combined with the fact that

, ( ) 2 = (1) ˚ ( ) 2 + , ( ) 2 + 2 ˚ ( ) ( -1 ) ( • -˚ ) ( )
ensure the desired convergence in point (ii). This closes the proof of Lemma 3.7.3. Now, let us start the proof of the CLT-type result for 1 ≥ and 1 ≥ . The proof is done by induction. The veri cation of step 0 is trivial by the central limit theorem for i.i.d. random variables. For step ≥ 1, we suppose that, for any test function ∈ B ( ), we have

∀0 ≤ ≤ -1, √ ( )1 ≥ -( ) d -----→ →∞ N 0, 2 ( -( )) .
Again, let us go back to the decomposition (3.31). First, we prove that

1 √ ( +1) =1 ( ) d -----→ →∞ N 0, 2 . (3.33)
In order to apply Theorem 2.3 in [START_REF] Mcleish | Dependent central limit theorems and invariance principles[END_REF], one needs to verify that

• The boundness of gives that

max 1≤ ≤ ( +1) 1 √ ( ) ≤ 3 √ ∞ , (3.34)
which shows that max 1≤ ≤ ( +1)

1 √ ( ) is uniformly bounded in L 2 -norm.
• From (3.34), one also gets that

max 1≤ ≤ ( +1) 1 √ ( ) P -----→ →∞ 0.
• For the asymptotic variance, we deduce that

( ) 2 = (1) 2 , ( ) 2 1 ≥ 1 ( ) + -1 (1) 2 , -1 ( , ) ( -1 ) 2 1 ≥ -1 2 ( ) -2 -1 (1) 2 -1 ( -1 ) , ( )
, -1 ( , ) ( -1 )1 ≥ 3 ( )
First, let us prove that 1

( +1) =1 1 ( ) . . -----→ →∞ =0
(1) ( 2 , ).

(3.35)

In fact, by the construction of the Feynman-Kac IPS, we have 1

( +1) =1 1 ( ) = =0 (1) ( 2 , ).
Hence, Theorem 3.2.1 gives the desired convergence (3.35). Second, for the term concerning 2 ( ), we would like to show that 1

( +1) =1 2 ( ) . . -----→ →∞ =0 -1 (1) -1 , -1 ( , ) 2 . (3.36)
Similar to the previous case, we deduce that 1

( +1) =1 2 ( ) = =0 -1 (1) -1 , -1 ( , ) 2 . = =0 -1 (1) 2 -1 , -1 ( , ) 2 .
The convergence (3.36) is then obtained by combining Theorem 3.2.1 and the point (ii) of Lemma 3.7.3. Then, for the term concerning 3 ( ), we prove that 1

( +1) =1 3 ( ) . . -----→ →∞ =0 -1 (1) -1 , -1 ( , ) 2 . (3.37)
Notice that

E 3 ( ) F -1 = -1 (1) 2 -1 ( -1 ) , -1 ( , ) ( -1 )E ( ) F -1 = -1 (1) 2 -1 ( -1 ) , -1 ( , ) ( -1 ) , -1 ( , ) ( -1 ) , -1 ( , ) ( -1 )
= 2 ( ).

Hence, by exploiting the already proved convergence (3.36), it is su cient to verify that 1

( +1) =1 3 ( ) -2 ( ) . . -----→ →∞ 0. (3.38)
Recall the ltration (E ; ≥ 0) de ned by

∀ ∈ [( + 1) ], E = F ∨ ( 1 , • • • , ).
It is readily checked that ( 3 ( ) -2 ( )) is a (E )-martingale di erence array. In addition, the boundness of ensures that

3 ( ) -2 ( ) ≤ 8 ∞ . . .
Thanks to Azuma-Hoe ding inequality, one obtains

∀α > 0, P ( +1) =1 3 ( ) -2 ( ) ≥ α ≤ 2 exp -α 2 32( + 1) 2 ∞
.

The almost sure convergence (3.38) is then followed from Borel-Cantelli lemma. In conclusion, by combining (3.35),(3.36) and (3.37), one gets

1 ( +1) =1 ( ) 2 . ./P -----→ →∞ 2 ( ). (3.39)
Next, the induction hypothesis, Lemma 3.7.5 and Theorem 3.2.1 ensure that

, ( ) = -1 ( -1 ) --1 ( -1 ) 1 ≥ -1 O p 1 √ -1 ( • -) ( , )1 ≥ -1 , p (1) (3.40)
whence

-1 =0 , ( ) = p 1 √ .
Slutsky's lemma then gives the CLT-type convergence for ( )1 ≥ . The CLT-type result for ( )1 ≥ is a direct consequence of Slutsky's lemma and the following decomposition

√ ( )1 ≥ -( ) 1 ≥ = 1 (1) √ ( -( ))1 ≥ -( -( )) 1 ≥ .
This ends the proof of Theorem 3.2.2.

Proof of Proposition 3.3.1

First, we notice that

( ) = (1) 2 -Γ ‡,( ) , 1 ≥ + Γ ‡,( ) , -Γ ( ) , 1 ≥ . . .
We start by study the rst term on the right-hand side of the equality above. Thanks to Theorem 3.6.1, and by considering the stochastic bound given in Proposition 3.7.7, we have, on the event

{ ≥ }, ( ) ⊗2 ( ⊗2 ) = -1 +1 Γ ‡,( ) , ( ⊗2 ) + 1 -1 =0 Γ ‡,( ) , ( ⊗2 ) + O L 2 1 2 .
Notice that

-1 = 1 -O 1 and -1 +1 -1 = - + 1 + O 1 2 , which yields (1) 2 -Γ ‡,( ) , 1 ≥ = =0 Γ ‡,( ) , ( ⊗2 ) -Γ ( ) , ( ⊗2 ) + O L 2 1 . (3.41)
Next, by applying the decomposition given in Proposition 3.6.4, we deduce that

Γ ‡,( ) , - Γ ( )
,

1 ≥ = 1 + 1 -1 -1 =0 Γ †,( ) , ( )1 ≥ .
Combining the two parts, we nally obtain the desired stochastic bound in (3.12). For (3.13), the reasoning is similar by the same algebraic manipulations. The only remark is that due to the "normalization" procedure, the stochastic bound w.r.t. L 2 -norm given by Proposition 3.7.7 and Proposition 3.7.8 will be replaced by a weaker version, namely,

Γ ‡, , ( )/ (1) 2 = O p (1) and Γ †, , ( )/ (1) 2 = O p (1)
. This is ensured by Theorem 3.2.1.

Proof of Theorem 3.6.1

On the event { < }, it is clear that both equalities hold. On the event { ≥ }, the particle system is well-de ned from level 0 to level . Since

-1 +1-| | 1 | | = =0 ( -1) 1- , we have -1 +1-| | 1 | | Γ , ( ) = -1 ( -1) +1 [2] 0: ∈ ( ( ) 2 ) ×( +1) =0 ( -1) 1- -1 =0 ( [2] +1 , [2] ) ( ) ( [ 2]
).

Enumerating all the possibilities for the coalescence indicator ∈ {0, 1} +1 leads to

∈ {0,1} +1 =0 ( -1) 1- Γ , ( ) = [2] 0 ∈ ( ) 2 • • • [2] -1 ∈ ( ) 2 -1 =0 1 1 { 1 +1 = 2 +1 = 1 ≠ 2 } + -1 1 { 1 +1 = 1 ≠ 2 +1 = 2 } -1 -1 2 (X n ) 0 ( ) + 1 2 (X n ) 1 ( ) .
To conclude, one just has to observe that, for each 0 ≤ ≤ -1,

[2] ∈ ( ) 2 1 1 { 1 +1 = 2 +1 = 1 ≠ 2 } + -1 1 { 1 +1 = 1 ≠ 2 +1 = 2 } = -1 , . . while, by (3.2), -1 2 (X n ) 0 ( ) + 1 2 (X n ) 1 ( ) = ⊗2 (X n ) ( ) = ( ) ⊗2 ( ).
Multiplying both sides by (1) 2 gives the corresponding relation for ( ) ⊗2 ( ).

3.7.7 Proof of Proposition 3.6.5

For all ≥ 2, let us consider the following auxiliary random matrix:

               H ‡,0|0 := H †,0 ; H ‡,1|0 := H †,1 ; H ‡,0|1 := 0; H ‡,1|1 := H †,1 .
and

               H ‡,0|0 ,( ) := H †,0 ; H ‡,1|0
,(

) := 1 -1 H †,1 ; H ‡,0|1 ,( ) := 0; H ‡,1|1 ,( ) := H †,1
.

Using the partial semigroup structure, we de ne, for any coalescence indicators and ,

∀ ∈ B ( 2 ), Γ ‡, | , ( ) := Λ ‡, | (F ) = 2 ( [ ]) • H ‡, | 1 • H ‡, | 2 • • • H ‡, | (F ), with F [ [2] ] := ( [ 2]
).

Similarly, we also de ne

∀ ∈ B ( 2 ), Λ ‡, | ,( ) (F ) = 2 ( [ ]) • H ‡, | 1,( ) • H ‡, | 2,( ) • • • H ‡, | ,( ) (F ).
Remark that 0 = 0 × 1 -1 . Hence, by de nition, we have

Λ ‡, | ,( ) (F ) = 1 -1 | -| Λ ‡, | (F ). . .
Next, we consider the binary decomposition w.r.t. a coalescence indicator . More precisely,

Λ ‡, (F ) = ∈S( ) Λ ‡, | ,( ) (F ) = Λ ‡, | ,( ) (F ) + ∈ S( ) Λ ‡, | ,( ) (F ) = Λ ‡, | (F ) + ∈ S( ) Λ ‡, | ,( ) (F ) = Λ †, (F ) + ∈ S( ) 1 -1 | -| Λ ‡, | (F ).

PROOFS

Therefore, it su ces to verify that for any coalescence indicator and , we have

Λ ‡, | (F ) = O L 2 (1).
By de nition, if there exists 0 ≥ 0 such that 0 = 0 and 0 = 1, we have Λ ‡, | (F ) ≡ 0.

If not, let us consider the mapping : {0, 1} 2 ↦ → {0, 1} de ned by

(0, 0) = 0, (1, 0) = 1 and (1, 1) = 1.
We also denote := ( 0 , 0 ), ( 1 , 1 ), . . . , ( , ) . It is then easily checked that

Λ ‡, | (F ) = Λ †, (F ) = Γ †, ( ).
As a consequence, thanks to Proposition 3.7.8, we have

Λ ‡, | (F ) = O L 2 (1).
This is su cient to end the proof of Proposition 3.6.5.

Technical results

In this section, we list some technical results in support of the proofs given in the following sections. We remark that Lemma 3.7.5 serves as a technical lemma, designed to prove Proposition 3.7.4 by induction. Since the latter one is proved to be true, the hypothesis in Lemma 3.7.5 can thus be removed. This is why in the Proposition 3.7.5, it can be used without induction argument.

In the proof of Proposition 3.7.7, we do not give the nest analysis, which is done later in the proof of Lemma 3.7.10. This organization is due to the complication of the notation in the present work.

Since the rougher analysis in the proof of Proposition 3.7.7 is more straightforward than the ner version in Lemma 3.7.10, we consider it to be a good warm-up to the techniques involved in this section, which are highly repetitive in regard of the application of the pivotal decomposition (3.63).

Lemma 3.7.4. For any test function , ∈ B ( 0 ) and for both 0 = 0 and 0 = 1, we have

( 0 ) ⊗2 0 ( ⊗ )1 ≥0 -⊗2 0 0 ( ⊗ ) = O L 2 1 √ .
Proof. For the case 0 = 1, it is su cient to verify that

∀ ∈ B ( 0 ), 0 ( )1 ≥0 -( ) = O L 2 1 √ ,
which is clear for the bounded i.i.d. random variables. More precisely, we have

E 0 ( ) (1 ≥0 + 1 -1) -0 ( ) =E 0 ( ) (1 ≥0 -1) ≤ ∞ P 0 ( 0 ) = 0 .
Thanks to Cauchy-Schwarz inequality, one derives , (1)

L 1 = -1 (1) -1 ( • -˚ ) , (1)
-1 ( -1 ) --1 ( -1 ) 1 ≥ -1

L 1 ≤ -1 (1) -1 ( • -˚ ) , (1) 1 ≥ -1 L 2 -1 ( -1 ) --1 ( -1 ) 1 ≥ -1 L 2 .
In addition, we also have

-1 (1) -1 ( • -˚ ) , (1) 1 ≥ -1 L 2 ≤ -1 (1) -1 • , (1) 1 ≥ -1 --1 • , (1) 1 ≥ -1 L 2 + -1 (1) -1 ˚ , (1) 1 ≥ -1 --1 ˚ , (1) 1 ≥ -1 L 2 .
Therefore, consider the hypothesis, and the fact that -1 (1) ≤ 1, . .

we get

, (1)

L 1 = O 1 ,
which yields

P (1)1 ≥ < (1) 2 = O 1 . (3.44)
Next, since

{ < } ⊂ { ≤ } ={ ≤ -1} ∪ (1) = 0 ⊂ { ≤ -1} ∪ (1)1 ≥ < (1) 2
, one derives that

P ( ≤ ) ≤P ( ≤ -1) + P (1)1 ≥ < (1) 2
.

By applying the inequality above recursively from to 0, one nally obtains

P ( ≤ ) = O 1 .
Proposition 3.7.4. For any test function ∈ B ( ), we have

( )1 ≥ -( ) = O L 2 1 √ .
In particular, one also has

( )1 ≥ -( ) = O L 2 1 √ .
Proof. The proof is done by induction. For = 0, the stochastic bound is clear for the bounded i.i.d. random variables. which is guaranteed by Lemma 3.7.4. For step ≥ 1, we suppose that

∀ ∈ B ( ), ( )1 ≥ -( ) = O L 2 1 √ .
We consider the event Ω ⊂ Ω de ned by

Ω := (1)1 ≥ ≥ (1) 2
.

By the de nition of the absorbing time , one has

Ω = (1)1 ≥ ≥ (1) 2 and ≥ ⊂ { ≥ }, whence 1 Ω ≤ 1 ≥ .
Then, by the fact that

1 ≥ = 1 ≥ -1 Ω + 1 Ω ≤ 1 ≥ -1 + 1 -1 Ω + 1 Ω ≤ 1 -1 ≥ + 1 -1 Ω + 1 Ω ≤ 21 (Ω ) c + 1 Ω , (3.45)
we obtain

( ) -( ) 1 ≥ ≤ ( ) -( ) 1 Ω + 4 ∞ 1 (Ω ) c . . .
Then, by applying the induction hypothesis, and thanks to a by-product (3.44) of Lemma 3.7.5, the inequality above leads to

( ) -( ) 1 ≥ L 2 ≤ ( ) -( ) 1 Ω L 2 + 4 ∞ P Ω c O 1 √ .
It is thus su cient to verify that

( ) -( ) 1 Ω L 2 = O 1 √ . (3.46)
By the fact that 1 Ω 1 ≥ = 1 Ω , we have the following equality:

( ) -( ) 1 Ω = 1 (1) ( )1 ≥ -( ) 1 Ω - ( ) (1) (1)1 ≥ -(1) 1 Ω .
Then, by the de nition of the event Ω , we have

1 (1) ( )1 ≥ -( ) 1 Ω ≤ 2 (1) ( )1 ≥ -( ) 1 Ω . . .
As a consequence, to prove that

1 (1) ( )1 ≥ -( ) 1 Ω = O L 2 1 √ , and
( ) (1) (1)1 ≥ -(1) 1 Ω = O L 2 1 √ .
One only needs to verify that

( )1 ≥ -( ) = O L 2 1 √ . (3.47)
According to the bias-martingale decomposition (3.31), we have

( )1 ≥ -( ) = 1 ( +1) =1 ( ) + =1 , ( ), whence ( )1 ≥ -( ) 2 = 1 ( +1) =1 ( ) 2 + =1 , ( ) 2 + 2 ( +1) =1 ( ) =1 , ( ).
By de nition, we have , ( )

L 2 = -1 (1) -1 ( • -˚ ) , ( ) -1 ( -1 ) --1 ( -1 ) 1 ≥ -1 L 2 ≤2 ∞ -1 ( -1 )1 ≥ -1 --1 ( -1 ) 1 ≥ -1 L 2 .
By applying the induction hypothesis, one obtains

, ( ) = O L 2 1 √ , which gives =1 , ( ) = O L 2 1 √ ,
and, by Cauchy-Schwarz inequality, =1 , ( )

2 = O L 1 1 . Meanwhile, since ( ) 1≤ ≤ ( +1)
is a martingale di erence array, we have

1 E       ( +1) =1 ( ) 2       = 1 ( +1) =0 E ( ) 2 -----→ →∞ 2 ( ) < +∞,
where the convergence is a by-product (3.39) of the proof of Theorem 3.2.2 and dominated convergence theorem. Hence, we obtain

( +1) =1 ( ) = O L 2 √ .
In summary, we have

E ( )1 ≥ -( ) 2 = 1 ( +1) =0 E ( ) 2 + 2E               ( +1) =0 ( ) O L 2 ( √ ) =1 , ( ) O L 2 1 √               + 2 E             =1 , ( ) 2 O L 1 ( 1 )            
, which, thanks to Cauchy-Schwarz inequality, leads to

( )1 ≥ -( ) L 2 = O 1 √ . (3.48)
This ends the veri cation of (3.47) and the proof of this proposition.

Proposition 3.7.5 (L 2 -propagation of chaos). For any test function , ∈ B ( ), we have

( ) 2 ( ⊗ )1 ≥ -⊗2 ( ⊗ ) = O L 2 1 √ .
Proof. Before starting the proof, let us mention that by Minkowski's inequality, for two random variables and , one has

= O L 2 1 √ and = O L 2 1 √ =⇒ + = O L 2 1 √ .
Notice that

( ) 2 ( ⊗ )1 ≥ -⊗2 ( ⊗ ) = ( ) 2 ( ⊗ ) -⊗2 ( ⊗ ) 1 ≥ + ⊗2 ( ⊗ ) (1 -1 ≥ ).
Thanks to Proposition 3.7.4 and Lemma 3.7.5, one derives

E ⊗2 ( ⊗ ) (1 -1 ≥ ) 2 ≤ ∞ ∞ E 1 -1 ≥ ≤ ∞ ∞ P ( < ) O( 1 )
, which implies that

⊗2 ( ⊗ ) (1 -1 ≥ ) = O L 2 1 √ .
Next, considering the decomposition (3.2), we deduce that

( ) 2 ( ⊗ ) -⊗2 ( ⊗ ) 1 ≥ = -1 ( ) ⊗2 ( ⊗ ) -⊗2 ( ⊗ ) 1 ≥ + 1 -1 ( ) + ( ) ( ) 1 ≥ .
Concerning the term at the right-hand side of the equality above, we noticed that

1 -1 ( ) + ( ) ( ) 1 ≥ L 2 ≤ 2 -1 ∞ ∞ .
In addition, since

( ) ⊗2 ( ⊗ ) -⊗2 ( ⊗ ) 1 ≥ = ( ) ( ) -( ) + ( ) ( ) -( ) 1 ≥ ≤2 ∞ ∨ ∞ ( ) -( ) ∨ ( ) -( ) 1 ≥ , . .
it is then su cient to verify that

∀ ∈ B ( ), ( ) -( ) 1 ≥ = O L 2 1 √ . (3.49)
which is guaranteed by Proposition 3.7.4 since

( ) -( ) 1 ≥ = ( )1 ≥ -( ) 1 ≥ . . .
The proof is then nished.

Proposition 3.7.6 (Biasedness). For any test function ∈ B ( ), we have

E ( )1 ≥ -( ) = O 1 .
In particular, we also have

E ( )1 ≥ -( ) = O 1 .
Remark. Di erent from the other technical results, the order of the bias given in this proposition will not be used to prove the consistency of the variance estimator. They are put in this section simply because we think the order of bias is important but not as relevant in the present work, where the most results we discussed are "short term" asymptotic properties of the IPS. In addition, we want to mention that by the same strategy, one can obtain an explicit bound w.r.t. both and for the bias. The main di erence from the classic Feynman-Kac particle models discussed in [START_REF] Moral | Feynman-Kac formulae: Genealogical and interacting particle systems with applications[END_REF] is the "lack-of-martingale" or, said di erently, the bias-martingale structure (cf. (3.31)). As a consequence, the decay rate of the absorbing time is not exponential w.r.t. any more. Instead, it is replace by O(1/ ), as stated in Lemma 3.7.5. This is why the order of bias w.r.t.

is not a ected. The "lack-of-martingale" structure also requires an induction in order to deal with the bias term encountered in the bias-martingale decomposition (3.31). This technique is frequently used in the adaptive SMC context (cf. [DG19](Chapter 2)). 123 and we have 1 Ω ≤ 1 ≥ ≤ 21 (Ω ) c + 1 Ω . Notice that, by Lemma 3.7.5 and the de nition of , one has

(1) (1) -1 ( ) -( ) 1 (Ω ) c ≤ ( )1 ≥ -( ) 1 (Ω ) c + ( ) 1 ≥ -( ) 1 (Ω ) c ≤4 ∞ 1 (Ω ) c = O L 1
1 .

(3.52)

In addition, by de nition of Ω , one gets

- (1)1 ≥ -(1) (1) ( ) 1 ≥ -( ) 1 Ω ≤ 2 (1) (1)1 ≥ -(1) ( ) 1 ≥ -( ) . . .
Thus, thanks to a by-product (3.48) in the proof of Proposition 3.7.4, we obtain

(1) (1) -1 ( ) -( ) 1 Ω = O L 1 1 . (3.53)
Finally, combining both (3.52) and (3.53) terminates the veri cation of (3.51), which also ends the proof of Proposition 3.7.6.

Before proceeding further, we recall and introduce some notation that is used frequently in the following technical results. For ∈ N * , we denote In particular, we denote ( ) := [ ] . We also write

( ) 2 × := ( ) 2 × ( ) 2 × • • • × ( ) 2 times . (3.55)
With a slight abuse of notation, we admit that (( , ), ) = ( , , ). and (( , ), ( , )) = ( , , , ).

We also adopt the notation introduced in Section 3.6.5. Fixing some ∈ {0, 1} +1 , we denote

Λ ‡, [ [2] ] := 1 ( -1) [2] 0: -1 ∈ ( ( ) 2 ) × -1 =0 G ‡ p (X p ) ( [2] +1
, [2] ) , (3.56) with the convention

Λ ‡, 0 [ [2] 0 ] := 1 ( -1)
.

It is readily checked that

Λ ‡, [ [2] ] = [2] -1 ∈ ( ) 2 Λ ‡, -1 [ [2] -1 ]G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ).
This allows an alternative representation of Γ ‡, , :

∀ ∈ B ( 2 ), Γ ‡, , ( ) = [2] ∈ ( ) 2 Λ ‡ [ [2] ] ( ) ( [2]
), (3.57) which covers the case = 0. Similarly, we also denote

Λ †, [ [2] ] := 1 ( -1) [2] 0: -1 ∈ ( ( ) 2 ) × -1 =0 G †, p ( [2] : +1 , B p , X p ) ( ) ( [2] +1
, [2] ) , (3.58)

with the convention

Λ †, 0 [ [2] 0 ] := 1 ( -1)
.

We also have the decomposition

Λ †, [ [2] ] = [2] -1 ∈ ( ) 2 Λ †, -1 [ [2] -1 ] G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ). (3.59)
As is shown in the previous case (3.57), † Γ , admits the following alternative representation:

∀ ∈ B ( 2 ), Γ †, , ( ) = [2] ∈ ( ) 2 Λ †, [ [2] ] ( [ 2]
),

which covers the case = 0.

Proposition 3.7.7. For any coalescence indicator ∈ {0, 1} +1 , we have

Γ ‡, , (1)1
≥ = O L 2 (1).
In particular, for any test function ∈ B ( 2 ), we also have

Γ ‡, , ( )1 ≥ = O L 1 (1).
Proof. The proof is done by induction. For the step = 0, it is clear since ‡ Γ 0, (1) = 1. For step ≥ 1, we suppose that sup

>0 E Γ ‡, -1, (1) 2 1 ≥ -1 < +∞.
By the alternative representation (3.57), for all ≥ 4, we have

E Γ ‡, , (1)
2 1 ≥ G -1 = ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ ‡, -1 [ [2] -1 ]Λ ‡, -1 [ [2] -1 ]1 ≥ -1 E       ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1       , (3.61) since the de nition of G ‡ n-1 gives E       ( [2] , [2] ) ∈ ( ( )) ×2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1       1 = -1 = 0.
In order to simplify the notation, we may omit 1 ≥ -1 , which is G -1 -measurable, in the rest of the proof. Before proceeding, we recall the conditional distribution of the selection step. Given G -1 , we have

∀ ∈ [ ], -1 ∼ -1 ( -1 ) (•) + (1 --1 ( -1 )) =1 -1 ( -1 ) (X n-1 ) ( -1 ) (•).
Hence, for any ∈ [ ], we have

(X n-1 ) ( -1 ) =1 P -1 = G -1 = (X n-1 ) ( -1 ) -1 ( -1 ) + =1 (1 --1 ( -1 )) -1 ( -1 ) = -1 ( -1 ) ≤ 1. . . (3.62)
With the notation introduced in (3.54) and (3.55), for ≥ 4, we have the decomposition

(( ) 2 ) ×2 = (( ) 2 ) ×2 ∩ [ ] 4 2 ∪ (( ) 2 ) ×2 ∩ [ ] 4 3 ∪ ( ) 4 . (3.63)
The rest of the proof consists in studying the term

E       ( [2] , [2] ) ∈ ( ( )) ×2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1     
 with respect to the decomposition above and the bound given in (3.62).

(i) Case:

( [2]
, [2] ) ∈ ( ) 2 ×2 ∩ [ ] 4 2 : In this case, there are only two distinct random variables in the tuple

1 -1 , 2 -1 , 1 -1 , 2 -1 .
Due to the symmetry of the particles, we rst calculate the number of choices to assign two di erent random variables in the tuple above. More precisely, in this case, the two possible assignments are

1 -1 = 1 -1 , 2 -1 = 2 -1 and 1 -1 = 2 -1 , 2 -1 = 1 -1 .
Without loss of generality, we suppose that

[2]
-1 are two distinct random variables, with one of the two assignments above. Then, when [2] varies freely in ( ) 2 , the values of

[2]
-1 will be a.s. determined by the chosen assignment and the value of In addition, since varies on the interval [0, 1], we have

0 ≤ (X n ) ( ) ≤ 1. . . Now, let us deduce that E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1       ≤E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) G -1       ≤ -1 2 E       [2] ∈ ( ) 2 (X n-1 ) ( ) 2 1 1 -1 = 1 -1 1 2 -1 = -1 1 -1 +(1--1 ) 2 -1 G -1      
.

By the conditional independence between 1 -1 and 2 -1 , one deduces

E       [2] ∈ ( ) 2 (X n-1 ) ( ) 2 1 1 -1 = 1 -1 1 2 -1 = -1 1 -1 +(1--1 ) 2 -1 G -1       = [2] ∈ ( ) 2 E       (X n-1 ) ( )1 1 -1 = 1 -1 G -1       E       (X n-1 ) ( )1 2 -1 = -1 1 -1 +(1--1 ) 2 -1 G -1       ≤ 1 ∈ [ ] E       (X n-1 ) ( )1 1 -1 = 1 -1 G -1       2 ∈ [ ] E       (X n-1 ) ( )1 2 -1 = -1 1 -1 +(1--1 ) 2 -1 G -1       . (3.65)
Combined to (3.62), one gets

E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) G -1       ≤ 1 2 × -1 . . . (3.66)
which gives

E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) G -1       ≤ -1 2 . . . (3.67)
Considering the choices of assignments mentioned above, one nally gets

E        ( [2]
, [2] ) ∈ ( ( )

) ×2 ∩[ ] 4 2 G ‡ n (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1        ≤ 2 × -1 2 . . . (ii) Case: ( [2] , [2] ) ∈ ( ) 2 ×2 ∩ [ ] 4 3 :
In this case, there are three distinct random variables in the tuple

1 -1 , 2 -1 , 1 -1 , 2 -1 .
Similar to the previous case, we calculate the number of choices to assign three di erent random variables in the tuple above. Let us x one assignment. We suppose that [2] and 1 are three distinct numbers. Then, whilst ( [2] , 1 ) varies freely in ( ) 3 , the value of 2 -1 is a.s. determined by the chosen assignment and the values of (

E       ( [2] , 1 ) ∈ ( ) 3 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1       ≤E       ( [2] , 1 ) ∈ [ ] 3 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 )1 1 -1 = 1 -1 G -1       ≤ -1 2 E       ( [2] , 1 ) ∈ [ ] 3 (X n-1 ) ( -1 ) 3 -1 ( [2] -1 , [2] -1 )1 1 -1 = 1 -1 G -1       = -1 2 ( [2]
, 1 ) ∈ ( )

3 E       (X n-1 ) ( -1 )1 1 -1 = 1 -1 G -1       E       (X n-1 ) ( -1 )1 2 -1 = -1 1 -1 +(1--1 ) 2 -1 G -1       E       (X n-1 ) ( -1 )1 1 -1 = 1 -1 G -1      
Again, combined with (3.62), one deduces

E       ( [2]
, 1 ) ∈ ( ) 3

G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1       ≤ 1 3 × -1 2 .
Considering the number of assignments, we obtain

E        ( [2] , [2] ) ∈ ( ( )) ×2 ∩[ ] 4 3 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1        ≤ 4 × -1 2 .
(iii) Case:

( [2]
, [2] ) ∈ ( ) 4 :

In this case, all the random variables

1 -1 , 2 -1 , 1 -1 , 2 -1
are distinct. Similarly, by the conditional independence of A n-1 and the bound given in 3.7. PROOFS

129

(3.62), one gets

E       ( [2] , [2] ) ∈ ( ) 4 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1       ≤ -1 2 ( [2] , [2] ) ∈ ( ) 4 E (X n-1 ) ( ) 4 -1 ( [2] -1 , [2] -1 ) G -1 = -1 2 ( [2] , [2] ) ∈ ( ) 4 E       (X n-1 ) ( -1 )1 1 -1 = 1 -1 G -1       E       (X n-1 ) ( -1 )1 2 -1 = -1 1 -1 +(1--1 ) 2 -1 G -1       E       (X n-1 ) ( -1 )1 1 -1 = 1 -1 G -1       E       (X n-1 ) ( -1 )1 2 -1 = -1 1 -1 +(1--1 ) 2 -1 G -1       ≤1 4 × -1 2
Combining the three cases discussed above, we safely deduce that

E       ( [2] , [2] ) ∈ ( ( )) ×2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1       ≤ (2 + 4 + 1) =7 × -1 2 .
. . Now, let us go back to (3.61), by taking expectation on both sides, we have, for ≥ 4,

E Γ ‡, , (1) 2 1 ≥ ≤7 -1 2 E Γ ‡, -1, (1) 2 1 ≥ -1 ≤ 112 9 E Γ ‡, -1, (1) 2 1 ≥ -1 < +∞.
This closes the proof of Proposition 3.7.7.

Proposition 3.7.8. For any coalescence indicator ∈ {0, 1} +1 , we have

Γ †, , (1)1
≥ = O L 2 (1).
In particular, for any test function ∈ B ( 2 ), we also have This implies that, on the event { ≥ }, we have 1

Γ †, , ( )1 ≥ = O L 1 (1).
+1 (1 - 2 +1 ) (X n ) ( ) ( 1 ) (X n ) ( ) -(X n ( 2 )) ≠ 1 1 -( ) / = 1 +1 (1 - 2 +1 ) (X n ) ( ) ( 1 ) (X n ) ( ) -(X n ( 2 )) ≠ 1 1 -( ) / 1 ≠ 1 (1-( ))>0 .
The conclusion follows.

Lemma 3.7.8. For any [2] ∈ ( ) 2 , and for any coalescence indicator ∈ {0, 1} +1 and ∈ {0, 1}, we have almost surely

sup >1 E        [2] +1 ∈ [ ] 2 G †, n ( [2] : +1 , B n , X n ) ( [2] +1 , [2] )1 ≥ W        < +∞.
In particular, we have

sup >1 E        [2] +1 ∈ ( ) 2 G †, n ( [2] : +1 , B n , X n ) ( ) ( [2] +1 , [2] )1 ≥ W        < +∞.
Remark. We remark that in the de nition of Γ †, , , we do not need to investigate for a di erent coalescence indicator in general. The reason that is not set to be ( ) lies in the fact that Lemma 3.7.8 is applied in the proof of Lemma 3.7.9.

Proof. If not mentioned otherwise, the calculations of the random variables is only valid on the event { ≥ }. Recall that, given [2] ∈ ( ) 2 ,

( [2] +1 , [2] ) = 1 1 +1 = 1 1 2 +1 = 1 +(1-) 2 ,
and given W , one has

∀ ∈ [ ], ∼ (•) + (1 -) =1 ( ) (X n ) ( ) (•).
By de nition, one has

G †,1 n ( [2] : +1 , B n , X n ) ≤ 1 +1 2 +1 (X n ) ( 2 ) + 1 +1 (1 - 2 +1 ) (X n ) ( ) ( 1 ) (X n ) ( ) -(X n ( 2 )) ≠ 1 1 -( ) / + 2 +1 (1 - 1 +1 ) (X n ) ( ) ( 2 ) (X n ) ( ) -(X n ( 2 )) ≠ 2 1 -( ) / , whence G †,1 n ( [2] : +1 , B n , X n ) ≤ 1 +1 2 +1 (X n ) ( 2 ) + 2 1 +1 (1 - 2 +1 ) (X n ) ( ) ≠ 1 1 -( ) / + 2 2 +1 (1 - 1 +1 ) (X n ) ( ) ≠ 2 1 -( ) /
.

Since 1 and 2 are both {0, 1}-valued, we deduce that

E        [2] +1 ∈ [ ] 2 G †,1 n ( [2] : +1 , B n , X n ) ( [2] +1 , [2] )1 ≥ W        ≤ 1 1 +(1-) 2 (X n ) ( 2 )1 ≥ + 1 2 +1 =1 (1 - 2 +1 ) 2 ( 1 +(1-) 2 ) ≠ 1 1 -( ) 1 ≥ + 1 +(1-) 2 1 +1 =1 (1 - 1 +1 ) 2 ( 1 ) ≠ 2 1 -( ) 1 ≥ .
By applying Lemma 3.7.6 and considering the convention (3.1), we have

E        [2] +1 ∈ [ ] 2 G †,1 n ( [2] : +1 , B n , X n ) ( [2] +1 , [2] )1 ≥ W        ≤ 1 1 +(1-) 2 (X n ) ( 2 )1 ≥ + 1 2 +1 =1 (1 - 2 +1 ) 2 ( 1 +(1-) 2 ) + 1 -( 1 )
=1 1 -( )

1 ≥ + 1 +(1-) 2 1 +1 =1 (1 - 1 +1 ) 2 ( 1 ) + 1 -( 2 ) =1 1 -( ) 1 ≥ .
The simple fact 0

= (1 -)1 ≥ ≤ ( ) (1 -( )1 ≥ ) for any ∈ [ ] yields E        [2] +1 ∈ [ ] 2 G †,1 n ( [2] : +1 , B n , X n ) ( [2] +1 , [2] )1 ≥ W        ≤ ( 1 ) ( 2 ) ∨ ( 1 ) (X n ) ( 2 )1 ≥ + ( 1 ) 2 +1 =1
(1 -(

+1 )) 2 ( 1 +(1-) 2 ) + 1 -( 1 ) =1 1 -( ) 1 ≥ + ( 1 +(1-) 2 ) 1 +1 =1 (1 -( 1 +1 )) 2 ( 1 ) + 1 -( 2 ) =1 1 -( ) 1 ≥ = ( 1 ) (X n ) ( 2 ) + 4 ( 1 ) ( 1 +(1-) 2 ) + 2 1 ≥ ≤ 7. . . 2

By de nition, since

G †,0 n ( [2] : +1 , B n , X n ) = G ‡ n (X n ) + 1 -1 G †,1 n ( [2] : +1 , B n , X n ) ,
the analysis for the case = 0 is the combination of the case = 1 and the similar reasoning in (3.65), namely, a direct consequence of (3.62). This terminates the proof of Lemma 3.7.8. Lemma 3.7.9. For any [2] ∈ ( ) 2 and for any coalescence indicator ∈ {0, 1} +1 , we have almost surely

∀ 2 +1 ∈ [ ], sup >1 E       1 +1 ∈ [ ] G †, n ( [2] : +1 , B n , X n ) 1 1 +1 = 1 1 ≥ W       < +∞, (3.69)
as well as

∀ 1 +1 ∈ [ ], sup >1 E       2 +1 ∈ [ ] G †, n ( [2] : +1 , B n , X n ) 1 2 +1 = 2 1 ≥ W       < +∞.
Proof. By symmetry of the de nition of G †, n ( [2] : +1 , B n , X n ), it su ces to verify (3.69). In fact, by simple observation, one has

1 1 +1 = 1 = 1 1 +1 = 1 1 1 +1 = 1 ≤ 1 1 +1 = 1 2 +1 =1 1 2 +1 = 1 .
. .

Therefore, one only needs to show that sup

>1 E        [2] +1 ∈ [ ] 2 G †, n ( [2] : +1 , B n , X n ) ( ) ( [2] +1 , [2] )1 ≥ W        < +∞,
which, by taking = ( ), is guaranteed by Lemma 3.7.8. This terminates the proof of Lemma 3.7.9.

Proof of Proposition 3.7.8. Now, we start the proof by induction. It is trivial for the step = 0 as Γ †, 0, (1) = 1. For step ≥ 1, we suppose that sup

>0 E Γ †, -1, (1) 2 1 ≥ -1 < +∞.
By the alternative representation (3.60), for all ≥ 4, we have

E Γ †, , (1)
2 1 ≥ W -1 = ( [2] -1 , [2] -1 ) ∈ ( ( )) ×2 Λ †, -1 [ [2] -1 ] Λ †, -1 [ [2] -1 ]1 ≥ -1 E       ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 , (3.70)
since by de nition of G †, -1 n-1 , one has

E       ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 1 = -1 = 0.
As we have mentioned several times, in the following part of the proof, we omit the notation 1 ≥ -1 . Similar as in the proof of Proposition 3.7.7, the rest of the reasoning relies on the decomposition (3.63).

(i) Case:

( [2]
, [2] ) ∈ ( ) 2 ×2 ∩ [ ] 4 2 : In this case, there are only two distinct random variables in the tuple

1 -1 , 2 -1 , 1 -1 , 2 -1 .
As we have already mentioned in (3.64), there are 2 possible assignments such that we can x two distinct random variables within the tuple above. Without loss of generality, we suppose that 1 -1 and 1 -1 are two distinct random variables, and we x one of these two assignments. Then, when ( 1 , 1 ) varies freely in ( ) 2 , the values of 2 -1 and 2 -1 will be a.s. determined by the chosen assignment and the values of

1 -1 and 1 -1 . E        ( [2]
,

[2] ) ∈ ( ( )

2 ) ×2 ∩[ ] 4 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 =E       ( 1 , 1 ) ∈ ( ) 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 (3.71)
Given W -1 , by the conditional independence of 1 -1 and 1 -1 under the chosen assignment, we have

E       ( 1 , 1 ) ∈ ( ) 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 ≤E       1 ∈ [ ] G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) 1 1 -1 = 1 -1 W -1       E       1 ∈ [ ] G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) 1 1 -1 = 1 -1 W -1       . (3.72)
Then, by applying Lemma 3.7.9, one gets

E       ( 1 , 1 ) ∈ ( ) 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 = O . . (1).
Since the number of di erent assignments 2 does not depend on , one deduces that

E        ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 = O . . (1).
(ii) Case:

( [2]
, [2] ) ∈ ( ) 2 ×2 ∩ [ ] 4 3 : In this case, there are three distinct random variables in the tuple

1 -1 , 2 -1 , 1 -1 , 2 -1 .
As is calculated in (3.68), the number of assignment is 4 at this time. Let us x one assignment. We suppose that

[2] -1 and 1 -1 are three distinct random variables. Then, whilst ( [2] , 1 ) varies freely in ( ) 3 , the value of 2 -1 is a.s. determined by the chosen assignment and the values of (

E        ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 3 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 =E       ( [2] , 1 ) ∈ ( ) 3 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 ≤E       [2] ∈ ( ) 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1       E       1 ∈ [ ] G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) 1 { 1 -1 = 1 -1 } W -1       . (3.73)
Thanks to Lemma 3.7.8 and Lemma 3.7.9, we get

E       ( [2] , 1 ) ∈ ( ) 3 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 = O . . (1).
Again, since the number of the di erent assignments 4 does not depend on , we obtain

E        ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 3 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 = O . . (1).
(3.74) (iii) Case:

( [2]
, [2] ) ∈ ( ) 4 : In this case, all the random variables

1 -1 , 2 -1 , 1 -1 , 2 -1
are distinct. This time, Lemma 3.7.8 gives directly

E       ( [2] , [2] ) ∈ ( ) 4 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 ≤E       [2] ∈ [ ] 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1       E       [2] ∈ [ ] 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1       = O . . (1).
(3.75)

Combining the three cases above, we nally obtain

sup >0 E       ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 < +∞. . .
Returning to (3.70) and induction hypothesis, the veri cation of step is then nished, so as the proof of Proposition 3.7.8.

Proposition 3.7.9. For any test function ∈ B ( 2 ), we have

E Γ ‡, , ( )1 ≥ G -1 = Γ ‡, -1, Q ‡, -1 ˆ ( )1 ≥ -1 .
Proof. First, by the alternative representation we have introduced in (3.57), we have

E Γ ‡, , ( )1 ≥ G -1 = ( [2] -1 ) ∈ ( ) 2 Λ ‡, -1 [ [2] -1 ]1 ≥ -1 E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) G -1       , (3.76)
Thus, it su ces to show that

E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) G -1       1 ≥ -1 =Q ‡, -1 ˆ ( ) ( [2] -1 -1 )1 ≥ -1 .
(3.77)

Alghough it may seem unnecessary, we recall that we have almost surely

-1 G ‡ n-1 (X n-1 ) = 1 -1 2 -1 (X n-1 ) ( -1 ) 2 + 1 -1 (1 - 2 -1 ) (X n-1 ) ( -1 ) 2 + 2 -1 (1 - 1 -1 ) (X n-1 ) ( -1 ) 2 + (1 - 1 -1 ) (1 - 2 -1 ) (X n-1 ) ( -1 ) 2 .
Given W -1 , for any ∈ [ ], the de nition of the Feynman-Kac IPS gives

( -1 , ) ∼ -1 ( -1 ) • ( -1 , ) + (1 --1 ) =1 ( -1 ) ˚ ( -1 , ) (X n-1 ) ( -1 ) . (3.78) (i) Case -1 = 0: Notice that -1 E       [2] ∈ ( ) 2 G ‡ n (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       = 1 -1 -1 2 -1 -1 ⊗2 (X n-1 ) ( ⊗2 -1 ) • ⊗2 ( ) ( [2] -1 -1 ) + 1 1 -1 -1 2 ≠ 1 -1 (1 - 2 -1 ) (X n-1 ) ( -1 ) ( • ⊗ ˚ ) ( ) ( [2] -1 -1 ) + 1 2 -1 -1 1 ≠ 2 -1 (1 - 1 -1 ) (X n-1 ) ( -1 ) ( ˚ ⊗ • ) ( ) ( [2] -1 -1 ) + 1 2 [2] ∈ ( ) 2 (1 - 1 -1 ) (1 - 2 -1 ) ˚ ⊗2 ( ) ( [2] -1 -1 ), which yields E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       = -1 ⊗2 (X n-1 ) ( ⊗2 -1 ) • ⊗2 ( ) ( [2] -1 -1 ) + 1 -1 2 ≠ 1 -1 (1 --1 ( 2 -1 )) (X n-1 ) ( -1 ) ( • ⊗ ˚ ) ( ) ( [2] -1 -1 ) + 1 -1 1 ≠ 2 -1 (1 --1 ( 1 -1 )) (X n-1 ) ( -1 ) ( ˚ ⊗ • ) ( ) ( [2] -1 -1 ) + 1 ( -1) [2] ∈ ( ) 2 (1 --1 ( 1 -1 )) (1 --1 ( 2 -1 )) ˚ ⊗2 ( ) ( [2] -1 -1 ), (3.79)
First, by decomposition (3.2), we noticed that

-1 ⊗2 (X n-1 ) ( ⊗2 -1 ) = 2 (X n ) ( ⊗2 -1 ) + 1 -1 (X n-1 ) ( 2 -1 ). (3.80)
Then, we deduce that

1 -1 2 ≠ 1 -1 (1 --1 ( 2 -1 )) (X n-1 ) ( -1 ) = -1 ⊗2 (X n-1 ) ( -1 ⊗ (1 --1 )) - 1 -1 (1 --1 ( 1 -1 -1 )) (X n-1 ) ( -1 ) = 2 (X n-1 ) ( -1 ⊗ (1 --1 )) + 1 ( -1) =1 -1 ( 1 -1 -1 ) --1 ( -1 ) -1 ( -1 ) = 2 (X n-1 ) ( -1 ⊗ (1 --1 )) + 1 -1 -1 ( 1 -1 -1 ) (X n-1 ) ( -1 ) -(X n-1 ) ( 2 -1 ) .
(3.81)

The exactly same manipulations also give

1 -1 1 ≠ 2 -1 (1 --1 ( 1 -1 )) (X n-1 ) ( -1 ) = 2 (X n-1 ) ( -1 ⊗ (1 --1 )) + 1 -1 -1 ( 2 -1 -1 ) (X n-1 ) ( -1 ) -(X n-1 ) ( 2 -1 ) .
(3.82)

PROOFS

Now, let us put (3.80), (3.81) and (3.82) back into (3.79). One derives

E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       = 2 (X n-1 ) ( ⊗2 -1 ) • ⊗2 + 2 (X n-1 ) ( -1 ⊗ (1 --1 )) • ⊗ ˚ + 2 (X n-1 ) ( -1 ⊗ (1 --1 )) ˚ ⊗ • + 2 (X n-1 ) ((1 --1 ) ⊗2 ) ˚ ⊗2 + 1 -1 (X n-1 ) ( -1 ) ( -1 × • ) ⊗ ˚ + ˚ ⊗ ( -1 × • ) + (X n-1 ) ( 2 -1 ) • ⊗2 - • ⊗ ˚ -˚ ⊗ • ( ( )) ( [2] -1 -1 ), (3.83)
which, by de nition, turns out to be the following equality:

E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       1 ≥ -1 = Q 0 ˆ + 1 -1 Q †,0 ˆ ( ) ( [2] -1
-1 ).

(3.84) (ii) Case -1 = 1:

By the similar calculations done in the previous case, one has

-1 E       [2] ∈ ( ) 2 G ‡ n (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       = 0 + 1 1 -1 -1 2 ≠ 1 -1 (1 - 2 -1 ) (X n-1 ) ( -1 ) ( • ⊗ ˚ ) ( ) ( (1,1) -1 -1 ) + 1 1 -1 -1 1 ≠ 1 -1 (1 - 1 -1 ) (X n-1 ) ( -1 ) ( ˚ ⊗ • ) ( ) ( (1,1) -1 -1 ) + 1 2 [2] ∈ ( ) 2 (1 - 1 -1 ) (1 - 2 -1 ) ˚ ⊗2 ( ) ( (1,1) -1 -1 ), which yields E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       = 0 + 1 -1 2 ≠ 1 -1 (1 --1 ( 2 -1 )) (X n-1 ) ( -1 ) 1 • ⊗ ˚ ( ) ( [2] -1 -1 ) + 1 -1 1 ≠ 1 -1 (1 --1 ( 1 -1 )) (X n-1 ) ( -1 ) 1 ˚ ⊗ • ( ) ( [2] -1 -1 ) + 1 ( -1) [2] ∈ ( ) 2 (1 --1 ( 1 -1 )) (1 --1 ( 2 -1 )) 1 ˚ ⊗2 ( ) ( [2] -1
-1 ).

(3.85)

Then, taking into account the equality (3.81), one obtains a similar equation as (3.83):

E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       1 ≥ -1 = 2 (X n-1 ) ( ⊗2 -1 ) 1 • ⊗2 + 2 (X n-1 ) ( -1 ⊗ (1 --1 )) 1 • ⊗ ˚ + 2 (X n-1 ) ( -1 ⊗ (1 --1 )) 1 ˚ ⊗ • + 2 (X n-1 ) ((1 --1 ) ⊗2 ) 1 ˚ ⊗2 - 2 (X n-1 ) ( ⊗2 -1 ) 1 • ⊗2 + 1 -1 (X n-1 ) ( -1 ) 1 ( -1 × • ) ⊗ ˚ + ˚ ⊗ ( -1 × • ) -(X n-1 ) ( 2 -1 ) 1 • ⊗ ˚ + ˚ ⊗ • ( ( )) ( [2] -1 -1 )1 ≥ -1 .
(3.86) By de nition, we nally obtain that on the event { ≥ -1},

E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       = Q 1 ˆ - 2 (X n-1 ) ( ⊗2 -1 )1 ≥ -1 1 • ⊗ + 1 -1 Q †,1 ˆ ( ) ( [2] -1 -1 ) = Q †,1 ˆ + 1 -1 Q †,1 ˆ ( ) ( [2] -1
-1 ).

(3.87)

Combining the two cases, we conclude that we have proved that

E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       1 ≥ -1 =Q ‡, -1 ˆ ( ) ( [2] -1 -1 )1 ≥ -1 . . . (3.88)
In addition, since

Q ‡, -1 ˆ ( ) ( [2] -1 -1 )1 ≥ -1
is G -1 -measurable, the veri cation of (3.77) is then nished, so as the proof of Proposition 3.7.9.

Proposition 3.7.10. For any test function ∈ B ( 2 ), we have

E Γ †, , ( )1 ≥ W -1 = Γ †, -1, Q †, -1 ˆ ( )1 ≥ -1 .
Proof. By combining Proposition 3.7.9 and the fact that

G †,0 n ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) = G ‡ n ( [2] -1: , B n-1 , X n-1 ) - 1 -1 G †,1 n ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ), . .
it su ces to check the case -1 = 1. Similar as in the proof of Proposition 3.7.9, we consider the alternative representation (3.60), which gives

E Γ †, , ( )1 ≥ W -1 = ( [2] -1 ) ∈ ( ) 2 Λ †, -1 [ [2] -1 ]1 ≥ -1 E       [2] ∈ ( ) 2 G †,1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       , (3.89)
Thus, it su ces to show that

E       [2] ∈ ( ) 2 G †,1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) W -1       1 ≥ -1 = Q †,1 ˆ ( ) ( [2] -1 -1 )1 ≥ -1 .
(3.90)

We omit the notation 1 ≥ -1 in the rest of the proof. Recall that

G †,1 n-1 ( [2] -1: , B n-1 , X n-1 ) = 1 -1 2 -1 (X n-1 ) ( 2 -1 ) + 1 -1 (1 - 2 -1 ) (X n-1 ) ( -1 ) -1 ( 1 -1 -1 ) (X n-1 ) ( -1 ) -(X n-1 ) ( 2 -1 ) ≠ 1 -1 -1 ( -1 ) + 2 -1 (1 - 1 -1 ) (X n-1 ) ( -1 ) -1 ( 2 -1 -1 ) (X n-1 ) ( -1 ) -(X n-1 ) ( 2 -1 ) ≠ 2 -1 -1 ( -1 )
.

Hence, with the de nition (3.78), standard calculation gives

E       [2] ∈ ( ) 2 G †,1 n ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1       = 1 -1 -1 2 -1 -1 (X n-1 ) ( 2 -1 ) • ⊗2 + 1 -1 -1 2 ≠ 1 -1 (1 - 2 -1 ) -1 ( 1 -1 -1 ) (X n-1 ) ( -1 ) -(X n-1 ) ( 2 -1 ) ≠ 1 -1 (1 --1 ( -1 )) • ⊗ ˚ + 2 -1 -1 1 ≠ 2 -1 (1 - 1 -1 ) -1 ( 2 -1 -1 ) (X n-1 ) ( -1 ) -(X n-1 ) ( 2 -1 ) ≠ 2 -1 (1 --1 ( -1 )) ˚ ⊗ • ( ) ( [2] -1 -1 ), whence E       [2] ∈ ( ) 2 G †,1 n ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1       = (X n-1 ) ( 2 -1 ) • ⊗2 + -1 ( 1 -1 -1 ) (X n-1 ) ( -1 ) -(X n-1 ) ( 2 -1 ) • ⊗ ˚ + -1 ( 2 -1 -1 ) (X n-1 ) ( -1 ) -(X n-1 ) ( 2 -1 ) ˚ ⊗ • ( ) ( [2] -1 -1 ), which yields E       [2] ∈ ( ) 2 G †,1 n ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1       1 ≥ -1 = (X n-1 ) ( -1 ) ( -1 × • ) ⊗ ˚ + ˚ ⊗ ( -1 × • ) + (X n-1 ) ( 2 -1 ) • ⊗2 - • ⊗ ˚ -˚ ⊗ • ( ) ( [2] -1 -1 )1 ≥ -1 .
(3.91) By de nition, this gives the desired equality (3.90).

Lemma 3.7.10. For any test function ∈ B ( 2 ) and any coalescent indicator ∈ {0, 1} +1 , we have

Γ ‡, , ( )1 ≥ -Γ ‡, -1, Q ‡, -1 ˆ ( )1 ≥ -1 = O L 1 1 √ .
Proof. Thanks to Cauchy-Schwarz inequality and Proposition 3.7.9, it su ces to verify that

E Γ ‡, , ( ) 2 1 ≥ -Γ ‡, -1, Q ‡, -1 ˆ ( ) 2 1 ≥ -1 = O 1 .
Next, thanks to the alternative representation (3.57), one derives that

E Γ ‡, , ( ) 2 1 ≥ G -1 = ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ ‡, -1 [ [2] -1 ]Λ ‡, -1 [ [2] -1 ]1 ≥ -1 E       ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) ( ) ⊗2 ( [2] , [ 2]
)

G -1       .
To simplify the notation, we denote

1 ( ) := ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 E Λ ‡, [ [2] ]Λ ‡, [ [2] ] ( ) ⊗2 ( [2] , [ 2]
) G -1 1 ≥ , and 2 ( ) :=

( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 E Λ ‡, [ [2] ] ( ) ( [2]
)

G -1 E Λ ‡, [ [2] ] ( ) ( [2]
) G -1 1 ≥ , as well as 3 ( ) :=

( [2] , [2] ) ∈ ( ( ) 2 ) ×2 E Λ ‡, [ [2] ] ( ) ( [2] ) G -1 E Λ ‡, [ [2] ] ( ) ( [ 2]
) G -1 1 ≥

Thanks to the conditional independence between

[2] -1 and

[2] -1 given G -1 , we have 3 ( ) -2 ( ) = ( [2] , [2] ) ∈ ( ) 4 E Λ ‡, [ [2] ] ( ) ( [2] ) G -1 E Λ ‡, [ [2] ] ( ) ( [2] ) G -1 1 ≥ = ( [2] , [2] ) ∈ ( ) 4 E Λ ‡, [ [2] ]Λ ‡, [ [2] ] ( ) ⊗2 ( [2] , [ 2]
) G -1 1 ≥ , whence we deduce that

E Γ ‡, , ( ) 2 1 ≥ G -1 = 1 ( ) -2 ( ) + 3 ( ).
. .

In addition, since one of the by-product (3.88) in the proof of Proposition 3.7.9 yields

Q ‡, -1 ˆ ( ) ⊗2 ( [2] , [2] ) = ( [2] ) ∈ ( ( ) 2 ) ×2 E G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [ 2]
) G -1

( [2] ) ∈ ( ( ) 2 ) ×2 E G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [ 2]
)

G -1 = ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 E G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) G -1 E G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) G -1 , one has 3 ( ) = Γ ‡, -1, Q ‡, -1 ˆ ( ) 2 1 ≥ -1 , . . which guarantees that E [ 3 ( )] = E Γ ‡, -1, Q ‡, -1 ˆ ( ) 2 1 ≥ -1 .
Therefore, it su ces to verify that

E [ 1 ( )] = O 1 and E [ 2 ( )] = O 1 . (3.92)
Together, we prove both of the two convergence above by induction, as the proofs share the same mechanism. Without loss of generality, we suppose that ≡ 1. For = 0, standard calculations give

1 ( ) = 2 ( ) = 4 -6 ( -1) = O 1 .
For ≥ 1, we suppose that

E        ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ ‡, -1 [ [2] -1 ]Λ ‡, -1 [ [2] -1 ]1 ≥ -1        = O 1 .
Now, it is time to go back to the decomposition (3.63). As is mentioned for many times, we may omit the notation 1 ≥ -1 .

(i) Case:

( [2] , [2] ) ∈ ( ) 2 ×2 ∩ [ ] 4
2 : As we have seen in the proof of Proposition 3.7.7, there are 2 di erent assignments

1 -1 = 1 -1 , 2 -1 = 2 -1 and 1 -1 = 2 -1 , 2 -1 = 1 -1 , such that two distinct random variables can be found in 1 -1 , 2 -1 , 1 -1 , 2 -1 .
Let us x one assignment. This time, in order to execute a ner analysis, we suppose that 1 and 1 vary freely in ( ) 2 and the values of 2 -1 and 2 -1 will be almost surely determined by the values of 1 -1 and 1 -1 . Since the potential function G ‡ n and indicator functions are both nonnegative, we extend ( ) 2 to [ ] 2 , which gives the following inequality:

E       ( 1 , 1 ) ∈ ( ) 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1       ≤E       1 ∈ [ ] G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) G -1       E       1 ∈ [ ] G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) G -1       1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} .
(3.93) Now, we explain why there is an indicator function

1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}}
at the r.h.s. above. For the case -1 = 0, one has

-1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) = 1 1 -1 = 1 -1 ≠ 2 -1 = 2 -1 1 1 -1 = 1 -1 ≠ 2 -1 = 2 -1 . Hence, since ( [2]
, [2] ) ∈ (( ) 2 ) ×2 \( ) 4 , we have

#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 } = #{ 1 -1 -1 , 2 -1 -1 , 1 -1 -1 , 2 -1 -1 } ≤ #{ 1 , 2 , 1 , 2 } < 4.
At the same time, when -1 = 1, one has

-1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) = 1 1 -1 = 1 -1 = 2 -1 ≠ 2 -1 1 1 -1 = 1 -1 = 2 -1 ≠ 2 -1
.

Since 1 ≠ 2 and 1 ≠ 2 , we have

∃ , ∈ {1, 2}, . . = .
In no matter which case mentioned above, it is necessary that 1

-1 = 1 -1 . Therefore, one also gets #{ 1 -1 , 2 -1 , 1 -1 , 2 -1 } < 4.
The arguments above will be applied repeatedly in the rest of the proof. Next, by the same procedure in the proof of Proposition 3.7.7, we obtain

E        ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 2 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1        ≤ 1 × 1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} ,
where 1 denotes positive constant which does not depend on . Meanwhile, by the same procedure, we also have

( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 2 E -1 ( [2] -1 , [2] -1 ) G -1 G ‡ n-1 (X n-1 ) 2 E -1 ( [2] -1 , [2] -1 ) G -1 ≤ 1 × 1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}}
, where 1 is also a positive constant which does not depend on .

(ii) Case:

( [2]
, [2] ) ∈ ( ) 2 ×2 ∩ [ ] 4 3 : Again, similart in the proof of Proposition 3.7.7, there are 4 di erent assignments such that there are 3 distinct random variables within

1 -1 , 2 -1 , 1 -1 , 2 -1 .
Let us x one assignment. As is done in the previous case, to conduct a ner study, we suppose that ( [2] , 1 ) vary freely in ( ) 3 and the values of 2 -1 will be almost surely determined by the choice of assignment and the values of -1 and 1 -1 . This time, by extending ( )

3 to ( ) 2 ∪ [ ], we get E       ( 1 , 1 ) ∈ ( ) 3 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) ( ) ⊗2 ( [2] , [ 2]
)

G -1 ≤E       [2] ∈ ( ) 2 G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) G -1       E       1 ∈ [ ] G ‡ n-1 (X n-1 ) -1 ( [2] -1 , [2] -1 ) ( ) ( [2] ) G -1       1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} .
(3.94)

Again, by similar argument given in the proof of Proposition 3.7.7, we get

E        ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 3 G ‡ n-1 (X n-1 ) 2 -1 ( [2] -1 , [2] -1 ) -1 ( [2] -1 , [2] -1 ) G -1        ≤ 2 × 1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} , and
( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 3 E -1 ( [2] -1 , [2] -1 ) G -1 G ‡ n-1 (X n-1 ) 2 E -1 ( [2] -1 , [2] -1 ) G -1 ≤ 2 × 1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}}
, where 2 and 2 are positive constant which does not depend on .

Combining both cases, one gets

1 ( ) ≤ ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ ‡, -1 [ [2] -1 ]Λ ‡, -1 [ [2] -1 ]1 ≥ -1 ( 1 + 1 )1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} = ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ ‡, -1 [ [2] -1 ]Λ ‡, -1 [ [2] -1 ]1 ≥ -1 ( 1 + 1 ), . . and 2 ( ) ≤ ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ ‡, -1 [ [2] -1 ]Λ ‡, -1 [ [2] -1 ]1 ≥ -1 ( 2 + 2 )1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} = ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ ‡, -1 [ [2] -1 ]Λ ‡, -1 [ [2] -1 ]1 ≥ -1 ( 2 + 2 ). . .
The desired convergence (3.92) are therefore guaranteed by the induction hypothesis by taking the expectation on both sides of the inequalities above. This is the end of the proof of Lemma 3.7.10.

Lemma 3.7.11. For any test function ∈ B ( 2 ) and any coalescent indicator ∈ {0, 1} +1 , we have

Γ †, , ( )1 ≥ -Γ †, -1, Q †, -1 ˆ ( )1 ≥ -1 = O L 1 1 √ .
Proof. Before starting, we mention that this proof bears a resemblance to the one of Lemma 3.7.10. Thanks to Cauchy-Schwarz inequality, it is su cient to verify that

E Γ †, , ( ) 2 1 ≥ -Γ †, -1, Q †, -1 ˆ ( ) 2 1 ≥ -1 = O 1 .
Again, similar to the equation (3.70), by the alternative representation (3.60) and decomposition (3.59), we deduce that

E Γ †, , ( ) 2 1 ≥ W -1 = ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ †, -1 [ [2] -1 ] Λ †, -1 [ [2] -1 ]1 ≥ -1 E ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) ⊗2 ( [2] , [ 2]
) W -1 .

Similar to the previous case, we denote 1 ( ) :=

( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 E Λ †, [ [2] ] Λ †, [ [2] ] ⊗2 ( [2] , [ 2]
) G -1 1 ≥ , and 2 ( ) :=

( [2] , [2] ) ∈ ( ( ) 2 ) ×2 \( ) 4 E Λ †, [ [2] ] ( [2] ) W -1 E Λ †, [ [2] ] ( [ 2]
) W -1 1 ≥ , as well as

3 ( ) := ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 E Λ †, [ [2] ] ( [2] ) W -1 E Λ ‡, [ [2] ] ( [ 2]
) W -1 1 ≥ Thanks to the conditional independence between

[2] -1 , [2]
-1 and

[2] -1 , [2] -1 given W -1 , we have 3 ( ) -2 ( ) = ( [2] , [2] ) ∈ ( ) 4 E Λ †, [ [2] ] ( [2] ) W -1 E Λ †, [ [2] ] ( [2] ) W -1 1 ≥ = ( [2] , [2] ) ∈ ( ) 4 E Λ †, [ [2] ] Λ †, [ [2] ] ⊗2 ( [2] , [ 2]
) W -1 1 ≥ , from which we get

E Γ †, , ( ) 2 1 ≥ W -1 = 1 ( ) -2 ( ) + 3 ( ).
. .

Notice that Proposition 3.7.10 gives, on the event { ≥ -1},

( [2] , [2] ) ∈ ( ( ) 2 ) ×2 E G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( [2] ) W -1 E G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( [2] ) W -1 = Q †, -1 ˆ ( ) ⊗2 ( [2] -1 -1 , [2] -1 -1 ).
Hence, one has

3 ( ) = Γ †, -1, Q †, -1 ˆ ( ) 2 1 ≥ -1 , . . which guarantees that E [ 3 ( )] = E Γ †, -1, Q †, -1 ˆ ( ) 2 1 ≥ -1 .
Therefore, it su ces to verify that

E [ 1 ( )] = O 1 and E [ 2 ( )] = O 1 . (3.95)
Without loss of generality, we suppose that ≡ 1. The rest of the proof is done by induction. For = 0, by de nition, we have

1 ( ) = 2 ( ) = 4 -6 ( -1) = O 1 .
For ≥ 1, we suppose that

E        ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ †, -1 [ [2] -1 ] Λ †, -1 [ [2] -1 ]1 ≥ -1        = O 1 .
As is stated many times, we may omit the notation 1 ≥ -1 since it is W -1 -measurable. Once again, let us return to the decomposition (3.63). Since the essential idea is highly repetitive w.r.t. the reasoning in the proof of Proposition 3.7.10, we skip some of the unnecessary details in the rest of the proof.

(i) Case:

( [2] , [2] ) ∈ ( ) 2 ×2 ∩ [ ] 4
2 : By the same procedure given in the proof of Proposition 3.7.8 and Lemma 3.7.10, we obtain

E        ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 2 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) ⊗2 ( [2] , [ 2]
)

W -1 1 ≥ -1 ≤ 1 1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} , and
( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 2 E G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 E G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 1 ≥ -1 ≤ 1 1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} ,
where 1 and 1 are some constant that does not depend on .

(ii) Case:

( [2] , [2] ) ∈ ( ) 2 ×2 ∩ [ ] 4 3 : This time, we get E        ( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 3 G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) ( ) -1 ( [2] -1 , [2] -1 ) ⊗2 ( [2] , [ 2]
)

W -1 1 ≥ -1 ≤ 2 1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} , and
( [2] , [2] ) ∈ ( ( ) 2 ) ×2 ∩[ ] 4 3 E G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 E G †, -1 n-1 ( [2] -1: , B n-1 , X n-1 ) ( ) -1 ( [2] -1 , [2] -1 ) W -1 1 ≥ -1 ≤ 2 1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}}
, where 2 and 2 are some constant that does not depend on .

By combining the both cases, we establish that

1 ( ) ≤ ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ †, -1 [ [2] -1 ] Λ †, -1 [ [2] -1 ]1 ≥ -1 ( 1 + 1 )1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} = ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ †, -1 [ [2] -1 ] Λ †, -1 [ [2] -1 ]1 ≥ -1 ( 1 + 1 ), . . and 2 ( ) ≤ ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 Λ †, -1 [ [2] -1 ] Λ †, -1 [ [2] -1 ]1 ≥ -1 ( 2 + 2 )1 {#{ 1 -1 , 2 -1 , 1 -1 , 2 -1 <4}} = ( [2] -1 , [2] -1 ) ∈ ( ( ) 2 ) ×2 \( ) 4 Λ †, -1 [ [2] -1 ] Λ †, -1 [ [2] -1 ]1 ≥ -1 ( 2 + 2 ). . .
By taking the expectation on both sides of the inequalities above, the desired convergence (3.92) are then veri ed thanks to the induction hypothesis. The conclusion follows.

Lemma 3.7.12. For any test function ∈ B ( ) ⊗2 and any coalescent indicator ∈ {0, 1} +1 , we have

Γ ‡, -1, Q ‡, -1 ˆ ( ) -Γ ‡, -1, Q ‡, -1 ( ) = O L 1 1 √ , as well as Γ †, -1, Q †, -1 ˆ ( ) -Γ †, -1, Q †, -1 ( ) = O L 1 1 √ .
Proof. First, we noticed that for any test functions 1 , 2 ∈ B ( ) ⊗2 , Minkowski's inequality gives

Γ ‡, -1, (1)1 ≥ -1 ( -1 ) 2 ( 1 + 2 )1 ≥ -1 -⊗2 -1 ( 1 + 2 ) L 1 ≤ Γ ‡, -1, (1)1 ≥ -1 ( -1 ) 2 ( 1 )1 ≥ -1 -⊗2 -1 ( 1 ) L 1 + Γ ‡, -1, (1)1 ≥ -1 ( -1 ) 2 ( 2 )1 ≥ -1 -⊗2 -1 ( 2 ) L 1 .
(3.96)

Second, thanks to Cauchy-Schwarz inequality, Proposition 3.7.5 and Proposition 3.7.7, we deduce that

Γ ‡, -1, (1)1 ≥ -1 ( -1 ) 2 ( 1 )1 ≥ -1 -⊗2 -1 ( 1 ) L 1 ≤ Γ ‡, -1, (1)1 ≥ -1 L 2 ( -1 ) 2 ( 1 )1 ≥ -1 -⊗2 -1 ( 1 ) L 2 =O 1 √ .
(3.97)

PROOFS

Similarly, thanks to Proposition 3.7.8, we also have

Γ †, -1, (1)1 ≥ -1 ( -1 ) 2 ( 1 + 2 )1 ≥ -1 -⊗2 -1 ( 1 + 2 ) L 1 ≤ Γ †, -1, (1)1 ≥ -1 ( -1 ) 2 ( 1 )1 ≥ -1 -⊗2 -1 ( 1 ) L 1 + Γ †, -1, (1)1 ≥ -1 ( -1 ) 2 ( 2 )1 ≥ -1 -⊗2 -1 ( 2 ) L 1 . (3.98) and Γ †, -1, (1)1 ≥ -1 ( -1 ) 2 ( 1 )1 ≥ -1 -⊗2 -1 ( 1 ) L 1 ≤ Γ †, -1, (1)1 ≥ -1 L 2 ( -1 ) 2 ( 1 )1 ≥ -1 -⊗2 -1 ( 1 ) L 2 =O 1 √ .
(3.99)

Finally, returning to two pairs of decompositions (3.76), (3.86) and (3.89), (3.91), the boundness of -1 and the homogeneous structure in these two decompositions allow us to apply respectively (3.96), (3.97) and (3.98), (3.99). Note that (X n-1 ) ( -1 ) = 2 (X n-1 ) (1 ⊗ -1 ), and (X n-1 ) ( 2 -1 ) = 2 (X n-1 ) (1 ⊗ 2 -1 ). The desired L 1 -bound can therefore be obtained with some standard algebraic manipulations.

Chapter 4

Estimating Committor Function with Mondrian Forests abstract: In Molecular Dynamics, the rare event of interest can be modeled by the transition of some underlying Markov process between metastable states. In Transition Path Theory, a very important model reduction technique is to design a reaction coordinate, which is a function that measures the advance of a reactive trajectory towards a metastable state. Let and be two metastable states. A committor function is the perfect choice of reaction coordinate in the sense that it measures exactly the probability of reaching before . The committor function is also, in some sense, the perfect reaction coordinate for generalized Adaptive Multilevel Splitting (gAMS, see [BGG + 16]) and crucial to the performances of many other rare-event estimation algorithms. We investigate the performance of Mondrian Forests (MF) [START_REF] Balaji Lakshminarayanan | Mondrian forests: E cient online random forests[END_REF] to estimate the committor function, and we provide strategies to couple gAMS iteratively: rst, since gAMS can also provide information on the committor function, we can use this algorithm to generate training data for MF, and, conversely, to update gAMS by using the trained MF model as its reaction coordinate. As iterations go, the updated-gAMS to generate the training data of better quality, with with, MF is expected to be able to yield a good approximation of the committor function.

Introduction

In Transition Path Theory (TPT, see, e.g., [START_REF] Weinan | Transition-path theory and path-nding algorithms for the study of rare events[END_REF]), a typical problem is to sample the transition paths between a metastable state and another metastable state (see Figure 4.1). More precisely, let denote a state space in which the underlying dynamics is modeled by a Markov process X := ( ; ≥ 0). A metastable state of X is an open subset of such that when X is trapped in such set, it takes an extremely long time for X to escape (see Figure 4.1). Let , be two metastable sets in , then the committor function at point is the probability that X starting from , reaches before . By the metastability of and , when evaluating the values of the committor function close to , crude Monte Carlo typically fails because the process is very much likely to go back to rather than to go to : the probability to be evaluated is thus very small. This metastability problem is related to a time scale problem: although the molecular transitions are not so rare at the macroscopic timescale, the dynamics encoded in the Markov process can only be simulated w.r.t. femtosecond timescale. Therefore, in order to ensure a certain level of accuracy, the wallclock time of simulation is typically intractable.

One popular numerical approach to counter this is the gAMS framework [BGG + 16]. The basic idea is to generate an Interacting Particle System (IPS) based on an adaptive level updating strategy, where the trajectories that advance more survive. A level is calculated w.r.t. a reaction coordinate, which is of crucial importance to the performance of the algorithm. It is well-known (see, e.g., [START_REF] Bréhier | Analysis of adaptive multilevel splitting algorithms in an idealized case[END_REF][START_REF] Cérou | Adaptive multilevel splitting: Historical perspective and recent results[END_REF]) that the committor function is the optimal reaction coordinate for gAMS. In addition, since gAMS is able to evaluate e ciently the values of the committor function close to , the idea of designing a regressor to estimate the committor function is therefore natural.

An elementary approach is to split the state space or some compact subset based on a regular grid cells, and a natural approximation of the committor function can then be derived in each cell thanks to ensemble of paths generated by the gAMS algorithm. The reader is referred to [START_REF] Laura | Analysis of the adaptive multilevel splitting method on the isomerization of alanine dipeptide[END_REF] for a rich list of numerical experiments and a concrete application on alanine dipeptide. However, this natural construction will only work in low dimensional settings, since it is not possible to create a uniform mesh when the dimension of the state space is large. Intuitively, a possible generalization of such a method is to nd an intelligent way to create an adaptive mesh, such that it may also work in a high-dimensional setting.

Mondrian Forests (MF) [START_REF] Balaji Lakshminarayanan | Mondrian forests: E cient online random forests[END_REF], a variant of Random Forests (RF), i.e., an ensemble of randomized decision trees, are proposed as the regressor of committor function. The construction of MF are based on a stochastic process called Mondrian process (MP, see, e.g., [START_REF] Daniel | The mondrian process[END_REF]), taking values in guillotine partitions of an axis-aligned box. Roughly speaking, a Mondrian process is a high-dimensional generalization of the partition on an interval that is split by a Poisson process. Said di erently, MF provide an intelligent way to create "randomized uniform mesh" that is tractable in a high-dimensional setting. From a di erent angle, MF can be regarded as an RF-based regressor, such that online learning is available. More concretely, when new training data come, there is no need to retrain the model in order to have a better quality regressor. This property is crucial for our iterative updating strategy (cf. Unfortunately, the theoretical understanding of MF are still in its infancy. To the interested readers, we refer to [START_REF] Mourtada | Universal consistency and minimax rates for online mondrian forests[END_REF] and [START_REF] Mourtada | Minimax optimal rates for mondrian trees and forests[END_REF] for recent theoretical developments on MF, where a Purely Random Forests version of MF are proposed, along with a min-max convergence rate analysis. Therefore, we mainly provide numerical illustrations and ideas on how to use MF to design e cient strategy to estimate the committor function.

Setting

Overdampled Langevin dynamics

We consider an overdamped Langevin process X = ( ; ≥ 0) taking values in the state space = R de ned by = -∇ ( ) + 2 -1 , (4.1)

where ( ; ≥ 0) denotes a -dimensional Wiener process, denotes the associated energy and the inverse temperature ( ) -1 is denoted by . A metastable state is an open subset of such that when X is trapped in such set, it takes an extremely long time for X to escape. Let us denote and two metastable states in a state space . For the Markov process X := ( , ≥ 0), let and denote respectively the stopping times

On the choice of regressors

The committor function is a smooth function, taking values in [0, 1]. In this sense, any stateof-the-art machine learning model may serve as a regressor the committor function. However, when interacting with gAMS methods, certain practical aspects have to be taken into account when choosing the learning method:

• First, the regressor has to be fast in terms of prediction speed. In fact, gAMS algorithm requires that at each state of each trajectory, the reaction coordinate is evaluated. This is a huge amount of computation costs if the prediction of the regressor is complicated, and most of them would not be useful. In this sense, a sophisticated ne-tuned Deep Learning model may not be a relevant candidate.

• Second, the regressor must be able to execute online learning or incremental learning.

When new data come, the regressor should be able to update in order to have better accuracy. In this sense, classic Random Forests may not be a good choice.

• Finally, we expect it to be as adaptive and robust as possible in high dimension, since we do not have much insights on how to tune a speci c model to learn a high-dimensional function.

As such, we may consider two families of regressors in order to estimate the committor function e ciently. The advantages and disadvantages are listed as follows.

(i) Gradient-based learning algorithms, such as logistic regression, XGBoosting, or some shallow neural network:

(a) Online learning is available by a Stochastic Gradient Descent-based optimization algorithm;

(b) It is easy to add regularization modules, such as the "smoothness" of the prediction, etc;

(c) The parameters of the algorithm are hard to tune in general;

(d) Sometimes, the results are hard to interpret.

(ii) Ensemble-based learning algorithms, mainly variants of Random Forests:

(a) The ensemble methods are super robust in high-dimensional settings by the design of Monte Carlo-typed structure;

(b) Nearly no tuning is needed, which also means that the regularization modules are not easy to implement in general;

(c) Online learning is generally di cult to design;

(d) It is straightforward to understand and control possible dangerous situations. automatically since the minimum rectangle that contains one point degenerates to a point. In the nal stage, a randomized decision tree is therefore constructed, along with a random partition of the state space R 2 . In a high-dimensional setting, the orthogonal lines used to execute the splitting procedures are replaced by hyperplanes, and the rectangles are replaced accordingly by hypercubes. The basic mechanism remains the same. The prediction of MT is then provided following the decision tree. More precisely, the prediction on the point * is de ned as the average of the outputs of the training data that are in the same hypercube as * . The online learning of MT exploits the memoryless property of exponential distribution. Indeed, when new data come, one regenerates part of the decision tree such that the existing structure is not changed. It turns out that the online training of a Mondrian tree does not change its posterior distribution given the same data (cf. (4.4)), and the reader is referred to Section 5 of [START_REF] Balaji Lakshminarayanan | Mondrian forests: E cient online random forests[END_REF] for detailed algorithms.

A brief introduction to Mondrian Forests

Theoretical aspects of MF

In this section, we provide some theoretical properties of MF, highlighting the missing part in the theory. Hoe ding inequality and Borel-Cantelli lemma give the following law of large numbers as the prediction is uniformly bounded by 1 for the committor function estimation problem:

Pred T [ ] [ ] . . -----→ →∞ E Pred T 1 ( ) [ ] .
At the same time, we have

Var Pred T [ ] [ ] = 1 Var Pred T [ ]
.

The nal prediction is then more consistent than the prediction made by each regressor. Hence, can be set as large as possible in practice such that we only have to deal with the randomness introduced by a single MT.

Missing parts in the convergence analysis As is shown above, the variance brought by ensemble methods can be reduced simply by adding more MT in MF. However, for the expectation of the prediction provided by MT given data, the consistency is not guaranteed in general. Let us give a more detailed setting. Assume that the distribution of writes, (4.3)

The assumption that is Gaussian is quite natural since both crude Monte Carlo and gAMS provide estimation with normal limit distribution. Recent results [START_REF] Mourtada | Universal consistency and minimax rates for online mondrian forests[END_REF] show that when < +∞, there is no consistency in general. A more re ned analysis and a variant of Purely Random Forests version of MF are proposed in [START_REF] Mourtada | Minimax optimal rates for mondrian trees and forests[END_REF], with a min-max rate for -Hölder functions. However, we did not use this variant since the construction of the purely random Mondrian tree allows empty cell, and 0 is set to be the estimation when the evaluation is needed to be conducted in such cells. This induces problems in the implementation of gAMS algorithm since = 0 means a sudden death of a transition path, which makes the implementation of gAMS numerically unstable.

Since providing a huge amount of training data is not possible due to the computational cost of gAMS methods, we are more interested by the performance when relatively few training data is provided. Hence, asymptotic properties such as consistency are not the priority for the applications in order to improve the performance of gAMS, since the nal estimation of rareevent simulation is eventually estimated by gAMS, and even with non-converged estimation of committor function, we still have a theoretical-guaranteed unbiased estimator. In addition, since the construction of decision tree-type regressor ensures that the estimator can only take nite values, gAMS enters into Asymmetric SMC framework introduced in Chapter 3 and can also provide theoretical guaranteed consistent estimations.

Capability of recovering from a ruined model

By a "ruined model" we mean the model generated using extremely low quality training data. We start by training a MF model with largely perturbed data, and then, we continue by providing perfect data to the MF model, to see if it could recover from the ruined model. As presented in Figure 4.13, it may be di cult for an MF model to recover from a ruined one, meaning that a huge amount of high quality training data is required. According to Figure 4.10, a high quality training data of size 1000 is enough to provide an accurate approximation.

Conclusion of the numerical tests

In this section, we summarize the empirical knowledge on MF that we have collected in the numerical tests:

(i) MF do not need a huge sample size to give a reasonable approximation of committor functions;

(ii) MF prefer having less high quality data rather than more low quality data;

(iii) MF are not robust to largely perturbed data;

(iv) It is di cult for a ruined MF model to recover by updating through online learning with high quality training data. As a consequence, in the design of the iterative updating strategy mentioned in the previous sections, it is crucial to ensure the quality of the training data for MF. This intuition is contrary to the one in typical Machine Learning applications, where the size of the data is usually of top priority.

Iterative updating strategy with gAMS and MF

In this section, we discuss some possible combinations of MF and gAMS algorithm, and explain how they can help each other in order to improve accuracy. Some numerical illustrations are also provided.

Crude interaction between gAMS and MF

Let us start by using gAMS to estimate committor function in a crude way: we use the estimation provided by gAMS as the training data of MF, with some pre xed reaction coordinate such as the Euclidean distance to the state . The number of replicas is = 100, and the minimum number of replicas to be killed at each iteration is * = 20. The reaction coordinate 1 is the Euclidean distance to the point (-1, 0) with the threshold * = 1.8. The starting point 0 is sampled uniformly over the rectangle Ω. In particular, when the sampled point turns out to be in (resp. ), we provide directly 0 (resp. 1) as the output of gAMS. At each point, we simulate independently = 50 runs of gAMS, and the nal estimation is the average. The prediction given by the MF model is presented in Figure 4.14. Then, we use the trained MF model as the new reaction coordinate and the updated gAMS is therefore implemented to generate new training data for MF. The nal performance of the MF model is provided in Figure 4.15.

Clearly, the quality of approximation is improved. This procedure can be done repeatedly and we expect the approximation of the committor function to be more and more accurate.

Tempering

In the previous section, we need to have a reasonable reaction coordinate in order to run gAMS in the rst place. This is not always tractable, especially in high dimensional situations. Here, we provide a possible strategy that allows to "start out of nowhere". The idea is simple: when the temperature is high, the rare events associated to the energy barriers become less rare. We start with a small inverse temperature 0 , such that crude Monte Carlo provides reasonable estimations. Therefore, we estimate the associated committor function 0 by crude Monte Carlo. Next, we consider a tempering sequence (see Figure 4.16) 0 < 1 , < • • • < * -1 < * = * , where * denotes the target inverse temperature. For each ∈ [ ], we estimate by using -1 as reaction coordinate, until we get the target inverse temperature. Although this strategy is computationally intensive, it allows to design gAMS algorithm in a much more adaptive and optimal way.

Discussions

In this chapter, we mainly discussed the possibility of using MF to estimate committor functions. This paves the way to a wide range of possible interacting strategies of gAMS and MF. However, a more re ned study still needs to be conducted.

The rst remark is on the sample points of the training data. In high-dimensional settings, the uniform sampling strategy is typically intractable. In fact, one does not need to estimate the whole picture of the committor function if one only seeks to improve the performance of gAMS. A possible approach is to sample along the reactive trajectories generated by gAMS. In this way, it is expected that the prediction of MF to be more accurate along these transition paths, which yields a more accurate gAMS estimator. However, to develop a proper sampling strategy for the sample points is not trivial. For example, for the overdamped Langevin dynamics, the reactive trajectories stay longer in the well of the energy landscape, which means that by uniformly sampling the points along the reactive trajectories is not e cient in general. Hence, the details still need to be investigated.

The second remark is on hybrid approaches that combines gAMS and crude Monte Carlo. In fact, the implementation of gAMS is mainly due to the behaviors of committor function close to the metastable states and . We remark that when close to the rare event is of form 1 - * , namely the values of the committor function is very close to 1. This means that in a lot of places where the values of the committor function is around 0.5, crude Monte Carlo is already able to provide reasonable estimations. Therefore, the design of an adaptive hybrid approach may greatly increase the e ciency of the algorithm.

The nal remark is on Mondrian Forests. Since it is proved that nite lifetime parameter yields non-consistent estimator, it would be interesting to explore another alternative stopping criterion for the growth of MT, in order to improve its robustness against largely perturbed data. One possible choice is standard in a Random Forests context, that is to x a threshold to control the minimum number of data in each subset of the partition of the decision tree. Said di erently, we x a number min in N such that when a subset in the partition contains less data than min , the split is rejected. The ideal case is that we can somehow design a strategy such that this number min can be evaluated by the variance of the output of the data. Since variance estimation is available for both crude Monte Carlo and gAMS with MF as reaction coordinate, it would then be possible to develop more automated and advanced algorithms.

CONTENTS3

  Asymmetric Sequential Monte Carlo 65 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2 SMC framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2.2 Interacting particle system . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.3 Asymptotic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.4 Genealogy and survival history . . . . . . . . . . . . . . . . . . . . . . . 75 3.3 Variance estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.3.1 Asymptotic variance expansion . . . . . . . . . . . . . . . . . . . . . . . 76 3.3.2 Term by term asymptotic variance estimators . . . . . . . . . . . . . . . 77 3.3.3 Unbiased non-asymptotic variance estimator . . . . . . . . . . . . . . . 80 3.3.4 Connection between the estimators . . . . . . . . . . . . . . . . . . . . . 80 3.3.5 E cient asymptotic variance estimators . . . . . . . . . . . . . . . . . . 81 3.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.4.1 Unbiasedness condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.4.2 Other comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.5 Algorithms to compute variance estimators . . . . . . . . . . . . . . . . . . . . . 84 3.6 Coalescent tree-based expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.6.1 Original coalescent tree-based measures . . . . . . . . . . . . . . . . . . 90 3.6.2 Coalescent Feynman-Kac kernels . . . . . . . . . . . . . . . . . . . . . . 91 3.6.3 Binary decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.6.4 Coalescent tree occupation measures . . . . . . . . . . . . . . . . . . . . 96 3.6.5 Feynman-Kac measures ow in a random environment. . . . . . . . . . 99 3.6.6 E cient estimator of Γ †,( ) . . . . . . . . . . . . . . . . . . . . . . . . . 101 1.1 An illustration of Hidden Markov Models. . . . . . . . . . . . . . . . . . . . . . 1.2 The big picture of Subset Simulation. . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The construction of a P -invariant kernel. . . . . . . . . . . . . . . . . . . . . . 1.4 Feynman-Kac measures ow for Asymmetric SMC. . . . . . . . . . . . . . . . . 1.5 Schematic picture of a 2-dimensional toy example with the level sets of (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 An example of reaction coordinate : ( , ) ↦ → . . . . . . . . . . . . . . . . . . 1.7 An illustration of that enters into gAMS framework. . . . . . . . . . . . . . . 2.1 A representation of the coalescent tree-based measure Γ ( ) . . . . . . . . . . . . . 2.2 An IPS with + 1 = 7 levels and = 5 particles at each level. . . . . . . . . . . .

  5 = (1, 3). . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Illustration of the Feynman-Kac measures ow. . . . . . . . . . . . . . . . . . . 3.2 A representation of the original coalescent tree-based measure Γ ( ) . . . . . . . . 4.1 Schematic picture of transition paths. . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The illustration of iterative updating strategy. . . . . . . . . . . . . . . . . . . . 4.3 Representation of the three-hole energy landscape. . . . . . . . . . . . . . . . . 4.4 The numerical solution of (4.2) with inverse temperature = 1.67. . . . . . . . . v vi LIST OF FIGURES 4.5 First split in the construction of MT. . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Decision tree corresponding to the rst split. . . . . . . . . . . . . . . . . . . . . 4.7 Second split in the construction of MT. . . . . . . . . . . . . . . . . . . . . . . . 4.8 Second and third splits in the construction of MT. . . . . . . . . . . . . . . . . . 4.9 The decision tree with three splits. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Learning with the ground truth, where the training data is uniformly sampled over Ω = [-2, 2] × [-2, 2]. For the parameters of MF, we set = ∞ and = 50. The inverse temperature is = 1.67. . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Learning with slightly perturbed data, where ( ) are uniformly sampled over Ω = [-2, 2] × [-2, 2]. For the parameters of MF, we set = ∞ and = 50. The inverse temperature is = 1.67. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Learning with largely perturbed data, where ( ) are uniformly sampled over Ω = [-2, 2] × [-2, 2]. For the parameters of MF, we set = ∞ and = 50. The inverse temperature is = 1.67. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 The gure on the left represents the prediction made by a MF model trained by 100 largely perturbed data. The one on the right represents the model trained with 1000 more perfect data. ( ) are uniformly sampled over Ω = [-2, 2] × [-2, 2]. For the parameters of MF, we set = ∞ and = 50. The inverse temperature is = 1.67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Learning with the training data generated by gAMS algorithm. The reaction coordinate is set to be the distance to the state (i.e., 1 ). The inverse temperature is = 1.67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Learning with the data re-generated by gAMS algorithm. The reaction coordinate is the trained MF model. The inverse temperature is = 1.67. . . . . . . . . . . 4.16 Illustrations of committor functions for di erent temperatures. . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: An illustration of Hidden Markov Models.

=

  * : rare event of interest.

Figure 1 . 2 :

 12 Figure 1.2: The big picture of Subset Simulation.

.Figure 1 . 3 :

 13 Figure 1.3: The construction of a P -invariant kernel.

Figure 1 . 4 :

 14 Figure 1.4: Feynman-Kac measures ow for Asymmetric SMC.

Figure 1 . 6 :

 16 Figure 1.6: An example of reaction coordinate : ( , ) ↦ → .

Figure 1 . 7 :

 17 Figure 1.7: An illustration of that enters into gAMS framework.

  By exploiting some summary statistics :↦ → R , such that, for all ≥ 0, ( ) = * , we consider the adaptive Feynman-Kac particle models identi ed by the McKean kernel , ( , ) := , ( -1 ) ( )

Figure 2 . 2 :

 22 Figure 2.2: An IPS with + 1 = 7 levels and = 5 particles at each level.

Figure 2 . 3 :

 23 Figure 2.3: The rst family of [2]

Figure 2 . 4 :

 24 Figure 2.4: The second family of [2]

Figure 2 . 5 :

 25 Figure 2.5: An illustration of the duality formula recursively applied to a mini IPS of + 1 = 6 levels with 5 particles at each level. Every straight black or dotted arrow within the original IPS represents a Markov transition , and the black twisted ones pointing to the particles in the coupled particle block represent the Feynman-Kac transition kernels , . The red dotted bending arrows are identities. The indices

Figure 2 . 7 :

 27 Figure 2.7: An illustration of the duality formula recursively applied to a mini IPS of + 1 = 6 levels with 5 particles at each level. Every straight black or dotted arrow within the original IPS represents a Markov transition , and the black ones within the coupled particle block represent the Feynman-Kac transition kernels , . The red dotted bending arrows are identities. The indices of the original particles in the coupled particle block are

  Figure 3.1: Illustration of the Feynman-Kac measures ow.

Proposition 3.3. 1 .

 1 For any test function ∈ B ( ), we have sup >1 E ( ) -2 ( ) < +∞, (3.12) and ( -( )) -2 ( -( )) = O p 1 . (3.13)

T

  Allocate memory for IPS, GENE, SH and EVE; 3 T = 0; 4 SumG = 0; 5 for ∈ {1, 2, . . . , } do 6 IPS[0, ] ∼ 0 ; 7 SumG = SumG+ 0 (IPS[0, ]); 8 EVE[0, ] = ; 9 end 10 Iteration: 11 while SumG > 0 and T < do 12 SumG = 0; 13 for ∈ {1, 2, . . . , } do 14 U ∼ Uniform[0, 1]; 15 if U ≤ T (IPS[T, ]) then 16 ParentIndex = ; 17 IPS[T + 1, ] ∼ • T+1 (IPS[T, ParentIndex], •); 18 SH[T, ] = 1; 19 else 20 ParentIndex ∼ Categorical T ( 1 T ), T ( 2 T ), . . . , T ( T ) ; 21 IPS[T + 1, ] ∼ ˚ T+1 (IPS[T, ParentIndex], •); = T + 1; 29 end 30 T = max{0, T -1}1 T< + 1 T= .

  [START_REF] Janson | Probability asymptotics: notes on notation[END_REF] are intensely involved in our technical results. More precisely, we use frequently the notation O p , O L p and O . . . Let ( ; ∈ N) be a sequence of natural numbers, where represents the number of particles in the IPS. The notation = O p ( ) means that the sequence ( / ; ∈ N) is tight, namely, for any > 0, there exists 0 < < +∞, such that lim sup ∈N P (| / | > ) < . In particular, p (1) means convergence to 0 in probability. The notation = O L p ( ) means that the sequence ( / ; ∈ N) is uniformly bounded in L p -norm w.r.t. . The notation = O . . ( ) indicates that P ∈ Ω : sup ∈N | ( )/ | < +∞ = 1.

[

  ] := ( 1 , . . . , ) ∈ [ ] : Card{ 1 , . . . , } = .(3.54)

  By the fact that -1 is indicator function, we have 0 ≤ -1 ≤ 1.

  . Given G -1 , by the conditional independence between 1

we have 1 -( 2 +1)+1

 2 = 0 on the event { ≥ }. By de nition of 2

  . Given W -1 , by the conditional independence between 1

Figure 4 . 1 :

 41 Figure 4.1: Schematic picture of transition paths.

Figure

Figure 4 . 2 :

 42 Figure 4.2: The illustration of iterative updating strategy.

:

  = inf { ≥ 0 | ∈ } , and := inf { ≥ 0 | ∈ } .

Figure 4 . 3 :

 43 Figure 4.3: Representation of the three-hole energy landscape.

Figure 4 . 4 :

 44 Figure 4.4: The numerical solution of (4.2) with inverse temperature = 1.67.

MondrianFigure 4 . 6 :Figure 4 . 7 :Figure 4 . 8 :Figure 4 . 9 :

 46474849 Figure 4.6: Decision tree corresponding to the rst split.

  Properties of Ensemble methods MF are ensemble methods, meaning that many weak regressors are constructed independently and we use the average of their predictions as the nal estimation. Here, these weak regressors are Mondrian trees: they are variants of classic decision trees used in Random Forests. Denote by [ ] = ( ; 1 ≤ ≤ ) = ( , ; 1 ≤ ≤ ) a sequence of training data. We also denoteT 1 ( [ ] ), • • • , T ( [ ] ) a collectionof conditional i.i.d. Mondrian trees. We denote Pred T [ ] the prediction at point made by T ( [ ] ). Then, the nal prediction is Pred T [ ] [ ]

∼

  Unif (Ω); ∼ N (0, 2 ); = * ( ) + , there is no theoretical guarantee such that E Pred T 1 ( ) [ ] . . or P or ----------→ →∞ * ( ).

Figure 4 . 10 :

 410 Figure 4.10: Learning with the ground truth, where the training data is uniformly sampled over Ω = [-2, 2] × [-2, 2]. For the parameters of MF, we set = ∞ and = 50. The inverse temperature is = 1.67.

Figure 4 . 11 :

 411 Figure 4.11: Learning with slightly perturbed data, where ( ) are uniformly sampled over Ω = [-2, 2] × [-2, 2]. For the parameters of MF, we set = ∞ and = 50. The inverse temperature is = 1.67.

Figure 4 . 12 :

 412 Figure 4.12: Learning with largely perturbed data, where ( ) are uniformly sampled over Ω = [-2, 2] × [-2, 2]. For the parameters of MF, we set = ∞ and = 50. The inverse temperature is = 1.67.

Figure 4 . 13 :

 413 Figure 4.13: The gure on the left represents the prediction made by a MF model trained by 100 largely perturbed data. The one on the right represents the model trained with 1000 more perfect data. ( ) are uniformly sampled over Ω = [-2, 2] × [-2, 2]. For the parameters of MF, we set = ∞ and = 50. The inverse temperature is = 1.67.

Figure 4 . 14 :

 414 Figure 4.14: Learning with the training data generated by gAMS algorithm. The reaction coordinate is set to be the distance to the state (i.e., 1 ). The inverse temperature is = 1.67.

Figure 4 .

 4 Figure 4.15: Learning with the data re-generated by gAMS algorithm. The reaction coordinate is the trained MF model. The inverse temperature is = 1.67.

Figure 4 .

 4 Figure 4.16: Illustrations of committor functions for di erent temperatures.

  Figure A.2: An illustration for the estimation of asymptotic variance with 95% condential interval. The number of particles = 10 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators.

  of non-asymptotic variance Naive variance estimator Unbiased variance estimator with 95% CI Efficient (biased) asymptotic variance estimator with 95% CI

Figure A. 3 :

 3 Figure A.3: An illustration for the estimation of asymptotic variance with 95% condential interval. The number of particles = 50 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators.

  Figure A.4: An illustration for the estimation of asymptotic variance with 95% condential interval. The number of particles = 100 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators.

  of non-asymptotic variance Naive variance estimator Unbiased variance estimator with 95% CI Efficient (biased) asymptotic variance estimator with 95% CI

Figure A. 5 :

 5 Figure A.5: An illustration for the estimation of asymptotic variance with 95% condential interval. The number of particles = 200 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators.

  Figure A.6: An illustration for the estimation of asymptotic variance with 95% condential interval. The number of particles = 500 and = 0.01. The naive variance estimator is computed on the same sample of unbiased variance estimators.

Table 3

 3

.1: Time and space complexity under multinomial resampling scheme.

Estimation

Time complexity Space complexity

Acknowledgements

Algorithm 2: Computation of ( ) and ( ).

Require: absorbing time T, the associated interacting particle system IPS, test function .

Result: estimators ( ) and ( ). 1 if T < then Now, let us provide the e cient algorithms for the consistent asymptotic variance estimators (cf. Algorithm 5, 6). We need some auxiliary steps: generation of backward genealogy tracing matrix Θ and de nition of a special "star inner product" on R 3 . They are provided respectively in Algorithm 3 and Algorithm 4. With a slight abuse of notation, we use the notation M ×1 ( ) to denote the collection of all the array of length on the set . For A ∈ M ×1 ( ), we use A[ ] to denote the -th element of A. To simplify the notation, A = zeros( , ) means that we allocate memory for A ∈ M × (R) and let all the elements of A be 0. In addition, the -th row of A will be denoted by A[ , :]. Require:

Algorithm 5: Consistent asymptotic variance estimator for ( ).

Require: ( ; 0 ≤ ≤ -1), T, IPS, test function , Θ, SH, EVE and ( ). Result: asymptotic variance estimator ˆ 2 ( ).

Algorithm 6: Consistent asymptotic variance estimator for ( ).

Require: ( ; 0 ≤ ≤ -1), T, IPS, test function , Θ, SH, EVE and ( ).

Result: asymptotic variance estimator ˆ 2 ( -( )).

1 if T < then 2 return 0; 3 else 4 Allocate memory for MeanG and MeanG2 ∈ M ×1 (R);

Algorithm 7: Unbiased non-asymptotic variance estimator for ( ).

Require: ( ; 0 ≤ ≤ -1), T, IPS, test function , GENE, SH, EVE and ( ).

Result: non-asymptotic variance estimator ( ). 1 if T < then 2 return 0; 3 else 4 Allocate memory for MeanG, MeanG2 and MeanGdot2 ∈ M ×1 (R);

CHAPTER 3. ASYMMETRIC SEQUENTIAL MONTE CARLO Theorem 3.6.2 (Unbiasedness). Assume symmetric resampling, that is

Theorem 3.6.3 (Consistency). For any coalescence indicator ∈ {0, 1} +1 and any test function ∈ B ( ) ⊗2 , we have

Remark. The notation

The reader is referred to the beginning of Section 3.7 for details.

By linearity of signed measures, we have, on the event { ≥ },

as well as

Then, Theorem 3.2.1 gives

Finally, combined to Proposition 3.7.7 and Proposition 3.7.8, we have the following corollary.

Corollary 3.6.3.1. For any coalescence indicator ∈ {0, 1} +1 and any test function ∈ B ( ), we have

Proposition 3.7.1. For any test function ∈ B ( ), we de ne , := , ( ). Then, ( ( )) ≥1 de ned by

Proof. The measurability is clear by de nition. Since ∞ is bounded by 1, we have

which gives the integrability. Then, by the fact that

one writes

This ends the veri cation of Proposition 3.7.1.

Recall that, by de nition, we have

Hence,

The interest of the martingale di erence sequence de ned above lies in the following biasmartingale decomposition:

(3.31)

PROOFS

Since E[ 0 ( 0 )] = 0 ( 0 ), we have, thanks to Hoe ding's inequality for bounded i.i.d. random variables, P 0 ( 0 ) = 0 ≤ P 0 ( 0 ) < 0 ( 0 ) 2 exponential decay rate w.r.t. .

,

which guarantees

whence, by Cauchy-Schwarz inequality, 0 ( )1 ≥0 -0 ( )

For the case 0 = 0, since

The conclusion is also straightforward by considering the case 0 = 1. Lemma 3.7.5. If for all 0 ≤ ≤ -1, we have

then, we also have

Proof. By de nition, we have

For ≥ 1, thanks to the bias-martingale decomposition (3.31), the almost sure boundness (3.30) and Azuma-Hoe ding inequality, we have

(1) 4 (3.43)

Then, we verify that

By Markov's inequality, one has

L 1

(1) ≤

L 1

(1)

PROOFS

Proof. The proof is done by induction. Thanks to a by-product (3.42) of Lemma 3.7.4, we have

For step ≥ 1, we suppose that

By the bias-martingale decomposition (3.31), it gives

Next, standard calculations give

Remark that, by de nition, ( ) = 0.

Then, by applying Lemma 3.7.5, we noticed that

Mutatis mutandis, one also has

Therefore, considering the induction hypothesis (3.50), we only have to show that

(3.51)

Recall that the event Ω ⊂ Ω is de ned by

, Remark. Before starting the proof of Proposition 3.7.8, we would like to mention that the techniques involved are similar but a little bit di erent from the ones in the proof of Proposition 3.7.7. First, G †,1 n is not nonnegative in general except for the case where is indicator function for all ≥ 0. In addition, it is not obvious that

is bounded almost surely. In fact, it is easy prove that it is a.s. upper bounded by 3. However, there is no obvious reason that this term is a.s. lower bounded. This leads to the fact that

is not a.s. bounded in general, which is the main di cult part in the following technical results. Hence, unlike the previous case shown in the proof of Proposition 3.7.7, one should be extremely careful where dealing with the bound associated to the term

+1 , [2] )1 ≥ .

Therefore, we introduce the following Lemma in order to facilitate the proof.

Lemma 3.7.6. For any nonnegative real numbers , , ∈ R, if ≤ , > 0 and ≥ 0, then, we have ≤ + + .

Proof. Direct calculation gives

The conclusion follows.

Lemma 3.7.7. For any [2] and [2] +1 ∈ ( ) 2 , we have

≠ 2 1 -( ) / are both well-de ned on the event { ≥ }.

Proof. By symmetry of the de nition, we only show that 1

is always well-de ned on the event { ≥ } with the convention (3.1). In fact, when

PROOFS CHAPTER 4. ESTIMATING COMMITTOR FUNCTION WITH MONDRIAN FORESTS

The committor function * : \( ∪ ) ↦ → [0, 1] is de ned by * ( )

Our goal is to design an e cient strategy to estimate this function on some compact subset of the state space . In particular, we are interested in exploiting this estimation to improve the performance of gAMS and vice versa.

Ground truth

In this section, we show how one can obtain the reference values of a committor function in overdamped Langevin dynamics for some low-dimensional toy examples, which will be referred to as ground truth, namely the "real" values of the committor function. This method only works in low-dimensional setting. It is well known (see, e.g. [START_REF] Bréhier | Analysis of adaptive multilevel splitting algorithms in an idealized case[END_REF]) that * is the solution of the following elliptic Partial Di erential Equation (PDE):

with the boundary conditions

For su ciently ne grids, the numerical solution of (4.2) using Finite Di erence method is precise enough so that it can be regarded as the real committor function.

Three-hole potential The main example studied in this chapter is the following toy exmaple (see Figure 4.3):

We consider the following two metastable states respectively de ned by := {( , ) ∈ R : ( + 1) 2 + 2 < 0.1}, and

The reader is referred to [START_REF] Metzner | Illustration of transition path theory on a collection of simple examples[END_REF] for a wider list of toy examples. In order to solve the elliptic PDE (4.2) by Finite Di erence method, we consider a rectangular domain Ω = [-2, 2] × [-2, 2] and a uniform mesh with stepsize 0.01. We also add a Neumann boundary condition on the boundary Ω of the rectangular domain, i.e., ì = 0 where ì denotes the unit normal vector on the boundary. These ctitious boundary conditions do not a ect too much the quality of the result since when starting far from and , the di erence of the values of the committor function for two close points is negligible. This ensures that the Finite Di erence result yields an accurate approximation of the solution to (4.2)-(4.2'). The numerical solution of (4.2)-(4.2') for the inverse temperature = 1.67 is illustrated in Figure 4.4.

The mechanism of a Mondrian tree

In this section, we provide an intuitive interpretation on the construction of a Mondrian tree on a 2-dimensional toy example. For the generic algorithm, the reader is referred to [START_REF] Balaji Lakshminarayanan | Mondrian forests: E cient online random forests[END_REF]. Let us x a parameter ∈ R ∪ {+∞}, which de nes the lifetime of a Mondrian tree. Let us denote by ( , ; 1 ≤ ≤ ) the training data. Before starting, we remark that the partition de ned by a MT only depends on the inputs ( ; 1 ≤ ≤ ) of the training data.

Construction of MT in dimension 2

At step 0, we denote the minimum rectangle that covers 1 , 2 , . . . , (see Figure 4.5). Then, we sample an exponential random variable 0 with rate (| | + | |). If 0 < , a split on the side or will be executed. More precisely, we sample a uniform random variable on the interval

a splitting point will be uniformly sampled on the side ; otherwise, the split will be done on the side . After the splitting point is determined, a split will be executed orthogonally to the chosen side. For short, this amounts to say that an orthogonal split is performed uniformly on . In Figure 4.5, the splitting point is on the side and its abscissa is denoted by 0 . Therefore, the training data is vertically divided into two subgroups. At the same time, a node in a decision tree according to the split is therefore constructed (see Figure 4.6). By de nition, the condition node > 0 illustrated in Figure 4.6 can be determined by the pair of sets ({ > 0 }, { ≤ 0 }). We say that the lifetime of the node ({ > 0 }, { ≤ 0 }) is 0 . Note that a decision tree uniquely determines a partition on the whole state space R 2 .

Next, we perform the splitting for these two subgroups of data recursively. Let us start by the subgroup of data on the left-hand side (see Figure 4.7). Denote again by the minimum rectangle that covers all the data in this subgroup of data. We sample 1 1 w.r.t. an exponential distribution with rate (| | + | |). Now, if 0 + 1 1 < , we perform a uniform split on , in the same way as presented at step 0. The same mechanism is applied mutatis mutandis to the right-hand side (see Figure 4.8). The associated decision tree is also updated accordingly, and one tracks the lifetime of each node (see Figure 4.9).

The splitting procedure stops when the lifetime of the proposed condition node surpasses the pre xed lifetime . We also remark that, when a subset contains only 1 data, the splitting stops Self-consistency of MT The online training of each MT does not depend on the arrival orders of = ( , ). More precisely, we have

where M +1 represents the online updating strategy of Mondrian tree given the data +1 and T ( [ ] ) denotes the distribution of Mondrian tree with lifetime given the data [ ] . According to the author of [START_REF] Balaji Lakshminarayanan | Mondrian forests: E cient online random forests[END_REF], this is the only construction of decision tree available such that the property above is veri ed. However, this beautiful property comes with a cost: the splits of the MF do not use the value of the . Therefore, they are not the "optimal" splits given the training data used in classical decision tree's construction. Hence, the robustness over the "extremely bad quality data" will be a ected. This will be discussed in the numerical illustration in the next section.

Numerical illustrations

In this section, we provide some numerical illustrations and interpretations on the estimation of committor function. The energy is set to be the three-hole energy introduced in Section 4.2.2. In the following sections, the sample points are uniformly sampled in the rectangle Ω = [-2, 2] × [-2, 2], and by "perfect training data" we mean the numerical solution of (4.2) given by Finite Di erence methods.

Learning with perfect data

Unlike the typical estimation problems in statistics and machine learning context, the quality of the training data is indeed controllable in the committor function estimation problem. Although it is of no practical interest to do this kind of trade-o , we investigate the performance of training MF with perfect data (ground truth) to test its adequacy and to have an idea on how many data is needed to provide reasonable approximations. The results are given in Figure 4.10.

Learning with noisy data

Now, we consider adding some arti cial Gaussian noise to the training data. For a gAMS algorithm with = 100 and * = 20, the typical variance over the rectangle Ω is between 10 -6 to 10 -4 . Therefore, we consider adding a slightly larger centered normal noise with variance 4 × 10 -4 . The results are shown in Figure 4.11. Although it seems that MF cannot handle the situation perfectly, the approximation is still quite impressive. Since the added noises are i.i.d. normal random variables, the relative variance on the left ( ≤ 0) is noticeably larger than on the right ( ≥ 0). This is why the estimation quality on the left is worse than on the right in general. The situation where the noise is relatively large is presented in Figure 4.12. We use the possibly largest variance, 1, that can be made by crude Monte Carlo or gAMS. This time, unsurprisingly, MF failed to provide reasonable predictions.

Appendices

Appendix A

Generalized AMS under Asymmetric SMC framework

This appendix sets out to stress the implementation and numerical performance of the generalized AMS method ([BGG + 16]) with discrete reaction coordinate under Asymmetric SMC framework. The equivalence of AMS and Asymmetrical SMC is not straightforward to implement in practice-several adaptations should be taken into consideration, especially for the variance estimators. This implementation will simply be referred to as AMS for short in the following. An optimized and structured (readable) python package (powered by Cython [BBC + 11]) is provided in the following github repository: https://github.com/MGIMM/aAMS Several numerical experiments on the performance of di erent variance estimators will also be provided. In particular, we are interested in the behaviors of the Variance of di erent Variance estimators. The Variance of Variance estimators (VoV) will be used to assess the quality of di erent variance estimators (through the length of the 95% con dential interval). Finally, we provide some general suggestions on the application of AMS methods under Asymmetric SMC framework.

A.1 Setting

Before proceeding further, we remark that all the numerical experiments are based on the Overdamped Langevin dynamic with three-hole potential (see Section 4.2.2). The discretization step is xed to dt = 0.05. In order to make sure the reaction coordinate only take nite values, we introduce a new parameter (iota) that indicates the discretization step of the reaction coordinate. More precisely, we consider the reaction coordinate

The corresponding discretized version is de ned by

The metastable states are de ned as follows:

A.3 VoV behaviors

Let us continue to see the comportment of the VoV with di erent particle size varies from 10 to 500. We sample the IPS independently for 200 to 10000 times, and the con dential interval is obtained from the crude variance estimation of our variance estimators. The illustration can be found in Figure A.2-A.6. First, it is clear that when is small, the e cient asymptotic variance estimator is heavily biased (see Figure A.2-A.3). Then, it also shows that the average of the unbiased variance estimators over the independent IPS depends strongly on the crude variance estimator. Finally, we remark that from a relatively small particle size ( = 500, Figure A.6), the VoV of e cient variance estimator becomes noticeably smaller than the unbiased variance estimator. Hence, when using one particle system of large size for the rare-event estimation, the e cient variance estimator is expected to be more accurate and more e cient at the same time.