Siméon Valère

Bitseki Penda

S Val Ère

MODERATE DEVIATION PRINCIPLES FOR KERNEL ESTIMATOR OF INVARIANT DENSITY IN BIFURCATING MARKOV CHAINS MODELS

Keywords: Bifurcating Markov chains, bifurcating auto-regressive process, binary trees, density estimation Mathematics Subject Classification (2020): 62G05, 62F12, 60F10, 60J80

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Introduction

The study of bifurcating Markov chains (BMCs, for short) models has taken a special place in the literature these last years due to their links with the study of the cell dynamics (see for e.g. [START_REF] Bitseki Penda | Autoregressive functions estimation in nonlinear bifurcating autoregressive models[END_REF][START_REF] Delmas | Detection of cellular aging in a Galton-Watson process[END_REF][START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF][START_REF] Guyon | Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging[END_REF][START_REF] Hoffmann | Statistical estimation in a randomly structured branching population[END_REF]). The first model of BMC, named "symmetric" bifurcating autoregressive process (BAR, for short) were introduced by Cowan and Staudte [START_REF] Cowan | The bifurcating autoregression model in cell lineage studies[END_REF] in order to understand the cell division mechanisms of Escherichia Coli (E. Coli, for short). E. Coli is a rod shaped bacterium which reproduces by dividing in two, thus producing two new cells. One of type 1 which has the old end of the mother and the other of type 0 which has the new end of the mother. The age of a cell is thus given by the age of its old pole in the sense of the number of divisions from which this pole exists. This cell division mechanism raises several questions, among other that of the symmetry of the division. In order to give a rigorous answer to this question, Guyon [START_REF] Guyon | Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging[END_REF] has developed and studied the theory of BMCs. We note that to the best of our knowledge, the term BMC appears for the first time in the work of [START_REF] Basawa | Non-Gaussian bifurcating models and quasi-likelihood estimation[END_REF]. In particular, Guyon has studied an extension of the model introduced by Cowan and Staudte, named "asymmetric" BAR. In the conclusion of his study, Guyon concludes that aging has an impact on cell reproduction. We note that an extension of the model proposed by Guyon, named nonlinear BAR (NBAR, for short) were studied by Bitseki and Olivier in [START_REF] Bitseki Penda | Autoregressive functions estimation in nonlinear bifurcating autoregressive models[END_REF]. Another question of interest related to cell division is estimating the division rate at which cells divide. This question has been tackled recently in the work of Doumic & al. [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF] and Hoffman & Marguet [START_REF] Hoffmann | Statistical estimation in a randomly structured branching population[END_REF]. In all the previous work, the behaviour and the definition of parameters of interest are associated with the density of the invariant probability of an auxiliary Markov chain (see below for a precise definition). The estimation of this invariant density has recently been the subject of several studies. One can cite [START_REF] Bitseki Penda | Adaptive estimation for bifurcating Markov chains[END_REF][START_REF] Bitseki Penda | Local bandwidth selection for kernel density estimation in a bifurcating Markov chain model[END_REF] where adaptive methods have been proposed for the estimation of this invariant density. More recently, Bitseki and Delmas [START_REF] Bitseki Penda | Central limit theorem for kernel estimator of invariant density in bifurcating markov chains models[END_REF] have studied central limit theorem for kernel estimators of this invariant density. Our main objective in this paper is to complete the previous study by establishing a moderate deviation principle for these kernel estimators. Before going any further, let us recall the definition of the main concepts that we will use and study.

2. The model of bifurcating Markov chain and definition of the estimators 2.1. The regular binary tree associated to BMC models. We denote by N (resp. N * ) the space of (resp. positive) natural integers. We set T 0 = G 0 = {∅}, G k = {0, 1} k and T k = 0≤r≤k G r for k ∈ N * , and T = r∈N G r . The set G k corresponds to the k-th generation, T k to the tree up to the k-th generation, and T the complete binary tree. One can see that the genealogy of the cells is entirely described by T (each vertex of the tree designates an individual). For i ∈ T, we denote by |i| the generation of i (|i| = k if and only if i ∈ G k ) and iA = {ij; j ∈ A} for A ⊂ T, where ij is the concatenation of the two sequences i, j ∈ T, with the convention that ∅i = i∅ = i. For A ⊂ T, we denote by |A| the number of elements of A. Note that for all n ∈ N, |G n | = 2 n and |T n | = 2 n+1 -1.

2.2.

The probability kernels associated to BMC models. For our convenience, we set S = R d , d ≥ 1 and S is equipped with the Borel sigma-algebra S . For any q ∈ N * , we denote by B(S q ) (resp. B b (S q ), resp. C b (S q )) the space of (resp. bounded, resp. bounded continuous ) R-valued measurable functions defined on S q . For all q ∈ N * , we set S ⊗q = S ⊗ . . . ⊗ S . Let P be a probability kernel on (S, S ⊗2 ), that is: P(•, A) is measurable for all A ∈ S ⊗2 , and P(x, •) is a probability measure on (S 2 , S ⊗2 ) for all x ∈ S. For any g ∈ B b (S 3 ) and h ∈ B b (S 2 ), we set for x ∈ S:

(1) (Pg)(x) = S 2 g(x, y, z) P(x, dy, dz) and (Ph)(x) = S 2 h(y, z) P(x, dy, dz).

We define (Pg) (resp. (Ph)), or simply P g for g ∈ B(S 3 )(resp. Ph for h ∈ B(S 2 )), as soon as the corresponding integral (1) is well defined, and we have that Pg and Ph belong to B(S). We denote by P 0 , P 1 and Q respectively the first and the second marginal of P, and the mean of P 0 and P 1 , that is, for all x ∈ S and B ∈ S P 0 (x, B) = P(x, B × S), P 1 (x, B) = P(x, S × B) and Q = (P 0 + P 1 ) 2 .

Now let us give a precise definition of bifurcating Markov chain.

Definition 2.1 (Bifurcating Markov Chains, see [START_REF] Guyon | Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging[END_REF][START_REF] Bitseki Penda | Central limit theorem for kernel estimator of invariant density in bifurcating markov chains models[END_REF]).

We say a stochastic process indexed by T, X = (X i , i ∈ T), is a bifurcating Markov chain (BMC) on a measurable space (S, S ) with initial probability distribution ν on (S, S ) and probability kernel P on S × S ⊗2 if: -(Initial distribution.) The random variable X ∅ is distributed as ν.

-(Branching Markov property.) For any sequence (g i , i ∈ T) of functions belonging to B b (S 3 ) and for all k ≥ 0, we have

E i∈G k g i (X i , X i0 , X i1 )|σ(X j ; j ∈ T k ) = i∈G k Pg i (X i ).
Following [START_REF] Guyon | Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging[END_REF], we introduce an auxiliary Markov chain Y = (Y n , n ∈ N) on (S, S ) with Y 0 = X 1 and transition probability Q. The chain (Y n , n ∈ N) corresponds to a random lineage taken in the population. We shall write E x when X ∅ = x (i.e. the initial distribution ν is the Dirac mass at x ∈ S). We will assume that the Markov chain Y is ergodic and we denote by µ its invariant probability measure. Asymptotic and non-asymptotic behaviour of BMCs are strongly related to the knowledge of µ. In particular, Guyon has proved that if Y is ergodic, then for all f ∈ C b (S),

|A n | -1 u∈An f (X u ) ----→ n→+∞ µ, f in probability, where A n ∈ {G n , T n }.
But in most cases, the invariant probability µ is unknown, so its estimation from the data is of great interest. For that purpose, we do the following assumption.

Assumption 2.2. The transition kernel P has a density, still denoted by P, with respect to the Lebesgue measure.

Remark 2.3. Assumption 2.2 implies that the transition kernel Q has a density, still denoted by Q, with respect to the Lebesgue measure. More precisely, we have Q(x, y) = 2 -1 S (P(x, y, z) + P(x, z, y))dz. This implies in particular that the invariant probability µ has a density, still denoted by µ, with respect to the Lebesgue measure (for more details, we refer for e.g. to [START_REF] Duflo | Random iterative models[END_REF], chap 6).

Kernel estimator of the invariant density

µ. Recall that A n ∈ {G n , T n } and S = R d , d ≥ 1. Assume we observe X n = (X u , u ∈ A n ). Let (h n , n ∈ N)
be a sequence of positive numbers which converges to 0 as n goes to infinity. We will simply write h for h n if there is no ambiguity. Let the kernel function K : S → R such that S K(x)dx = 1. Then, for all x ∈ S, we propose to estimate µ(x) by ( 2)

µ An (x) = |A n | -1 h -d/2 n u∈An K hn (x -X u ),
where

K hn (•) = h -d/2 n K(h n •).
These estimators are strongly inspired from [START_REF] Masry | Recursive probability density estimation for weakly dependent stationary processes[END_REF][START_REF] Parzen | On estimation of a probability density function and mode[END_REF][START_REF] Roussas | Nonparametric estimation in Markov processes[END_REF]. They have been studied in [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF][START_REF] Bitseki Penda | Local bandwidth selection for kernel density estimation in a bifurcating Markov chain model[END_REF] (non asymptotic studies) and in [START_REF] Bitseki Penda | Central limit theorem for kernel estimator of invariant density in bifurcating markov chains models[END_REF] (central limit theorem).

2.4.

Moderate deviation principle and related topics. Our aim is to study moderate deviation principles for the estimators defined in [START_REF] Bitseki Penda | Central limit theorem for kernel estimator of invariant density in bifurcating markov chains models[END_REF]. Before we proceed, let us introduce the notion of moderate deviation principle. We give the definition in a general setting. Let (Z n ) n≥0 be a sequence of random variables with values in S endowed with its Borel σ-field S and let (s n ) n≥0 be a positive sequence that converges to +∞. We assume that Z n /s n converges in probability to 0 and that Z n / √ s n converges in distribution to a centered Gaussian law. Let I : S → R + be a lower semicontinuous function, that is for all c > 0 the sub-level set {x ∈ S, I(

-inf x∈ Å I(x) ≤ lim inf n→+∞ 1 b 2 n log P Z n b n √ s n ∈ A ≤ lim sup n→+∞ 1 b 2 n log P Z n b n √ s n ∈ A ≤ -inf x∈ Ā I(x),
where Å and Ā denote respectively the interior and the closure of A.

The following two concepts are closely related to the theory of MDP: super-exponential convergence and exponential equivalence. Let (Z n , n ∈ N), (W n , n ∈ N) be sequences of random variables and Z a random variable with value in a metric space (S, d).

Definition 2.5 (Super-exponential convergence). We say that (Z n ) n≥0 converges (b 2 n )-superexponentially fast in probability to Z and we note

Z n superexp = ==== ⇒ b 2 n Z if, for all δ > 0, lim sup n→+∞ 1 b 2 n log P d(Z n , Z) > δ = -∞.
Definition 2.6 (Exponential equivalence, see [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], Chap 4). We say that (Z n ) n≥0 and

(W n ) n≥0 are (b 2 n ) n≥0 -exponentially equivalent and we note Z n superexp ∼ b 2 n W n if for any δ > 0, lim sup n→+∞ 1 b 2 n log P d(Z n , W n ) > δ = -∞.
Remark 2.7. Note that for a determininistic sequence that converges to some limit , it also converges (b 2 n )-superexponentially fast to for any rate b n . We also note that if (Z n ) n≥0 and (W n ) n≥0 are (b 2 n ) n≥0 -exponentially equivalent and if (Z n ) n≥0 satisfies a MDP, then (W n ) n≥0 satisfies the same MDP (for more details, see for e.g [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], Chap 4).

The following result give a sufficient condition for super-exponential convergence of a sequence of random variables.

Remark 2.8. We assume that (S, d) is a metric space. Let (Z n ) n∈N be a sequence of random variables with values in S, Z a random variable with values in S. So if d(Z n , Z) is upper-bounded by a deterministic sequence which converges to 0, then, for all sequence (b n , n ∈ N) converging to +∞,

Z n superexp = ==== ⇒ b 2 n Z.
The moderate deviation principle has been proved in the i.i.d. setting for kernel density estimator, see for e.g. Gao [START_REF] Gao | Moderate deviations and large deviations for kernel density estimators[END_REF], Mokkadem & al. [20]. We refer also to [START_REF] Mokkadem | Confidence bands for densities, logarithmic point of view[END_REF] where Mokkaddem and Pelletier have constructed confidence bands for probability densities based on moderate deviation principles. In this paper, we will establish moderate deviation principle for µ An (x) following the martingale approach developed in [START_REF] Bitseki Penda | Central limit theorem for kernel estimator of invariant density in bifurcating markov chains models[END_REF]. We will need the following assumption. Assumption 2.9. There exists a positive real number M and α ∈ (0, 1) such that for all f ∈ B b (S):

(3)

|Q n f -µ, f | ≤ M α n f ∞ for all n ∈ N.
Remark 2.10. Assumption 2.9 is for example satisfy for nonlinear bifurcating autoregressive process under mild hypotheses on the autoregression functions (see [START_REF] Bitseki Penda | Moderate deviation principle in nonlinear bifurcating autoregressive models[END_REF] Lemma 9 for more details).

The others assumptions we will need are based on the following bias-variance type decomposition of the estimator µ An (x):

(4) µ An (x) -µ(x) = B hn (x) + V An,hn (x),
where for h > 0 and A ⊂ T finite:

B h (x) = h -d/2 K h µ(x) -µ(x) and V A,h (x) = |A| -1 h -d/2 u∈A K h (x -X u ) -K h µ(x) ,
and for h > 0 and u ∈ T, we set:

K h µ(x) = E µ [K h (x -X u )] = S K h (x -y)µ(y) dy.
To study the variance term V An,hn (x), we will introduce a more general sequence of functions (see Section 3.2).

The following assumptions on the kernel, the bandwidth and the regularity of the unknown density function are usual. Recall S = R d with d ≥ 1. (i) The kernel function K ∈ B(S) satisfies:

K ∞ < +∞, K 1 < +∞, K 2 < +∞, S K(x) dx = 1 and lim |x|→+∞ |x|K(x) = 0.
(ii) There exists γ ∈ (0, 1/d) such that the bandwidth (h n , n ∈ N) are defined by h n = 2 -nγ .

Assumption 2.12 (Further regularity on the density µ, the kernel function and the bandwidths).

Suppose that there exists an invariant probability measure µ of Q and that Assumptions 2.2 and 2.11 hold. We assume there exists s > 0 such that the following hold: (i) The density µ belongs to the (isotropic) Hölder class of order (s, . . . , s) ∈ R d :

The density µ admits partial derivatives with respect to x j , for all j ∈ {1, . . . d}, up to the order s and there exists a finite constant L > 0 such that for all x = (x 1 , . . . , x d ), ∈ R d , t ∈ R and j ∈ {1, . . . , d}:

∂ s µ ∂x s j (x -j , t) - ∂ s µ ∂x s j (x) ≤ L|x j -t| {s} ,
where (x -j , t) denotes the vector x where we have replaced the j th coordinate x j by t, with the convention

∂ 0 µ/∂x 0 j = µ. (ii) The kernel K is of order ( s , . . . , s ) ∈ N d : We have R d |x| s K(x) dx < ∞ and R x k j K(x) dx j =
0 for all k ∈ {1, . . . , s } and j ∈ {1, . . . , d}. For α > 1/2, we shall also assume the following. Assumption 2.13. Keeping the same notations as in (ii) of Assumption 2.11, we further assume that Assumption 2.9 holds with [START_REF] Bitseki Penda | Adaptive estimation for bifurcating Markov chains[END_REF] lim n→+∞ (2 1-dγ α) n = 0.

Remark 2.14. As consequence of Assumption 2.13 and (ii) of Assumption 2.11, for moderate deviation principle, the ergodicity rate α begins to have an impact on the choice of the bandwidth for α > 1/2. This is out of step with the central limit theorem where the ergodicity rate α begins to have an impact on the choice of the bandwidth for α > 1/ √ 2 (see [START_REF] Bitseki Penda | Central limit theorem for kernel estimator of invariant density in bifurcating markov chains models[END_REF] for more details).

In the sequel, we will consider the positive sequence (b n , n ∈ N) such that:

(6) lim n→+∞ b n = +∞; lim n→+∞ n 3/2 b n |G n |h d n = 0; lim n→+∞ b n |G n |h 2s+d n = +∞,
where s is the regularity parameter given in Assumption 2.12.

The paper is organised as follows. In Section 3.1 we state the main result for the moderate deviation principles of the estimators µ An (x) for x in the set continuity of µ and A n ∈ {T n , G n }. In Section 3.2, directly linked to the study of variance term V A,h (x) defined in (4), we study the moderate deviation principle for general additive functionals of BMCs. Sections 4 and 5 are devoted to the proofs of results. In Section 6, we recall some useful results.

Main result

3.1. Moderate deviation principle for µ An . First, we state a strong consistency result for the estimators µ An (x) for x in the set of continuity of µ. Its proof is given in Section 4.1.

Lemma 3.1. Let X be a BMC with kernel P and initial distribution ν such that Assumptions 2.9, 2.11 and 2.12 hold. Furthermore, if α > 1/2 then assume that Assumption 2.13 holds. Let (b n , n ∈ N) be a positive sequence with satisfies (6). Then, for all x in the set of continuity of µ and

A n ∈ {T n , G n } we have µ An (x) superexp = ==== ⇒ b 2 n µ(x).
The main result of this Section is the following theorem which state the moderate deviation principle for µ An (x) -µ(x) for x in the set of continuity of the function µ. Theorem 3.2. Under the hypothesis of Lemma 3.1, for all x in the set of continuity of µ and

A n ∈ {T n , G n }, b -1 n |A n |h d n ( µ An (x) -µ(x)
) satisfies a moderate deviation principle on R with speed b 2 n and rate function I defined by:

I(y) = y 2 /(2 K 2 2 µ(x)) for all y ∈ R, that is, for any A ⊂ R, -inf y∈ Å I(y) ≤ lim inf n→+∞ 1 b 2 n log P b -1 n |A n |h d n ( µ An (x) -µ(x)) ∈ A ≤ lim sup n→+∞ 1 b 2 n log P b -1 n |A n |h d n ( µ An (x) -µ(x)) ∈ A ≤ -inf y∈ Ā I(y),
where Å and Ā denote respectively the interior and the closure of A.

In order to obtain confidence intervals for µ(x), it would be interesting to replace µ(x) in the expression of the rate function I(•) by an estimator. In that direction, we have the following. Let A * n ∈ {G n , T n }. Obviously, A * n and A n can be the same. We consider the estimator µ A * n (x) of µ(x) defined with A * n instead of A n . Let ( n , n ∈ N) be a sequence of real numbers such that n → 0 as n → +∞. Then, we have the following result which the proof is given in Section 4.3. Theorem 3.3. Under the hypothesis of Lemma 3.1, for all x in the set of continuity of µ and

A n , A * n ∈ {T n , G n }, b -1 n ( K 2 µ A * n (x) ∨ n ) -1 |A n |h d n ( µ An (x) -µ(x)
) satisfies a moderate deviation principle on R with speed b 2 n and rate function I defined by: I (y) = y 2 /2 for all y ∈ R. In particular, using the contraction principle (see for e.g Dembo and Zeitouni [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], Chap 4), we have the following corollary of Theorem 3.3. Corollary 3.4. Under the hypothesis of Theorem 3.3, we have the following convergence for x in the set of continuity of µ and

A n , A * ∈ {T n , G n } : lim n→+∞ 1 b 2 n log P b -1 n K 2 µ A * n (x) ∨ n -1 |A n |h d n µ An (x) -µ(x) > δ = - δ 2 2 ∀δ > 0.
n |G n | 1/2 h 1/2 n µ Gn (x)-µ(x) , . . . , |G n-k | 1/2 h 1/2 n-k µ G n-k (x)-µ(x) t satisfies a moderate deviation principle on R k+1 with speed b 2
n and good rate function

J x : R k+1 → R defined by J x (z) = 2 K 2 2 µ(x) -1 z t Γ -1 z , z ∈ R k+1 ,
with Γ = diag(2 0 , . . . , 2 k ), where diag(•) denotes the diagonal matrix and z t stands for the transpose of vector z.

Remark 3.7. We deduce from Corollary 3.6 that the estimators

|G n-| 1/2 h d/2 n-( µ G n-(x) -µ(x)
) are asymptotically independent in the sense of moderate deviation for ∈ {0, . . . , k} and for any k ∈ N.

Moderate deviation principle for additive functionals of BMCs.

In order to study the variance term V An,hn (x), we give here a moderate deviation principle for a general additive functionals of BMCs. For that purpose, we introduce the following assumption. Assumption 3.8. For n ∈ N, let f n = (f ,n , n ≥ ≥ 0) be a sequence of functions defined on S such that f ,n = 0 if > n and there exists γ ∈ (0, 1/d) such that:

(i) sup 0≤ ≤n {2 -dγn/2 f ,n ∞ ; 2 dγn/2 Qf ,n ∞ ; Q(f 2 ,n ) ∞ ; 2 dγn P(f ,n ⊗ 2 ) ∞ } < +∞. (ii) sup 0≤ ≤n {2 dγn/2 µ, |f ,n | ; µ, f 2
,n } < +∞. (iii) The following limit exists and is finite:

(7) σ 2 = lim n→+∞ n =0 2 -f ,n 2
L 2 (µ) < +∞.
We will use the following notations. For a finite set A ⊂ T and a function f ∈ B(S), we set:

M A (f ) = i∈A f (X i ).
In this paper, we are interested in the cases A = G n and A = T n , that is the n-th generation and the first n generation of the tree. Recall µ the invariant probability of Q, transition probability of the auxiliary Markov chain (Y n , n ∈ N). For f ∈ L 1 (µ), we set:

f = f -µ, f .
Recall the sequence f n defined in Assumption 3.8. For n ∈ N, we set:

(8) N n,∅ (f n ) = |G n | -1/2 n =0 M G n-( f ,n ).
The notation N n,∅ means that we consider the average from the root ∅ to the n-th generation.

Remark 3.9. The definition of N n,∅ (f n ) in ( 8) is mainly motivated by the decomposition (4). It will allow us to threat the variance term of the estimator µ An (x) defined in [START_REF] Bitseki Penda | Central limit theorem for kernel estimator of invariant density in bifurcating markov chains models[END_REF]. Instead, for n ∈ N, we set f x n (•) = K hn (x -•). Then, we consider the sequences of functions (f id ,n , n ≥ ≥ 0) and (f 0 ,n , n ≥ ≥ 0) defined by: (9)

f id ,n = f x n and f 0 ,n = f x n 1 { =0} . It is not difficult to check that under Assumption 2.11, the sequence (f id ,n , n ≥ ≥ 0) and (f 0
,n , n ≥ ≥ 0) defined in (9) satisfy Assumption 3.8. In particular, let x be in the set of continuity of µ. Thanks to Lemma 6.3, we have: [START_REF] Delmas | Detection of cellular aging in a Galton-Watson process[END_REF] lim

n→+∞ f x n 2 L 2 (µ) = lim n→+∞ µ, (f x n ) 2 = µ(x) K 2 2 . If A n = G n , it suffices to consider the sequence f n = (f ,n , 0 ≤ ≤ n) with f ,n = f 0
,n and in that case, using [START_REF] Delmas | Detection of cellular aging in a Galton-Watson process[END_REF], the asymptotic variance defined in [START_REF] Bitseki Penda | Moderate deviation principle in nonlinear bifurcating autoregressive models[END_REF] is given by σ

2 = K 2 2 µ(x). If A n = T n , it suffices to consider the sequence f n = (f ,n , 0 ≤ ≤ n) with f ,n = f id
,n and in that case, using [START_REF] Delmas | Detection of cellular aging in a Galton-Watson process[END_REF], the asymptotic variance defined in [START_REF] Bitseki Penda | Moderate deviation principle in nonlinear bifurcating autoregressive models[END_REF] is given by σ 2 = 2 K 2 2 µ(x). For our convenience, we assume that the quantity γ which appears in Assumptions 2.11 and 3.8 is the same. The main result of this section is the following. Theorem 3.10. Let X be a BMC with kernel P and initial distribution ν such that Assumptions 2.9, 2.11 and 3.8 hold. Furthermore, if α > 1/2 then assume that Assumption 2.13 holds. Let (b n , n ∈ N) be a positive sequence with satisfies [START_REF] Bitseki Penda | Autoregressive functions estimation in nonlinear bifurcating autoregressive models[END_REF]. Then b -1 n N n,∅ (f n ) satisfies a moderate deviation principle on R with speed b 2 n and rate function I defined by: I(x) = x 2 /(2σ 2 ) for all x ∈ R, with the finite variance σ 2 defined in [START_REF] Bitseki Penda | Moderate deviation principle in nonlinear bifurcating autoregressive models[END_REF].

Remark 3.11. In particular, using the contraction principle (see for e.g Dembo and Zeitouni [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], Chap 4), Theorem 3.10 implies that

lim n→+∞ 1 b 2 n log P b -1 n N n,∅ (f n ) > δ = -I(δ) ∀δ > 0.
Remark 3.12. Unlike the results of Bitseki and Gorgui [START_REF] Bitseki Penda | Moderate deviation principles for bifurcating markov chains: case of functions dependent of one variable[END_REF], one can note that the different regimes disappear in Theorem 3.10. Moreover, we are able here to give the fluctuations if 2α 2 > 1 which is not the case in [START_REF] Bitseki Penda | Moderate deviation principles for bifurcating markov chains: case of functions dependent of one variable[END_REF].

4. Proof of Lemma 3.1, Theorems 3.2 and 3.3 and Corollary 3.6

We will denote by C any unimportant finite constant which may vary from line to line (in particular C does not depend on n ∈ N).

4.1.

Proof of Lemma 3.1. We begin the proof with A n = T n . Recall the decomposition (4) with T n instead of A. Using Lemma 6.3, we have lim n→+∞ |B hn (x)| = 0. From Remark 2.7, this implies that B hn (x)

superexp = ==== ⇒ b 2 n 0. Next, we set f n (•) = K hn (x -•) in such a way that we have |T n | -1 h d-/2 n u∈Tn K h (x -X u ) -K h µ(x) = |T n | -1 h d-/2 n n =0 M G ( fn ).
Following line by line the proof of (32) (where we take f ,n = f n for all ≤ n), we get

P |T n | -1 h d-/2 n n =0 M G ( fn ) > δ ≤ 2 exp 3δ c 1 + c 2 δ exp - 3δ 2 |T n |h d n c 1 + c 2 δ .
Taking the log, dividing by b 2 n and letting n goes to the infinity in the latter inequality, we get

|T n | -1 h d-/2 n u∈Tn K h (x -X u ) -K h µ(x) superexp = ==== ⇒ b 2 n 0.
It then follows from the decomposition (4) that µ Tn (x)

superexp = ==== ⇒ b 2 n µ(x).
We similarly get the result for A n = G n and this ends the proof of the lemma.

4.2.

Proof of Theorem 3.2. We begin the proof with A n = T n . We have the following decomposition:

b -1 n |T n |h d n µ Tn (x) -µ(x) = |G n | |T n | b -1 n N n,∅ (f n ) + |T n |h d n b n B hn (x),
where f n = (f ,n , n ≥ ≥ 0) with the functions f ,n = f id ,n defined in [START_REF] Cowan | The bifurcating autoregression model in cell lineage studies[END_REF] for n ≥ ≥ 0 and f ,n = 0 otherwise; N n,∅ (f n ) is defined in [START_REF] Bitseki Penda | Local bandwidth selection for kernel density estimation in a bifurcating Markov chain model[END_REF] and the bias term B hn (x) is defined in (4). Thanks to Theorem 3.10 applied to the sequence (f id ,n , n ≥ ≥ 0) and using that lim

n→+∞ |G n |/|T n | = 1/2, we get that |G n ||T n | -1 b -1 n N n,∅ (f n ) satisfies a moderate deviation principle in R with speed b 2 n
and rate function I defined by: I(y) = y 2 /(2 K 2 2 µ(x)) for all y ∈ R. To complete the proof of Theorem 3.2, it suffices to prove that [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] lim

n→+∞ |T n |h d n b n B hn (x) = 0.
Next, using that

µ(x -h n y) -µ(x) = d j=1 (µ(x 1 -h n y 1 , . . . , x j -h n y j , x j+1 , . . . , x d ) -µ(x 1 -h n y 1 , . . . , x j-1 -h n y j-1 , x j , x j+1 , . . . , x d )),
the Taylor expansion and Assumption 2.12, we get that, for some finite constant C > 0,

|T n | 1/2 h d/2 n B hn (x) = |T n |h d n R d h -d n K(h -1 n (x -y))µ(y)dy -µ(x) = |T n |h d n R d K(y)(µ(x -h n y) -µ(x)) dy ≤ C |T n |h d n d j=1 R d K(y) (h n |y j |) s s ! dy ≤ C |T n |h 2s+d n .
Now, [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] follows using the latter inequality and (6). This ends the proof of Theorem 3.2 for A n = T n . The proof is similar for A n = G n using f ,n = f 0 ,n .

4.3.

Proof of Theorem 3.3. . We begin the proof with A n = T n . We have the following decomposition:

(12) b -1 n |T n |h d n ( µ Tn (x) -µ(x)) K 2 µ A * n (x) ∨ n = T 1 (n) + T 2 (n)
where

T 1 (n) = ( K 2 µ(x)b n ) -1 |T n |h d n µ Tn (x) -µ(x) ; T 2 (n) = 1 K 2 µ A * n (x) ∨ n - 1
K 2 µ(x) b -1 n |T n |h d n µ Tn (x) -µ(x) .
First, we prove that

(13) T 2 (n) superexp = ==== ⇒ b 2 n 0.
Let δ > 0. For all r > 0, we have

P |T 2 (n)| > δ ≤ P b -1 n |T n |h d n µ Tn (x) -µ(x) > δ/r + P 1 K 2 µ A * n (x) ∨ n - 1 K 2 µ(x) > r .
This implies that (see for e.g [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], Lemma 1.2.15)

(14) lim sup n→+∞ 1 b 2 n log P |T 2 (n)| > δ ≤ max lim sup n→+∞ 1 b 2 n log P b -1 n |T n |h d n µ Tn (x) -µ(x) > δ/r ; lim sup n→+∞ 1 b 2 n log P 1 K 2 µ A * n (x) ∨ n - 1 K 2 µ(x) > r .
Using Theorem 3.2 and the contraction principle, we have

(15) lim sup n→+∞ 1 b 2 n log P b -1 n |T n |h d n µ Tn (x) -µ(x) > δ/r = - δ 2 2 K 2 µ(x)r 2 .
Following the step 1 of the proof of Theorem 6 in [START_REF] Bitseki Penda | Moderate deviation principle in nonlinear bifurcating autoregressive models[END_REF] and using Lemma 3.1, we can prove that

K 2 2 µ A * n (x) ∨ 2 n superexp = ==== ⇒ b 2 n K 2 2 µ(x).
Using Lemma B.2 in [START_REF] Bitseki Penda | Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application[END_REF], the latter convergence implies that

(16) 1 K 2 µ A * n (x) ∨ n superexp = ==== ⇒ b 2 n 1 K 2 µ(x) .
Using ( 14), ( 15) and ( 16), we get lim sup

n→+∞ 1 b 2 n log P |T 2 (n)| > δ ≤ - δ 2 2 K 2 µ(x)r 2 .
Since r can be taken arbitrarily close to 0, we get (13) and using [START_REF] Djellout | Moderate deviations for martingale differences and applications to o-mixing sequences[END_REF], this implies that

(17) b -1 n |T n |h d n ( µ Tn (x) -µ(x)) K 2 µ A * n (x) ∨ n superexp ∼ b 2 n T 1 (n).
Using Theorem 3.2 and the contraction principle, we get that T 1 (n) satisfies a moderate deviation principle on R with speed b 2 n and rate function I defined by: I (y) = y 2 /2 for all y ∈ R. Using (17) and Remark 2.7, we get the result of Theorem 3.3. 4.4. Proof of Corollary 3.6. Let a = (a 0 , . . . , a k ) t ∈ R k+1 . Let n > k. We consider the sequence f n = (f ,n , n ≥ ≥ 0) defined by f ,n = 2 /2 a K h n-(x -•) for all ∈ {0, . . . , k} and f ,n = 0 otherwise. We easily check that f n satisfies Assumptions 3.8. In particular, the asymptotic variance defined in [START_REF] Bitseki Penda | Moderate deviation principle in nonlinear bifurcating autoregressive models[END_REF] is given by σ

2 = k =0 2 a 2 K 2 2 µ(x). Observe that the linear combinaison M n (a), with coefficients a = (a 0 , . . . , a k ) t ∈ R k+1 , of the estimators |G n-| 1/2 h d/2 n-( µ G n-(x) -µ(x)), ∈ {0
, . . . , k} has the following decomposition:

(18) M n (a) = N n,∅ (f n ) + k =0 a |G n-| h d n- 1/2 B h n-(x),
where N n,∅ (f n ) is defined in [START_REF] Bitseki Penda | Local bandwidth selection for kernel density estimation in a bifurcating Markov chain model[END_REF] and the B h n-(x), ∈ {0, . . . , k}, are defined in (4). Applying Theorem 3.10, we get that b -1 n N n,∅ (f n ) satisfies a moderate deviation principle on R with speed b 2 n and rate function I x,a : R → R defined by ( 19)

I x,a (y) = y 2 2 k =0 2 a 2 K 2 2 µ(x)
, y ∈ R.

Using [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], we have that

lim n→+∞ 1 b n k =0 a |G n-| h d n- 1/2 B h n-(x) = 0.
Using Remark 2.7, this implies that

(20) 1 b n k =0 a |G n-| h d n- 1/2 B h n-(x) superexp = ==== ⇒ b 2 n 0.
Using [START_REF] Masry | Recursive probability density estimation for weakly dependent stationary processes[END_REF] and ( 20) we get that b -1 n M n (a) and b -1 n N n,∅ (f n ) satisfy the same moderate deviation principle. We then conclude that b -1 n M n (a) satisfies a moderate deviation principle on R with speed b 2 n and rate function I x,a defined in [START_REF] Mokkadem | Confidence bands for densities, logarithmic point of view[END_REF]. Since this is true for all vector a ∈ R k+1 , that is for all the linear combinaisons of the estimators

|G n-| 1/2 h d/2 n-( µ G n-(x) -µ(x)
), ∈ {0, . . . , k}, we get the result of Corollary 3.6.

Proof of Theorem 3.10

We begin with some notations. We will denote by C any unimportant finite constant which may vary from line to line (in particular C does not depend on n ∈ N nor on the considered sequence of functions

f n = (f ,n , n ≥ ≥ 0)). Let (p n , n ∈ N) be a non-decreasing sequence of elements of N * such that lim n→+∞ p 3 n b 2 n |G n-pn | -1 = 0.
When there is no ambiguity, we write p for p n . Let i, j ∈ T. We write i j if j ∈ iT. We denote by i ∧ j the most recent common ancestor of i and j, which is defined as the only u ∈ T such that if v ∈ T and v i, v j then v u. We also define the lexicographic order i ≤ j if either i j or v0 i and v1 j for v = i ∧ j. Let X = (X i , i ∈ T) be a BM C with kernel P and initial measure ν. For i ∈ T, we define the σ-field:

F i = {X u ; u ∈ T such that u ≤ i}.
By construction, the σ-fields (F i ; i ∈ T) are nested as F i ⊂ F j for i ≤ j.

We define for n ∈ N, i ∈ G n-pn and f n the martingale increments:

(21) ∆ n,i (f n ) = N n,i (f n ) -E [N n,i (f n )| F i ] and ∆ n (f n ) = i∈Gn-p n ∆ n,i (f n ), where (22)
N n,i (f n ) = |G n | -1/2 p =0 M iG p-( f ,n ) and iG p-= {ij, j ∈ G p-}.
We have:

i∈Gn-p n N n,i (f n ) = |G n | -1/2 pn =0 M G n-( f ,n ) = |G n | -1/2 n k=n-pn M G k ( fn-k,n ).
Using the branching Markov property, we get for i ∈ G n-pn :

(23) E [N n,i (f n )| F i ] = E [N n,i (f n )| X i ] = |G n | -1/2 pn =0 E Xi M G pn -( f ,n ) .
We have the following decomposition:

(24) N n,∅ (f n ) = ∆ n (f n ) + R 0 (n) + R 1 (n),
where ∆ n (f) is defined in [START_REF] Parzen | On estimation of a probability density function and mode[END_REF] and:

R 0 (n) = |G n | -1/2 n-pn-1 k=0 M G k ( fn-k,n ) and R 1 (n) = i∈Gn-p n E [N n,i (f n )| F i ] .
From (24), our goals will be achieved if we prove the following: Note that (25) and (26) mean that R 0 (n) and R 1 (n) are negligible in the sense of moderate deviations in such a way that using (24) and Remark 2.7, N n,∅ (f) and ∆ n (f) satisfy the same moderate deviation principle. To prove (27), the main method we will use is the moderate deviations for martingale (see [START_REF] Djellout | Moderate deviations for martingale differences and applications to o-mixing sequences[END_REF] for more details).

b -1 n R 0 (n) superexp = ==== ⇒ b 2 n 0; (25) b -1 n R 1 (n) superexp = ==== ⇒ b 2 n 0; (26) b -1 n ∆ n (f)
In the sequel, the sequence (2 -γn , n ∈ N) which appears in Assumption 3.8 will be denoted (h n , n ∈ N) in such a way that we have 2 -dγn/2 = h d/2 n . We have the following result. Proof. Let δ > 0. Using the Chernoff bound, we have, for all λ > 0, (28)

P b -1 n R 0 (n) > δ ≤ exp -λb n |G n | 1/2 δ E exp λ n-p-1 =0 M G ( fn-,n ) .
For all k ∈ {1, . . . , n -p} and for u ∈ T, we set

g p,k = k-1 r=0 2 r Q r fp+k-r,n and Z p,k (u) = g p,k (X u0 ) + g p,k (X u1 ) -2Qg p,k (X u ).
Then, using recursively the fact that

u∈G f (X u ) = u∈G -1 (f (X u0 ) + f (X u1 ) -2Qf (X u )) + u∈G -1 2Qf (X u ),
for all ≥ 1 and for some function f , we get

E exp λ n-p-1 =0 M G ( fn-,n ) = E exp λg p,n-p (X ∅ ) n-p-1 k=1 exp λ u∈G n-p-k-1 Z p,k (u) .
For all m ∈ {1, . . . , n -p -1}, we set

I m = E exp(λg p,n-p (X ∅ )) n-p-1 k=m exp(λ u∈G n-p-k-1 Z p,k (u)) .
Using the branching Markov property, we get the following decomposition:

I m = E exp(λg p,n-p (X ∅ )) J m n-p-1 k=m+1 exp(λ u∈G n-p-k-1 Z p,k (u)) , with J m = u∈Gn-p-m-1 E Xu exp(λZ p,m (u)) .
For all u ∈ G n-p-m-1 , we will upper bound the quantity E Xu [exp(λZ p,m (u))] and then J m . We claim that:

(29) |Z p,m (u)| ≤ M = C h -d/2 n ; (30) E Xu [Z p,m (u) 2 ] ≤ σ 2 m = C + C h d n m-1 r=0 (2α) r-1 2 1 {m>1} .
For that purpose, we plan to use the bound

(31) E exp(λZ) ≤ exp λ 2 σ 2 2(1 -λM/3) valid for any λ ∈ (0, 3/M ), any random variable Z such that |Z| ≤ M , E[Z] = 0 and E[Z 2 ] ≤ σ 2 .
For all u ∈ G n-p-m-1 and for all λ ∈ (0, Ch -d/2 n /3) we get, using ( 29)-( 31),

E Xu exp(λZ p,m (u)) ≤ exp λ 2 σ 2 m 2(1 -λM/3) .
For all m ∈ {1, . . . , n -p -1}, the latter inequality implies that

J m ≤ exp λ 2 σ 2 m |G n-p-m-1 | 2(1 -λM/3)
and

I m ≤ exp λ 2 σ 2 m |G n-p-m-1 | 2(1 -λM/3) I m+1 .
Recall that

I 1 = E exp λ n-p-1 =0
M G ( fn-,n ) . By recurrence, we get

E exp λ n-p-1 =0 M G ( fn-,n ) = I 1 ≤ exp λ 2 n-p-1 m=1 σ 2 m |G n-p-m-1 | 2(1 -λM/3) E exp λg p,n-p (X ∅ ) .
Using (i) and (ii) of Assumption 3.8 and (3), we have

|g p,n-p | ≤ | fn,n | + n-p-1 r=1 2 r |Q r-1 (Q fn-r,n )| ≤ Ch -d/2 n + Ch d/2 n n-p-1 r=1 (2α) r-1 .
This implies that

E exp λ n-p-1 =0 M G ( fn-,n ) ≤ exp λ 2 n-p-1 m=1 σ 2 m |G n-p-m-1 | 2(1 -λM/3) × exp λ C h -d/2 n + λ C h d/2 n n-p-2 r=0 (2α) r .
Distinguishing the cases 2α ≤ 1, 1/2 < 2α ≤ √ 2 and 2α > √ 2 and using (5) for 2α > 1, we get

E exp λ n-p-1 =0 M G ( fn-,n ) ≤ exp c 1 λ 2 |G n-p | 2(1 -c 2 λh -d/2 n /3) exp c 3 λh -d/2 n ,
where c 1 , c 2 and c 3 are some positive constants. The latter inequality and (28) imply that

P b -1 n R 0 (n) > δ ≤ exp -λb n |G n | 1/2 δ + c 1 λ 2 |G n-p | 2(1 -c 2 λh -d/2 n /3) exp c 3 λh -d/2 n .
Taking

1 λ = 3 b n |G n | 1/2 δ 2 c 2 b n |G n | 1/2 h -d/2 n δ + 3 c 1 |G n-p | ,
we are led to

P b -1 n R 0 (n) > δ ≤ C exp - 3 δ 2 b 2 n |G n | 2(c 2 δ b n |G n | 1/2 h -d/2 n + 3 c 1 |G n-p |) .
Since we can do the same thing for -f n instead of f n , we get that

(32) P b -1 n |R 0 (n)| > δ ≤ 2 C exp - 3 δ 2 b 2 n |G n | 2(c 2 δ b n |G n | 1/2 h -d/2 n + 3 c 1 |G n-p |) .
Finally, in the latter inequality, taking the log, dividing by b 2 n and letting n goes to infinity, we get the result of Lemma 5.1. Now, to end the proof, we will prove ( 29) and (30).

Proof of (29). Using Assumption 2.9, (i) and (ii) of Assumption 3.8 and Assumption 2.13, we get

|Z p,m (u)| ≤ C fp+1,n ∞ + C(1 + 2α)( m-1 r=1 (2α) r-1 Qf p+m-r,n ∞ )1 {m>1} ≤ C h -d/2 n + C h d/2 n m-1 r=0 (2α) r ≤ C h -d/2 n .
Proof of (30). Using the branching Markov property for the second inequality, Assumption 2.9 for the fourth inequality and (i) and (ii) of Assumption 3.8 for the last inequality, we get

E Xu [Z p,m (u) 2 ] ≤ E Xu [(g p,m (X u0 ) + g p,m (X u1 )) 2 ] ≤ CQ(g 2 p,m )(X u ) ≤ CQ( f 2 p+1,n )(X u ) + CQ m-1 r=1 2 r Q r-1 (Q fp+m-r,n ) 2 (X u ) 1 {m>1} ≤ C Q f 2 p+1,n ∞ + m-1 r=1 (2α) r-1 Qf p+m-r,n ∞ 2 1 {m>1} ≤ C + C h d n m-1 r=0 (2α) r 2 1 {m>1} .
Next, we have the following result.

Lemma 5.2. Under the assumptions of Theorem 3.10, we have b

-1 n R 1 (n) superexp = ==== ⇒ b 2 n 0.
1 In fact, we use the following. For α, β, γ > 0 and h(x) = -αx

+ βx 2 2(1-γx) we have h(x * ) = -α 2 2(β+αγ) for the choice x * = α 2αγ+β ∈ (0, 1/γ).
Proof. We have, using ( 23) and ( 49),

(33) R 1 (n) = |G n | -1/2 M Gn-p (g p,n ) where g p,n = p =0 2 p-Q p-f ,n .
We follow the same arguments that in the proof of Lemma 5.1. For all m ∈ {1, . . . , n -p} and for all u ∈ T, we set

Z p,m (u) = 2 m-1 Q m-1 g p (X u0 ) + 2 m-1 Q m-1 g p (X u1 ) -2 m Q m g p (X u ).
We also consider the following quantities for m ∈ {1, . . . , n -p} and λ > 0:

I m = E exp λ2 n-p Q n-p g p,n (X ∅ ) n-p k=m exp λ u∈G n-p-k Z p,k (u)
and

J m = u∈Gn-p-m E Xu exp λZ p,m (u) .
Note that using the branching Markov property, we have (34)

I m = E exp λ2 n-p Q n-p g p,n (X ∅ ) n-p k=m+1 exp λ u∈G n-p-k Z p,k (u) J m .
As for (29)-( 30), for all m ∈ {1, . . . , n -p} and u ∈ G n-p-m , one can prove that (35)

|Z p,m (u)| ≤ M = Ch -d/2 and E Xu Z p,m (u) 2 ≤ σ 2 m = C1 {m=1} + Ch d n p =0 (2α) p+m--2 2 .
Using (31) and (35), we have, for all u ∈ G n-p-m and for all λ ∈ (0, Ch -d/2 /3),

E Xu exp(λZ p,m (u)) ≤ exp λ 2 σ 2 m 2(1 -λM/3) .
The latter inequality and (34) imply that

I m ≤ exp λ 2 σ 2 m |G n-p-m | 2(1 -λM/3) I m+1 .
By recurrence, this implies that (36)

I 1 ≤ exp λ 2 n-p m=1 σ 2 m |G n-p-m | 2(1 -λM/3) E exp λ2 n-p Q n-p g p,n (X ∅ ) .
Using (i) and (ii) of Assumption 3.8 and Assumption 2.9, we get

(37) |g p,n | ≤ Ch d/2 n p =0 (2α) n-.
From (36), (37) and according to the value of α, we have, for some positive constants c 1 , c 2 and c 3 (recall the definition of M and σ 2 m given in (35)):

I 1 ≤ C exp λ 2 c 1 |G n-p | 2(1 -λc 2 h -d/2 n /3) if 2α ≤ 1; I 1 ≤ exp λc 3 (2α) n h d/2 n exp λ 2 c 1 |G n-p |(1 + (2α) 2p h d n ) 2(1 -λc 2 h -d/2 n /3) if 1 < 2α ≤ √ 2; I 1 ≤ exp λc 3 (2α) n h d/2 n exp λ 2 c 1 |G n-p |(1 + (2α) 2p h d n + 2 p (2α 2 ) n h d n ) 2(1 -λc 2 h -d/2 n /3) if 2α > √ 2.
Recall that

I 1 = E[exp(λM Gn-p (g p,n ))].
Using the Chernoff bound and (33), we have for all λ ∈ (0, Ch -d/2 n /3) and for all δ > 0,

P b -1 n R 1 (n) > δ ≤ exp -λb n |G n | 1/2 δ I 1 . Taking λ =          3bn|Gn| 1/2 δ 2c2bn|Gn| 1/2 h -d/2 n δ + 3c1|Gn-p| if 2α ≤ 1 3bn|Gn| 1/2 δ 2c2bn|Gn| 1/2 h -d/2 n δ + 3c1|Gn-p|(1+(2α) 2p h d n ) if 1 < 2α ≤ √ 2 3bn|Gn| 1/2 δ 2c2bn|Gn| 1/2 h -d/2 n δ + 3c1|Gn-p|(1+(2α) 2p h d n +2 p (2α 2 ) n h d n ) if 1 < 2α ≤ √ 2,
and since we can do the same things for -f n instead of f n , we get, if 2α ≤ 1 :

P b -1 n |R 1 (n)| > δ ≤ C exp - 3b 2 n |G n |δ 2 2(c 2 b n |G n | 1/2 h -d/2 n δ + 3c 1 |G n-p |) ; if 1 < 2α ≤ √ 2 : P b -1 n |R 1 (n)| > δ ≤ 2 exp c 3 (2α) n h d/2 n b n |G n | 1/2 2c 2 b n |G n | 1/2 h -d/2 n δ + 3c 1 |G n-p |(1 + (2α) 2p h d n ) × exp - 3b 2 n |G n |δ 2 2(c 2 b n |G n | 1/2 h -d/2 n δ + 3c 1 |G n-p |(1 + (2α) 2p h d n )) ; if 2α > √ 2 : P b -1 n |R 1 (n)| > δ ≤ 2 exp c 3 (2α) n h d/2 n b n |G n | 1/2 2c 2 b n |G n | 1/2 h -d/2 n δ + 3c 1 |G n-p |(1 + (2α) 2p h d n + 2 p (2α 2 ) n h d n ) × exp - 3b 2 n |G n |δ 2 2(c 2 b n |G n | 1/2 h -d/2 n δ + 3c 1 |G n-p |(1 + (2α) 2p h d n + 2 p (2α 2 ) n h d n ))
.

Finally, applying the log to each of these last three inequalities, dividing by b 2 n , letting n goes to infinity and using (6) and Assumption 2.13, we get the result of Lemma 5.2.

From (24), Lemmas 5.1 and 5.2, we have

(38) b -1 n N n,∅ (f n ) superexp ∼ b 2 n b -1 n ∆ n (f n ).
As a consequence, using Remark 2.7, b -1 n N n,∅ (f n ) and b -1 n ∆ n (f n ) satisfy the same moderate deviation principle.

We now study the martingale part ∆ n (f n ) of the decomposition (24). The bracket V (n) of ∆ n (f n ) is defined by:

V (n) = i∈Gn-p n E ∆ n,i (f n ) 2 |F i .
Using ( 22) and ( 21), we write:

(39) V (n) = |G n | -1 i∈Gn-p n E Xi   pn =0 M G pn-( f ,n ) 2   -R 2 (n) = V 1 (n) + 2V 2 (n) -R 2 (n), with: V 1 (n) = |G n | -1 i∈Gn-p n pn =0 E Xi M G pn-( f ,n ) 2 , V 2 (n) = |G n | -1 i∈Gn-p n 0≤ <k≤pn E Xi M G pn -( f ,n )M G pn -k ( fk,n ) , R 2 (n) = i∈Gn-p n E [N n,i (f n )|X i ] 2 .
We have the following result. Proof. Using the branching Markov property, we have

R 2 (n) = |G n | -1 M Gn-p (g p ) with g p = p =0 2 p-Q p-f ,n 2 .
Using Assumption 2.9 and (i) and (ii) of Assumption 3.8, we get

g p ∞ ≤ C fp,n 2 ∞ + C ( p-1 =0 2 p-Q p-f ,n ) 2 ∞ ≤ Ch -d n + C p-1 =0 (2α) p-h d/2 n 2 ≤ Ch -d n 1 {2α≤1} + C(h -d n + h d n (2α) 2p )1 {2α>1} . This implies that (40) R 2 (n) ≤ C|G n | -1 h -d n 1 {2α≤1} + C(|G n | -1 h -d n + (2α 2 ) p h d n |G n-p | -1 ) 1 {2α>1} .
Recall that h n = 2 -nγ with γ ∈ (0, 1/d). Using Assumption 2.13, we conclude from (40) that R 2 (n) is bounded by a deterministic sequence which converge to 0. As a consequence, using Remark 2.8, we get the result of Lemma 5.3.

Recall σ 2 given in [START_REF] Bitseki Penda | Moderate deviation principle in nonlinear bifurcating autoregressive models[END_REF]. We have the following result.

Lemma 5.4. Under the Assumptions of Theorem 3.10, we have

V 1 (n) superexp = ==== ⇒ b 2 n σ 2 .
Proof of (42). We set 3) and (i) and (ii) of Assumption 3.8, we get 5) and the fact that (h n , n ∈ N) converges to 0, we get that the sequence (c n , n ∈ N) converges to 0. Thus, we have that V 4 (n) is bounded by a deterministic sequence which converges to 0. Then (42) follows using Remark 2.8. Proof. Using (51), we get:

h (n) ,k = 2 k-Q p-1-( +k) P Q k f ,n ⊗ 2 and H 4,n = p-1 =0 p--1 k=0 h (n) ,k in such a that V 4 (n) = |G n-p | -1 M Gn-p (H 4,n ). Using, (
|h (n) ,k | ≤ 2 k-P(|Q k f ,n |⊗ 2 ) ≤ C2 k-h d n α 2k . This implies that |H 4,n | ≤ c n and then that |V 4 (n)| ≤ c n , where the sequence (c n , n ∈ N) is defined by c n = Ch d n 1 {2α 2 ≤1} + Ch d n (2α 2 ) p 1 {2α 2 >1} Using (
V 2 (n) = V 5 (n) + V 6 (n), with V 5 (n) = |G n | -1 i∈Gn-p 0≤ <k≤p 2 p-Q p-k fk,n Q k-f ,n (X i ), V 6 (n) = |G n | -1 i∈Gn-p 0≤ <k<p p-k-1 r=0 2 p-+r Q p-1-(r+k) P Q r fk,n ⊗ sym Q k-+r f ,n (X i ).
First, we set

h (n) k, ,r = 2 r-Q p-1-(r+k) P Q r fk,n ⊗ sym Q k-+r f ,n and H 6,n = 0≤ <k<p p-k-1 r=0 h (n) k, ,r
in such a way that V 6 (n) = |G n-p | -1 M Gn-p (H 6,n ). Using, (3) and (i) and (ii) of Assumption 3.8, we get

|h (n) k, ,r | ≤ Ch d n (2α 2 ) r α k-. This implies that |H 6,n | ≤ c n and then that V 6 (n) ≤ c n , where the sequence (c n , n ∈ N) is defined by c n = Ch d n 1 {2α 2 ≤1} + C(2α 2 ) p h d n 1 {2α 2 >1}
. Since the sequence (c n , n ∈ N) is deterministic and converges to 0, it follows, using Remark 2.8, that V 6 (n

) superexp = ==== ⇒ b 2 n 0.
Next, for the term V 5 (n), we have for all k > :

2 -Q p-k fk,n Q k-f ,n ≤ 2 -Q p-k | fk,n ||Q k-f ,n | ≤ C2 -h d/2 n α k-Q p-k (| fk,n |) ≤ Cα p (2α) -1 {k=p} + Ch d n (2α) -α k 1 {k≤p-1} ,
where we used (3) for the second inequality and (i) and (ii) of Assumption 3.8 for the second and the last inequality. Using the latter inequality in V 5 (n), we get

|V 5 (n)| ≤ C 2 -p 1 {2α<1} + α p 1 {2α≥1} + h d n .
We thus have that V 5 (n) is bounded by a deterministic sequence which converges to 0. It then follows from Remark 2.8 that

V 5 (n) superexp = ==== ⇒ b 2 n 0.
From the foregoing, we get the result of Lemma since

V 2 (n) = V 5 (n) + V 6 (n).
As a direct consequence of (39) and the Lemmas 5.3, 5.4 and 5.5, we have the following result.

Lemma 5.6. Under the Assumptions of Theorem 3.10, we have

V (n) superexp = ==== ⇒ b 2 n σ 2 .
We now study the 4th-order exponential moment condition. We stress that this condition imply in particular the exponential Lindeberg condition (condition (C3) in Proposition 6.1). We have the following result.

E[∆ n,i (f n ) 4 |F i ] > δ = -∞ ∀δ > 0.
Proof. For all i ∈ G n-p , we have

(43) E ∆ n,i (f n ) 4 |F i ≤ 16(p + 1) 3 2 -2n p =0 E Xi M G p-( f ,n ) 4 ,
where we have used the definition of ∆ n,i (f n ), the inequality ( r k=0 a k ) 4 ≤ (r + 1) 3 r k=0 a 4 k and the branching Markov property. Using (43), we get

(44) b 2 n i∈Gn-p E[∆ n,i (f n ) 4 |F i ] ≤ Cb 2 n p 3 2 -2n p =0 i∈Gn-p h n, (X i ), where h n, (x) = E x [M G p-( f ,n ) 4
]. We will now prove that the right hand side of (44) converges superexponentially to 0 at the speed b 2 n , that is lim sup

n→+∞ 1 b 2 n log P Cb 2 n p 3 2 -2n p =0 i∈Gn-p h n, (X i ) > δ = -∞.
For that purpose, we will treat the case = p, = p -1 and finally the case ∈ {0, . . . , p -2}. First, we treat the case = p. Set g p,n = f 4 p,n . We have

(45) b 2 n p 3 2 -2n i∈Gn-p h n,p (X i ) = b 2 n p 3 2 -2n i∈Gn-p gp,n (X i ) + b 2 n p 3 2 -2n |G n-p | µ, g p,n . Since b 2 n p 3 2 -2n |G n-p | µ, g p,n ≤ p 3 2 -p b 2 n (|G n |h d n ) -1
→ 0 as n → 0, it suffices to prove that the first term of the right hand side in (45) converges superexponentially to 0 at the speed b 2 n , that is, for all δ > 0, (46) lim sup

n→+∞ 1 b 2 n log P b 2 n p 3 2 -2n | i∈Gn-p gp,n (X i )| > δ = -∞.
As in the proof of Lemma 5.2, we can prove that

P b 2 n p 3 2 -2n | i∈Gn-p gp,n (X i )| > δ ≤ C exp - δ 2 |G n | 2 h 2d n Cp 3 b 2 n (δ + Cp 3 b 2 n (|G n+p |h d n ) -1 )
.

Taking the log and dividing by b 2 n , we get (46). Next, for ∈ {0, . . . , p-1}, we plan to prove that the quantity b 2 n p 3 2 -2n p-1 =0 i∈Gn-p h n, (X i ) is bounded by a deterministic sequence which converges to 0. First, for = p -1, using the branching Markov property, (i) and (ii) of Assumption 3.8, we have, for all i ∈ G n-p,

h n,p-1 (X i ) = E Xi [M G1 ( fp-1,n ) 4 ] ≤ CQ( f 4 p-1,n ) ≤ Ch -d n .
Using [START_REF] Bitseki Penda | Autoregressive functions estimation in nonlinear bifurcating autoregressive models[END_REF], this implies that

b 2 n p 3 2 -2n i∈Gn-p h n,p-1 (X i ) ≤ C b 2 n 2 -p p 3 (|G n |h d n ) -1 → 0 as n → +∞.
Now we consider the case ∈ {0, . . . , p -2}. From Lemma 6.4 with f replaced by f ,n and ν by the Dirac mass at X i (δ Xi ), we have

(47) b 2 n p 3 2 -2n p-2 =0 i∈Gn-p h n, (X i ) ≤ b 2 n |G n | -2 p 3 p-2 =0 i∈Gn-p 9 j=1 |ψ j,p-|(X i ).
For all j ∈ {1, . . . , 9}, we will upper bound each term of the right hand side in (47) by a deterministic sequence which converges to 0.

Upper bound of b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 1,p-|(X i ).
Using (i) of Assumption 3.8, we have

|ψ 1,p-| ≤ C2 p-Q p-(f 4 ,n ) ≤ C2 p-h -d n .
Using [START_REF] Bitseki Penda | Autoregressive functions estimation in nonlinear bifurcating autoregressive models[END_REF], this implies that

b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 1,p-|(X i ) ≤ Cb 2 n p 3 (|G n |h d n ) -1 → 0 as n → +∞. Upper bound of b 2 n |G n | -3 p 3 p =0 i∈Gn-p |ψ 2,p-|(X i ).
Using Assumption 2.9 and (i) and (ii) of Assumption 3.8 for the second inequality, we get

|ψ 2,p-| ≤ C2 2(p-) p--1 k=0 2 -k Q k P(|Q p-k-1-( f 3 ,n )| ⊗ sym |Q p--k-2 (Q f ,n )|) ≤ C2 2(p-) p--1 k=0 2 -k α p--k ≤ C2 p-1 {2α<1} + (p -)1 {2α=1} + (2α) p-1 {2α>1} .
Using ( 6) and ( 5), this implies that

b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 2,p-|(X i ) ≤ Cb 2 n |G n | -1 p 4 1 {2α≤1} + (2α) p 1 {2α>1} → 0 as n → +∞. Upper bound of b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 3,p-|(X i ).
Using (i) and (ii) of Assumption 3.8 for the second inequality, we get

|ψ 3,p-| ≤ 2 2(p-) p--1 k=0 2 -k Q k P(Q p--k-1 ( f 2 ,n )⊗ 2 ) ≤ C2 2(p-) p--1 k=0 2 -k ≤ C2 2(p-) .
Using [START_REF] Bitseki Penda | Autoregressive functions estimation in nonlinear bifurcating autoregressive models[END_REF], this implies that

b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 3,p-|(X i ) ≤ C b 2 n p 3 2 -n+p → 0 as n → +∞. Upper bound of b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 4,p-|(X i ).
Using Assumption (2.9) and (i) and (ii) of Assumption 3.8 for the second inequality, we get

|ψ 4,p-| ≤ C2 4(p-) P P |Q p--2 f ,n | ⊗ 2 ⊗ 2 ≤ C2 4(p-) α 4(p--2) h 2d
n . Using ( 6) and ( 5), this implies that

b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 4,p-|(X i ) ≤ C (b 2 n p 4 2 -n-p h 2d n 1 {2α 2 ≤1} + b 2 n p 3 2 -n+p (2α 2 ) 2p h 2d n 1 {2α 2 >1} ) → 0 as n → +∞. Upper bound of b 2 n |G n | -3 p 3 p =0 i∈Gn-p |ψ 5,p-|(X i ).
Using Assumption (2.9) and (i) and (ii) of Assumption 3.8 for the second inequality, we get

|ψ 5,p-| ≤ C 2 4(p-) p--1 k=2 k-1 r=0 2 -2k-r Q r P Q k-r-1 P |Q p--k-1 f ,n | ⊗ 2 ⊗ 2 ≤ C 2 4(p-) p--1 k=2 k-1 r=0 2 -2k-r h 2d n α 4(p--k) ≤ C h 2d n 2 2(p-) 1 {2α 2 <1} + (p -)1 {2α 2 =1} + (2α 2 ) 2(p-) 1 {2α 2 >1}
. Using ( 6) and ( 5), this implies that

b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 5,p-|(X i ) ≤ C (b 2 n p 4 2 -n+p h 2d n 1 {2α 2 ≤1} + b 2 n p 3 2 -n+p (2α 2 ) 2p h 2d n 1 {2α 2 >1} ) → 0 as n → +∞. Upper bound of b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 6,p-|(X i ).
Using Assumption (2.9) and (i) and (ii) of Assumption 3.8 for the second inequality, we get

|ψ 6,n | ≤ C 2 3(p-) p--1 k=1 k-1 r=0 2 -k-r Q r P Q k-r-1 P |Q p--k-1 f ,n | ⊗ 2 ⊗ sym Q p--r-1 ( f 2 ,n ) ≤ C 2 3(p-) p--1 k=1 k-1 r=0 2 -k-r h d n α 2(p--k) ≤ C h d n 2 2(p-) 1 {2α 2 <1} + (p -) 1 {2α 2 =1} + (2α 2 ) p-1 {2α 2 >1} .
Using ( 6) and ( 5), this implies that

b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 6,p-|(X i ) ≤ C (b 2 n p 4 2 -n+p h d n 1 {2α 2 ≤1} + b 2 n p 3 2 -n+p (2α 2 ) 2p h d n 1 {2α 2 >1} ) → 0 as n → +∞. Upper bound of b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 7,p-|(X i ).
In the same way as for ψ 6,p-, we have

b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 7,p-|(X i ) ≤ C (b 2 n p 4 2 -n h d n 1 {2α≤1} + b 2 n p 3 2 -n+p (2α 2 ) 2p h d n 1 {2α>1} ) → 0 as n → +∞. Upper bound of b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 8,p-|(X i ).
Using Assumption (2.9) and (i) and (ii) of Assumption 3.8 for the second inequality, we get

|ψ 8,p-| ≤ C 2 4(p-) p--1 k=2 k-1 r=1 r-1 j=0 2 -k-r-j Q j P Q r-j-1 P |Q p--1-r f ,n | ⊗ 2 ⊗ sym Q k-j-1 P |Q p--1-k f ,n | ⊗ 2 ≤ C 2 4(p-)

Appendix

We recall here a simplified version of Theorem 1 in [START_REF] Djellout | Moderate deviations for martingale differences and applications to o-mixing sequences[END_REF]. the real martingale (M n , n ∈ N) with respect to the filtration (H n , n ∈ N) and we denote ( M n , n ∈ N) its bracket. We have the following many-to-one formulas. Ideas of the proofs can be found in [START_REF] Guyon | Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging[END_REF] and [START_REF] Bitseki Penda | Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application[END_REF].

Lemma 6.2. Let f, g ∈ B(S), x ∈ S and n ≥ m ≥ 0. Assuming that all the quantities below are well defined, we have:

E x [M Gn (f )] = |G n | Q n f (x) = 2 n Q n f (x), (49)
E x M Gn (f ) 2 = 2 n Q n (f 2 )(x) + n-1 k=0 2 n+k Q n-k-1 P Q k f ⊗ Q k f (x), (50) E x [M Gn (f )M Gm (g)] = 2 n Q m gQ n-m f (x) (51) + m-1 k=0 2 n+k Q m-k-1 P Q k g ⊗ sym Q n-m+k f (x).
We recall the following result due to Bochner (see [START_REF] Parzen | On estimation of a probability density function and mode[END_REF]Theorem 1A] which can be easily extended to any dimension d ≥ 1). We also give some bounds on E x M Gn (f ) 4 , see the proof of Theorem 2.1 We will use the notation: g⊗ 2 = g ⊗ g.

Lemma 6.4. There exists a finite constant C such that for all f ∈ B(S),n ∈ N and ν a probability measure on S, assuming that all the quantities below are well defined, there exist functions ψ j,n for 1 ≤ j ≤ 9 such that:

E ν M Gn (f ) 4 = 9 j=1
ν, ψ j,n , and, with h k = Q k-1 (f (notice that either |ψ j | or | ν, ψ j | is bounded), writing νg = ν, g : 2 -k-r-j Q j |P Q r-j-1 |P h n-r ⊗ sym Q k-r-1 P h n-k ⊗ 2 ⊗ sym h n-j |.

|ψ 1,n | ≤ C 2 n Q n (f 4 ), |νψ 2,n | ≤ C 2 2n n-1 k=0 2 -k |νQ k P Q n-k-1 (f 3 ) ⊗ sym h n-k |, |ψ 3,n | ≤ C2 2n n-1 k=0 2 -k Q k P Q n-k-1 (f 2 )⊗ 2 , |ψ 4,n | ≤ C 2 4n P |P(h n-1 ⊗ 2 ) ⊗ 2 | , |ψ 5,n | ≤ C 2 4n n-1 k=2 k-1 r=0 2 -2k-r Q r P Q k-r-1 |P(h n-k ⊗ 2 )|⊗ 2 ,

Assumption 2 . 11 (

 211 Regularity of the kernel function and the bandwidth).

Remark 3 . 5 . 4 Corollary 3 . 6 .

 35436 Corollary 3.4 yields a simple confidence interval for µ(x), of decreasing size b n / |A n | h d n and with level asymptotically close to 1 -exp(-(b 2 n δ 2 )/2). Using the structure of the asymptotic variance σ 2 in (7), we can prove the following multidimensional result which the proof is given in Section 4.Under the hypothesis of Theorem 3.2, we have, for x in the set of continuity of µ and for all k ≥ 0, b -1

  satisfies a MDP on S with speed b 2 n and rate function I. (27)

Lemma 5 . 1 .

 51 Under the assumptions of Theorem 3.10, we have b -1 n R 0 (n)

Lemma 5 . 3 .

 53 Under the Assumptions of Theorem 3.10, we have R 2 (n)

Lemma 5 . 5 .

 55 Under the Assumptions of Theorem 3.10, we have V 2 (n)

Lemma 5 . 7 .

 57 Under the Assumptions of Theorem 3.10, we have

3 p 3 p≤ C (b 2 n p 5 2 -n+p h 2d n 1 {2α 2 ≤1} + b 2 n p 3 2 2 n p 3 2 -2n p- 1 =0 1 =0b 2 n 2 n=

 33222221122 -r-j h 2d n α 4(p-)-2r-2k ≤ C h 2d n 2 2(p-) 1 {2α 2 <1} + (p -) 2 1 {2α 2 =1} + (2α 2 ) 2(p-) 1 {2α 2 >1}. Using (6) and[START_REF] Bitseki Penda | Adaptive estimation for bifurcating Markov chains[END_REF], this implies thatb 2 n |G n | -2 p =0 i∈Gn-p |ψ 8,p-|(X i ) ≤ C (b 2 n p 5 2 -n+p h 2d n 1 {2α 2 ≤1} + b 2 n p 3 2 -n+p (2α 2 ) 2p h 2d n 1 {2α 2 >1} ) → 0 as n → +∞. Upper bound of b 2 n |G n | -2 p 3 p =0 i∈Gn-p |ψ 9,p-|(X i ).In the same way as for ψ 8,p-, we haveb 2 n |G n | -2 p =0 i∈Gn-p |ψ 9,p-|(X i ) -n+p (2α 2 ) 2p h 2d n 1 {2α 2 >1}) → 0 as n → +∞. Putting together all the upper bounds for ∈ {0, . . . , p -1} and using (43) and (47), we deduce that b i∈Gn-p h n, (X i ) is bounded by a deterministic sequence which converges to 0. As a consequence, it follows, using Remark 2.i∈Gn-p h n, (X i ) > δ = -∞. Finally, using (43), (44), (46), we get lim sup n→+∞ 1 log P i∈Gn-p E[∆ n,i (f n ) 4 |F i ] > δ b -∞ ∀δ > 0.

Proposition 6 . 1 .E

 61 Let (b n ) a sequence satisfying b n is increasing, b n -→ +∞, b n √ n -→ 0, such that c(n) := √ n/b n is non-decreasing, and define the reciprocal function c -1 (t) by c -1 (t) := inf{n ∈ N : c(n) ≥ t}.Under the following conditions: (C1) there existsQ ∈ R * + such that for all δ > 0k -M k-1 | > b n √ n H k-1 ) = -∞, (C3) for all a and for all δ > 0,|M k -M k-1 | 2 1 {|M k -M k-1 |≥a √ n bn } H k-1 > δ = -∞, (M n /(b n √ n)) n∈Nsatisfies the MDP on R with the speed b 2 n /n and rate function I(x) = x 2 2Q .

Lemma 6 . 3 .

 63 Let (h n , n ∈ N) be a sequence of positive numbers converging to 0 as n goes to infinity. Let g : R d → R be a measurable function such thatR d |g(x)|dx < +∞. Let f : R d → R be a measurable function such that f ∞ < +∞, R d |f (y)| dy < +∞ and lim |x|→+∞ |x|f (x) = 0. Define g n (x) = h -d n R d f (h -1 n (x -y))g(y)dy. Then, we have at every point x of continuity of g, lim n→+∞ g n (x) = g(x) R f (y)dy.

222

  -k-r Q r |P Q k-r-1 P h n-k ⊗ 2 ⊗ sym Q n-r-1 (f 2 ) |, |νψ 7,n | ≤ C 2 3n -k-r |νQ r P Q k-r-1 P h n-k ⊗ sym Q n-k-1 (f 2 ) ⊗ sym h n-r |, |ψ 8,n | ≤ C 2 4n -k-r-j Q j P |Q r-j-1 P h n-r ⊗ 2 | ⊗ sym |Q k-j-1 P h n-k ⊗ 2 | , |ψ 9,n | ≤ C 2 4n

Proof. We have the following decomposition which is a consequence of (50):

Now, the result of Lemma 5.4 is a direct consequence of the following:

To end the proof, we will now prove (41) and (42).

Proof of (41). Set

Following the same arguments that in the proof of Lemmas 5.1 and 5.2, we get after studious calculations: if 2α ≤ 1,

;

Taking the log, dividing by b 2 n , letting n goes to the infinity and using (6) and Assumption 2.13, we get lim sup

Next, using (iii) of Assumption 3.8, we get lim n→+∞ H

[n] 3 (f n ) = σ 2 . This ends the proof of (41) since (H

For Chen-Ledoux type condition, we have the following result.

Proof. For all i ∈ G n-p , using [START_REF] Parzen | On estimation of a probability density function and mode[END_REF] we have (48)

with N n,i (f) defined in [START_REF] Roussas | Nonparametric estimation in Markov processes[END_REF]. Following the proof of (32), we get

, we have

, where we used (49) and the branching Markov property for the first equality, Chernoff bound for the first inequality and (3) for the last inequality. Doing the same thing for -f instead of f, we get

From the foregoing, we get, using (48),

Finally, taking the log and dividing by b 2 n in the latter inequality, we get the result of Lemma 5.8.

We can now use Proposition 6.1 to deduce from Lemmas 5.6, 5.7 and 5.8 that ∆ n (f n ) satisfies a moderate deviation principle with speed b 2 n and rate function I defined by: I(x) = x 2 /(2σ 2 ) for all x ∈ R, with the finite variance σ 2 defined in [START_REF] Bitseki Penda | Moderate deviation principle in nonlinear bifurcating autoregressive models[END_REF]. Using (38) and Remark 2.7, we then deduce Theorem 3.10.


Najia Bouha
email: najia.bouha@gmail.com

Gildas Morvan
email: gildas.morvan@univ-artois.fr

Hassane Abouaïssa
email: hassane.abouaissa@univ-artois.fr

Yoann Kubera
email: yoann.kubera@univ-artois.fr

A FIRST STEP TOWARDS DYNAMIC HYBRID TRAFFIC MODELING

Keywords: Agent-based modeling, Multi-level modeling, Intelligent transportation systems, Simulation, Traffic flow

Hybrid traffic modeling and simulation provide an important way to represent and evaluate large-scale traffic networks at different levels of details. The first level, called "microscopic" allows the description of individual vehicles and their interactions as well as the study of driver's individual behavior. The second, based on the analogy with fluidic dynamic, is the "macroscopic" one and provides an efficient way to represent traffic flow behavior in large traffic infrastructures, using three aggregated variables: traffic density, mean speed and traffic volume. An intermediate level called "mesoscopic" considers a group of vehicles sharing common properties such as a same origin and destination. The work conducted in this paper presents a first step allowing simulation of wide area traffic network on the basis of dynamic hybrid modeling, where the representation associated to a network section can change at runtime. The proposed approach is implemented in a simulation platform, called JAM-FREE.

INTRODUCTION

Severe congestion is a daily problem which leads to a continuously growth of direct and indirect cost. Traffic congestion which can be recurrent typical to rush hours or non recurrent due to accidents, works, . . . represents a major preoccupation of many transportation institutions and practitioners, calls for an efficient and intelligent dynamic management. Accordingly, several works were undertaken to study the traffic phenomena, to implement effective strategies for an optimal use of the existing infrastructure and to minimize the congestion effects as well as a high quality of service. Nevertheless, the implementation of traffic measurement and control algorithms calls for a deep understanding of the traffic phenomena. In this context, traffic flow modeling and simulation play an important role and constitute efficient tools to perform tasks such as traffic prediction and monitoring, traffic control and forecasting, the repercussion of the construction of new parts on infrastructure onto the global behavior of the traffic flow, . . . According to the defined objective, several models have been developed and can be classified into microscopic, mesoscopic and macroscopic models. However, to simulate large-scale road networks, it can be interesting to integrate different representations in the same framework which leads to the so-called "hybrid modeling" as shown on fig. 1. Note that the concept of hybrid modeling has different meanings according to the studied domain. Here, hybrid modeling means the coupling of different models.

In this paper, the first step is devoted to the integration of microscopic and macroscopic models into a single framework. The concept of hybrid modeling has been developed by several authors. Hence, some existing hybrid micro-macro traffic models are shown in Table I. The models presented in Table I share the same limitation: connections between levels are fixed a priori and cannot be changed at runtime. Therefore, to be able to observe some emerging phenomena such as congestion formation or to find the exact location of a jam in a large macro section, a dynamic hybrid modeling approach is needed.

The work presented in this paper is devoted to overcome these shortcomings and proposes the first step towards the development and implementation of dynamic hybrid models. Such a new approach provides an efficient way to change the model micro model macro model [START_REF] Magne | Towards an hybrid macro-micro traffic flow simulation model[END_REF] SITRA-B+ SIMRES [START_REF] Poschinger | Coupling of concurrent macroscopic and microscopic traffic flow models using hybrid stochastic and deterministic disaggregation, Transportation and Traffic Theory for the 21st century[END_REF] IDM Payne [START_REF] Bourrel | Mixing micro and macro representations of traffic flow: a hybrid model based on the LWR theory[END_REF] optimal velocity LWR [START_REF] Mammar | Hybrid model based on second-order traffic model[END_REF] ARZ [START_REF] Espié | Hybrid traffic model coupling macro-and behavioral microsimulation[END_REF] ARCHISM SSMT El hmam (2006) generic ABM LWR, ARZ, Payne [START_REF] Joueiai | Generic solutions for consistency problems in multi-scale traffic flow models -analysis and preliminary results[END_REF] IDM LWR In the following sections, we identify the main issues related to dynamic hybrid traffic modeling and then propose some generic solutions to them. Then, the JAM-FREE simulator is presented, as well as some simulation results. Finally, we conclude and suggest some research perspectives.

DYNAMIC HYBRID TRAFFIC MODELING: MOTIVATIONS AND EXISTING SOLUTIONS

Motivations

Traffic simulation is generally used to: 1) simulate the road traffic flow on very large and complex infrastructures including both city and highways while 2) being able to test the effects of different traffic signs, for example, or traffic assignment and dynamic routing strategies. It provides also a simple way to evaluate the impact of dynamic traffic control (ramp-metering, dynamic speed limit, . . . ) on the whole behavior of the traffic, 3) assess their precise influence on the traffic flow and 4) understand how and why traffic perturbations leading to appearance of congestions do occur. The first goal is easily achieved using macroscopic simulation models at the expense of the fourth goal. Conversely, the fourth goal is easily carried-out using microscopic simulation models at the expense of the first goal. To deal with this paradox, and benefit from both approaches, a dynamic hybrid model can be used.

The advantages of this hybrid approach include the ability:

• To obtain both quantitative and qualitative information about the road traffic, using respectively macroscopic and microscopic representations in the same simulation using the clusters principle.

• to switch between these different levels of representation locally depending:

on the simulation needs; for instance understanding the source of a traffic congestion, on the computation constraints; for instance managing the CPU load,

• to experiment both macroscopic and microscopic dynamic routing strategies, i.e. road load balancing strategies, and others dynamic trafic control.

Use cases examples

In order to assess the relevance of our approach for dynamic hybrid simulation, use cases demonstrating the limits of the existing solutions depicted in Table I are presented in this section.

Reduce the CPU load during peak hours: The hybrid model can change the traffic representation of a portion of the road network dynamically. This feature can be used to switch from a microscopic representation to a macroscopic one, when the CPU is overused in order to reduce its load. Such situations include cases where the number of simulated vehicles becomes too high to be managed satisfyingly by the CPU (for instance during peak hours). This case is illustrated in figure 2.

Balance the CPU load between clusters: The hybrid model can change the traffic representation of a portion of the road network dynamically. This feature can be used to switch from:

• a microscopic representation to a macroscopic one when the CPU is overused in order to reduce its load, • inversely (from macroscopic to microscopic), in order to detect the reason for the occurrence of traffic congestion.

These features can be used jointly to balance the CPU load between two clusters. Such situations include cases where a traffic jam (or congestion) appears in a cluster using a macroscopic model and where the number of simulated vehicles in another cluster is significantly reduced (for instance during off-peak hours). Such situations also include cases where a traffic jam appears in a cluster using a microscopic model and where the number of simulated vehicles in another cluster is significantly reduced (for instance during fluidic period of the flow) as illustrated in figure 3.

Find out the reason of the occurence of the traffic jam: Figure 4 shows an example of the occurence of traffic congestions. It demonstrates the relevance of dynamic hybrid simulations to switch between two levels of representation in order to find out the causes of the appearance of these phenomena.

Follow moving macroscopic phenomena: The static division of the road network into clusters that can be found in most of the existing approaches causes precision losses. Indeed, a traffic flow simulator has to be able to reproduce macroscopic phenomena such as shockwaves, capacity drop, . . . Yet, the static decomposition found in static hybrid models will cause eventually a loss of quantitative information. To avoid this issue, a dynamic hybrid model gives the ability to move, resize, split and merge the clusters of the road network (see figure 5).

Limits of the existing solutions

To the best of our knowledge, the only other work describing a dynamic hybrid model is [START_REF] Sewall | Interactive hybrid simulation of large-scale traffic[END_REF]. However, authors focus on the visualization of data rather than accurate simulation of traffic flow: zooming in a part of the network triggers a micro representation on this area. Thus, there is no bi-directional information exchange between the micro and macro representations.

Main issues

First, we identify the main issues to address in order to simulate dynamic hybrid traffic models accurately.

1) How can the simulated network be structured to switch between different models at runtime? 2) How to switch between different models (particularly from macro to micro)? 3) How the temporalities of the different interacting models can be managed to avoid bias? In the next section, we present possible generic solutions to these issues.

Proposed solutions

Dynamic decomposition of the network into autonomous clusters: The network can be dynamically decomposed into autonomous clusters. Each cluster operates autonomously and simulates the traffic flow using a model of its own. However, cutting the road network is not arbitrary, to avoid incoherent situation like having an area of a few square meters. To ensure the integrity of the simulation, a minimum cut is deducted from sensors: the entry and exit points of a cluster are necessarily sensors. Thus, the road network can be viewed as a directed graph of interconnected sensors. Each arc of the graph represents a path joining two sensors directly, without going through an intermediate sensor.

A cluster is then characterized by input and output sensors and eventually sensors situated inside the cluster. The minimum cutting of the road network is then defined by:

• a cluster representing the outside of the simulated network, • clusters representing minimal subsets in the road network; a cluster is minimal if there is no sensor inside the cluster. Clusters of a simulation are necessarily disjoint combination of minimum contiguous clusters. An example of minimal cutting is depicted in figure 6. Each cluster transmits mean speed and flow information from its bounding sensors to the upstream and downstream clusters. In the case of a macro cluster, this information is integrated using flow conservation and prediction formulas to compute the flow at next time.

In the case of a micro cluster, it consists in changing the frequency of generation of vehicles and their speed based on data from the sensors, and measuring the number of vehicles passing the sensors and their speed, for a given time interval to deduct the sensor data.

Aggregation and disaggregation of traffic variables: When running the simulation, an agent manages the decomposition of the cluster network. This agent encapsulates rules, determining whether the model used in a cluster must be changed (for example from micro to macro) and determining whether the area of a cluster must be enlarged or reduced according to the needs of the simulation (see the previous section). Switching from micro to macro is straightforward: the mean speed and flow inside the cluster can be directly computed from sensors.

On the other hand, switching from macro to micro requires to perform a local warm-up so the speed, density and flow of vehicles can be accurate.

Formal multi-level agent-based meta-model: Regular multiagent based simulation meta-models lack the structure to manage such hybrid approaches: their representation of the agents, the environment and the temporal dynamics of the system is designed to support a single viewpoint. Multi-level agent-based modeling is an interesting approach to simulate such systems. Indeed it offers a large range of techniques to dynamically adapt the level of detail of simulations, couple heterogenous models or detect and reify emergent phenomena [START_REF] Gaud | Holonic multilevel simulation of complex systems : Application to real-time pedestrians simulation in virtual urban environment[END_REF][START_REF] Gil-Quijano | From biological to urban cells: Lessons from three multilevel agent-based models, Principles and Practice of Multi-Agent Systems[END_REF][START_REF] Picault | An interaction-oriented model for multi-scale simulation[END_REF].

Many meta-models and simulation engines dedicated to multi-level agent-based modeling have been proposed in the literature. See e.g. [START_REF] Morvan | Multi-level agent-based modeling -a literature survey[END_REF] for a complete review. Managing multiple viewpoints on the same phenomenon induces the use of heterogeneous time models, thus raising issues related to time and consistency. However, simulation bias can be controlled by rules that constraint perceptions, influence production and reaction computation, according to causality and coherence principles [START_REF] Morvan | IRM4MLS: the influence reaction model for multi-level simulation[END_REF]. To deal with this issue, we developed a generic approach called SIMILAR to design simulations (see [START_REF] Morvan | SIMILAR documentation[END_REF] for more detailed information about the main principles of SIMILAR). Indeed, this approach relies on a multi-level, influence-reaction and agent-based knowledge representation, more fitting to multiple viewpoints [START_REF] Ferber | Influences and reaction: a model of situated multiagent systems, 2nd International Conference on Multi-agent systems[END_REF][START_REF] Michel | The IRM4S model: the influence/reaction principle for multiagent based simulation[END_REF][START_REF] Soyez | A methodology to engineer and validate dynamic multilevel multi-agent based simulations[END_REF].

THE JAM-FREE SIMULATOR Architecture

JAM-FREE is based on a multi-level architecture (see e.g., figure 7):

• the infrastructure level models the components of the road network; the naming and identification of road network components follow the Sétra (Service d'études sur les transports, les routes et leurs am'enagements, is a French technical service of the Ministère de l' Écologie, du Développement Durable et de l' Énergie dedicated to transportation issues) specification [START_REF] Sétra | Identification et localisation sur le reseau routier national[END_REF].

• the control level is the core of the dynamic hybrid approach: it manages the dynamic decomposition of the network into clusters.

• the simulation dependent levels model the traffic dynamic using a dedicated representation; by default, three traffic levels are considered: microscopic, macroscopic and datadriven (i.e., using real data).

Models used in JAM-FREE

Road network structure: We consider that the road network can be divided dynamically into subsets called clusters. The traffic is simulated on each cluster using various heterogeneous models depending on the situation: either a microscopic model or a macroscopic model. For computation time efficiency reasons, we choose not to model the road network at a physical level. Indeed, such a model requires the vehicles to interpret a large continuous zone of asphalt with blank lines as separated lanes. Such computations are unnecessary complex, since in our use case (France) rational drivers usually never drive over the blank lines separating the lanes if they are not overtaking. To keep the model simple, the road network is only modeled at a semantic level.

The road network is modeled as a set of interconnected roads. Each road contains a set of lanes (usually from 1 to 5). Roads also contain vertical signs (e.g. stop sign, speed limit sign) that are bond either to all the lanes or to specific lanes (e.g. extraction lane in the highway, slow vehicles lane). In our model, we support a specific subset of the french vertical signs. The connection points of the roads are called nodes. We distinguish different types of nodes:

• Crossroads nodes • Roundabout nodes • Highway insertion node
• Highway extraction node Microscopic level models: The behavior of a vehicle can be seen as the sum of three different parts:

• A navigation behavior when the current lane of the vehicle does not lead to the desired destination of the vehicle. This behavior consists in changing the lane of the vehicle until the current lane of the vehicle leads to the desired destination. • An overtaking behavior when the vehicle either wants to increase its speed by changing its lane, or when the vehicle has to swerve because a faster vehicle is tailing it.

• A behavior when no lane change is required (acceleration model). JAM-FREE implements the highly used Intelligent-Driver Model (IDM) to model the acceleration behavior of vehicles [START_REF] Kesting | Enhanced intelligent driver model to access the impact of driving strategies on traffic capacity[END_REF].

IDM is a microscopic traffic flow model, i.e., each vehicledriver combination constitutes an active "particle" in the simulation. Such models characterize the traffic state at any given time by the positions and speeds of all simulated vehicles. In case of multi-lane traffic, the lane index complements the state description. More specifically, IDM is a car-following model. In such models, the decision of any driver to accelerate or to brake depends only on his or her own speed, and on the position and speed of the "leading vehicle" immediately ahead. The model structure of IDM can be described as follows:

• the influencing factors (model input) are the own speed v, the bumper-to-bumper gap s to the leading vehicle, and the relative speed (speed difference) of the two vehicles (positive when approaching), • the model output is the acceleration chosen by the driver for this situation, • the model parameters describe the driving style, i.e., whether the simulated driver drives slow or fast, careful or reckless.

Lane changes takes place, if:

• the potential new target lane is more attractive, i.e., the "incentive criterion" is satisfied, • and the change can be performed safely, i.e., the "safety criterion" is satisfied. JAM-FREE implements the lane changing model MOBIL [START_REF] Kesting | General lanechanging model MOBIL for car-following models[END_REF].

Chosen acceleration and lane changing models are presented in detail in the JAM-FREE documentation.

Macroscopic level model: Any macroscopic traffic flow model can be used in JAM-FREE since it defines generic connectors to convert micro/macro representations. By default, a generic implementation of the METANET model is provided [START_REF] Messner | Metanet: A macroscopic simulation program for motorway networks[END_REF]. METANET is based on a second order macroscopic model and uses the following traffic flow equations:

ρ i (k + 1) = ρ i (k) + T s L i [q i-1 (k) -q i (k)] (1)
where, ρ i , defines the traffic density in (veh/km/lane). L i , k and T s represent, the segment length (km), the simulation step and its duration, respectively. q i in (vehicles/h) defines the traffic flow in the segment i:

q i = ρ i v i (2)
The speed v i in (km/h) represents the velocity of vehicles in segment i at time kT s .

v i = V e (ρ i ) (3)
where V e (ρ i ) is the static speed-density characteristic called a fundamental diagram. The momentum equation is obtained by introducing two small constant parameters in (3). Such equation reads:

v i (k + 1) = T s τ [V e (ρ i ) -v i (k)] relaxation + T s η L i v i (k)[v i-1 (k) -v i+1 (k)] convection - T s ν τL i ρ i+1 (k) -ρ i (k) ρ i (k) + κ anticipation (4)
The first term of equation ( 4) represents relaxation to the equilibrium. This is the most dominant term in the equation since the other terms manifest their effects only when there are (occasional) fluctuations in the traffic speed and density upstream or downstream. This relaxation term describes the fact that the drivers adjust their speeds to the equilibrium speed-density characteristic V e (ρ i ) according to the reaction time τ. The second term, represents the convection, i.e., the influence of the upstream traffic mean speed. The third term, called "anticipation", translates the effect of drivers reacting to downstream traffic density variations. Drivers tend to decelerate if the downstream density is higher and accelerate if it is lower. ν, κ are a model parameters.

Traffic generation: In our model, each input point is a special connector that will generate traffic on the road it is attached to. Note that such a connector has to be created and put on each lane where the traffic appears. The creation of vehicles is managed by a traffic input point agent. Various implementation of this agent currently exist:

• "Flow-mass traffic input point" agents, generating the traffic flow using a flow-mass parameter • "Scripted traffic input point" agents, generating the traffic flow using user-defined events. The created vehicles and the creation dates are manually specified by the users.

A first experimentation

To validate the models implemented in JAM-FREE, as well as the flow (dis)aggregation algorithms, we performed a first hybrid simulation using real data from of the A25 highway in France. We simulate a 4780m portion of the highway, decomposed into 11 clusters (see fig. 8). The clusters containing insertion or extraction lanes are simulated with a microscopic model, the others with a macroscopic model. The obtained results permit to validate the performance of JAM-FREE as well as its microscopic and macroscopic models. We cannot present them all in this paper for an evident question of space. The figure 9 shows some simulation results in cluster R8 (i.e., after (dis)aggregation processes), using typical model parameter values (i.e., without specific calibration).

CONCLUSION AND PERSPECTIVES

In this paper a first step towards the implementation of a dynamic hybrid model was proposed. The main objectives of this step were to identify use-cases, propose generic solutions to the dynamic hybrid simulation problem and study the core aspects of the multi-level simulation approach. These ideas have then been implemented in a simulator called JAM-FREE. The first experiments using real data allow us to validate several issues addressed in the paper and the implemented models. Further works will focus on the introduction of several traffic models allowing to the practitioners multiple choices in function of their simulation objective and the validation of the identified use-cases. Moreover, a more accurate macro/micro switching algorithm is being developed using a sophisticated continuous warm-up process. In addition, some control strategies will be added to JAM-FREE in order to evaluate their impact on the global behavior of the networks.

Fig. 1 .

 1 Fig. 1. Hybrid traffic simulation and control approach

Fig. 2 .

 2 Fig. 2. Illustration of the use case "Reduce the CPU load during peak hours".

Fig. 3 .

 3 Fig. 3. Illustration of the use case "Balance the CPU load between clusters".

Fig. 4 .

 4 Fig. 4. Illustration of the use case "Find out the reason of a traffic jam".

Fig. 5 .

 5 Fig. 5. Dynamic hybrid model of a shockwave phenomenon.

Fig. 6 .

 6 Fig. 6. Minimal cutting of a network

Fig. 8 .

 8 Fig. 8. Description of the simulated network

  Fig. 9. Simulation results in cluster R8