Déborah Ferré

Loïc Hervé
email: loic.herve@insa-rennes.fr

James Ledoux
email: jamesledoux@insa-rennes.fr

Dborah Ferr É

Herv É Loc

Limit theorems for stationary Markov processes with L2-spectral gap

Keywords: subject classification : 60J05, 60F05, 60J25, 60J55, 37A30, 62M05 Markov additive process, central limit theorems, Berry-Esseen bound, Edgeworth expansion, spectral method, ρ-mixing, M -estimator

come

Introduction

In this paper, we are concerned with the class of Markov Additive Processes (MAP). The discrete and continuous-time cases are considered so that the time parameter set T will denote N or [0, +∞). Let X be any set equipped by a σ-algebra X and let B(R d ) be the Borel σalgebra on R d (d ≥ 1). A (time homogeneous) MAP (X t , Y t ) t∈T is a (time homogeneous) Markov process with state space X × R d and transition semigroup (Q t ) t∈T satisfying: ∀t ∈ T, ∀(x, y) ∈ X × R d , ∀(A, B) ∈ X × B(R d ), Q t (x, y; A × B) = Q t (x, 0; A × By).

(1.1)

In other words, the transition semigroup is additive in the second component. It follows from the definition that the first component (X t ) t∈T of a MAP is a (time homogeneous) Markov process. The second component (Y t ) t∈T must be thought of as a process with independent increments given σ(X s , s ≥ 0). We refer to [START_REF]Markov additive processes Part II[END_REF] for the general structure of such processes. Note that a discrete-time MAP is also called a Markov Random Walk (MRW). In stochastic modelling, the first component of a MAP is usually associated with a random environment which drives or modulates the additive component (Y t ) t∈T . The MAPs have been found to be an important tool in various areas as communication networking (e.g. see [START_REF] Asmussen | Applied probability and queues[END_REF][START_REF] Pacheco | Markov-additive processes of arrivals[END_REF][START_REF] Pacheco | Markov-modulated processes & semiregenerative phenomena[END_REF]), finance (e.g. see [START_REF] Asmussen | Ruin probabilities[END_REF][START_REF] Asmussen | Russian and American put options under exponential phase-type Lévy models[END_REF][START_REF] Jobert | Option pricing with Markov-modulated dynamics[END_REF]), reliability (e.g. see [START_REF]The theory and applications of reliability, with emphasis on Bayesian and nonparametric methods[END_REF][START_REF] Gravereaux | Poisson approximation for some point processes in reliability[END_REF][START_REF] Limnios | Semi-Markov Processes and Reliability[END_REF][START_REF] Özekici | Reliability modeling and analysis in random environments[END_REF]), . . . Some important instances of MAP are:

• in discrete/continuous-time : (X t , Y t ) t∈T where (Y t ) t∈T is a R d -valued additive functional (AF) of the Markov process (X t ) t∈T . Therefore any result on the second component of a MAP applies to an AF. Basic discrete and continuous-time AFs are respectively

Y 0 = 0, ∀t ∈ N * , Y t = t k=1 ξ(X k ); ∀t ∈ [0, +∞[, Y t = t 0 ξ(X s ) ds (1.2)
where ξ is a R d -valued function satisfying conditions under which Y t is well-defined for every t ∈ T. When (X t ) t∈T is a regular Markov jump process, then any non-decreasing AF has the form (e.g. [START_REF]Introduction to stochastic processes[END_REF])

t 0 ξ 1 (X s ) ds + s≤t ξ 2 (X s-, X s )
where X t-= lim s→t,s<t X s , ξ 1 and ξ 2 are non-negative measurable functions such that ξ 2 (x, x) = 0 for every x ∈ X. General representations and properties of AFs may be found in [5, 77, and references therein]. Such AFs are basically introduced when some kind of "rewards" are collected along with the dynamics of the Markov process (X t ) t∈T through the state space X. Thus, Y t is the accumulated reward on the finite interval [0, t]. Even if the state space X is a finite set, the numerical computation of the probability distribution of such AFs is not an easy task (e.g. see [START_REF] Bladt | Distributions of reward functions on continuous-time Markov chains[END_REF][START_REF] Stefanov | Exact distributions for reward functions on semi-Markov and Markov additive processes[END_REF]).

• in discrete-time: the Markov renewal processes when the random variables Y t , t ∈ N, are non-negative; if we consider a hidden Markov chain (X t , Z t ) t∈N , where the so-called observed process (Z t ) t∈N is R d -valued (Z 0 = 0), then (X t , t k=1 Z k ) t∈N is a MAP.

• in continuous time: the Markovian Arrival Process where (X t ) t∈T is a regular jump process and (Y t ) t∈T is a point process (see [START_REF] Asmussen | Applied probability and queues[END_REF]), which includes the so-called Markov Modulated Poisson Process.

Seminal works on MAPs are [START_REF] Ezhov | Markov processes which are homogeneous in the second component[END_REF][START_REF] Ezhov | Markov processes which are homogeneous in the second component[END_REF][START_REF] Keilson | A central limit theorem for processes defined on a finite Markov chain[END_REF][START_REF] Neveu | Une gnralisation des processus accroissements positifs indpendants[END_REF][START_REF] Pinsky | Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain[END_REF] and are essentially concerned with a finite Markov process (X t ) t∈T as first component. The second component (Y t ) t∈T was sometimes called a process defined on a Markov process. When X is a finite set, the structure of MAPs are well understood and an account of what is known can be found in [START_REF] Asmussen | Applied probability and queues[END_REF]Chap XI]. In this paper, we are concerned with Gaussian approximations of the distribution of the second component Y t of a MAP. Central limit theorems for (Y t ) t∈T may be found in [START_REF] Bhattacharya | On the functional central limit theorem and the law of the iterated logarithm for Markov processes[END_REF][START_REF] Fukushima | On a class of Markov processes taking values on lines and the central limit theorem[END_REF][START_REF] Glynn | Limit theorems for cumulative processes[END_REF][START_REF] Hitsuda | The central limit theorem for additive functionals of Markov processes and the weak convergence to Wiener[END_REF][START_REF] Holzmann | Martingale approximations for continuous-time and discrete-time stationary Markov processes[END_REF][START_REF] Keilson | A central limit theorem for processes defined on a finite Markov chain[END_REF][START_REF] Kipnis | Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions[END_REF][START_REF] Pinsky | Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain[END_REF][START_REF] Steichen | A functional central limit theorem for Markov additive processes with an application to the closed Lu-Kumar network[END_REF][START_REF] Touati | Théorèmes de limite centrale fonctionnels pour les processus de Markov[END_REF] under various assumptions. Here, such results are derived when (X t ) t∈T has an invariant probability measure π, is stationary and has the L 2 (π)-spectral gap property (see conditions (AS1-AS2) below). Moreover, standard refinements of the central limit theorem (CLT) related to the convergence rate are provided. Before, notations and assumptions used throughout the paper are introduced.

Let (X t , Y t ) t∈T be a MAP with state space X×R d and transition semigroup (Q t ) t∈T . (X, X ) is assumed to be a measurable space equipped with a σ-algebra X . In the continuous-time case, (X t , Y t ) t∈T is assumed to be progressively measurable. (X t ) t∈T is also a Markov process with transition semigroup (P t ) t∈T given by P t (x, A) := Q t (x, 0; A × R d ).

Throughout the paper, we assume that (X t ) t∈T has a unique invariant probability measure denoted by π (∀t ∈ T, π•P t = π). We denote by L 2 (π) the usual Lebesgue space of (classes of) functions f :

X → C such that f 2 := π(|f | 2 ) = ( X |f | 2 dπ) 1/2 < ∞.
The operator norm of a bounded linear operator T on L 2 (π) is defined by T 2 := sup {f ∈L 2 (π): f 2 =1} T (f ) 2 . We appeal to the following conditions. AS 1. (X t ) t∈T is stationary (i.e. X 0 ∼ π). AS 2. The semigroup (P t ) t∈T of (X t ) t∈T has a spectral gap on L 2 (π):

lim t→+∞ P t -Π 2 = 0, (1.3)
where Π denotes the rank-one projection defined on L 2 (π) by: Πf = π(f )1 X . AS 3. The process (Y t ) t∈T satisfies the moment condition

sup t∈(0,1]∩T E π,0 [|Y t | α ] < ∞ (1.4)
where |•| denotes the euclidean norm on R d and E π,0 is the expectation when (X 0 , Y 0 ) ∼ (π, δ 0 ). In the discrete-time case, notice that the moment condition (1.4) reduces to (AS3d)

E π,0 [|Y 1 | α ] < ∞ (AS3d)
and that condition (AS2) is equivalent to the ρ-mixing property of (X t ) t∈N , with ρ-mixing coefficients going to 0 exponentially fast [START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF]. Condition (AS2) is also related to the notion of essential spectral radius (e.g. see [START_REF] Wu | Essential spectral radius for Markov semigroups. I. Discrete time case[END_REF]).

Under (AS1-AS2), we show that the second component (Y t ) t∈T of the MAP satisfies, in discrete and continuous time, the following standard limit theorems :

(a) the central limit theorem, under (AS3) with the optimal value α = 2; (b) the local limit theorem, under (AS3) with the optimal value α = 2 and the additional classical Markov non-lattice condition;

(c) the one-dimensional Berry-Esseen theorem, under (AS3) with the (almost) optimal value (α > 3);

(d) a one-dimensional first-order Edgeworth expansion, under (AS3) with the (almost) optimal value (α > 3) and the Markov non-lattice condition.

These results correspond to the classical statements for the sequences of independent and identically distributed (i.i.d.) random variables, with the same order α (up to ε > 0 in (c) and (d)). Such results are known for special MAPs satisfying (AS2) (comparison with earlier works is made after each statement), but to the best of our knowledge, the results (a)-(d) are new for general MAPs satisfying (AS2), as, for instance, for AF involving unbounded functionals.

Here, the main arguments are

• for the statement (a): the ρ-mixing property of the increments (Y t+1 -Y t ) t∈T of the process (Y t ) t∈T (see Proposition 3.1). This result, which has its own interest, is new to the best of our knowledge. The closest work to this part is a result of [START_REF] Grigorescu | Limit theorems for J -X processes with a general state space[END_REF] which, by using φ-mixing properties, gives the CLT for MAPs associated with uniformly ergodic driving Markov chains (i.e. (P t ) t∈T has a spectral gap on the usual Lebesgue space L ∞ (π)). Condition (AS2) is less restrictive than uniform ergodicity (which is linked to the so-called Doeblin condition).

• For the refinements (b-d) : the Nagaev-Guivarc'h spectral method. The closest works to this part are, in discrete-time the paper [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] in which these refinements are obtained for the AF: Y t = t k=1 ξ(X k ), and in continuous-time the work of Lezaud [62] which proves, under the uniform ergodicity assumption, a Berry-Esseen bound for the integral additive functional (1.2). Here, in discrete-time, we borrow to a large extent the weak spectral method of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]: this is outlined in Proposition 4.2, which gives a precise expansion (close to the i.i.d. case) of the characteristic function of Y t . For continuous-time MAPs, similar expansions can be derived from the semigroup property of the Fourier operators of the MAP. Proposition 4.2, and its continuous-time counterpart Proposition 4.4, are the key results to establish limit theorems (as for instance the statements (b-d)) with the help of Fourier techniques.

The classical (discrete and continuous-time) models for which the spectral gap property (AS2) is met, are briefly reviewed in Subsections 2.2-2.4. The above limit theorems (a)-(d) are valid in all these examples and open up possibilities for new applications. First, our moment conditions are optimal (or almost optimal). For instance, in continuous time, the Berry-Esseen bound in [START_REF] Lezaud | Chernoff and Berry-Esseen inequalities for Markov processes[END_REF] requires that ξ in the integral (1.2) is bounded, while our statement (c) holds true under the condition π(|ξ| 3+ε ) < ∞. Second, our results are true for general MAPs. For instance, they apply to Y t = t k=1 ξ(X k-1 , X k ). This fact enables us to prove a Berry-Esseen bound for M -estimators associated with ρ-mixing Markov chains, under a moment condition which greatly improves the results in [START_REF] Rao | On the rate of convergence of estimators for Markov processes[END_REF].

The paper is organised as follows. The L 2 (π)-spectral gap assumption for a Markov process is briefly discussed in Section 2 and connections to standard ergodic properties are pointed out. In Section 3, the CLT for (Y t ) t∈T under (AS1)-(AS3) with α = 2 is derived. The functional central limit theorem (FCLT) is also discussed. Section 4 is devoted to refinements of the CLT. First, the Fourier operator is introduced in Subsection 4.1, the characteristic function of Y t is investigated in Subsection 4.2, and our limit theorems are proved for discrete-time MAPs in Subsection 4.3. Their extension to the non-stationary case is discussed in Subsection 4.4. The continuous-time case is studied in Subsection 4.5. The statistical application to M -estimators for ρ-mixing Markov chains is developed in Section 5.

Finally, we point out that the natural way to consider the Nagaev-Guivarc'h method in continuous-time is the semigroup property of the Fourier operators of the MAP (see Subsection 4.1 for details). To the best of our knowledge, this property, which is closely related to the additivity condition (1.1) defining a MAP, has been introduced and only exploited in [START_REF] Hitsuda | The central limit theorem for additive functionals of Markov processes and the weak convergence to Wiener[END_REF].

2 The L 2 (π)-spectral gap property (AS2)

Basic facts on property (AS2)

We discuss the condition (AS2) for the semigroup (P t ) t∈T of (X t ) t∈T . It is well-known that (P t ) t∈T is a contraction semigroup on each Lebesgue-space L p (π) for 1 ≤ p ≤ +∞, that is: we have P t p ≤ 1 for all t ∈ T, where • p denotes the operator norm on L p (π). Condition (AS2), introduced by Rosenblatt [START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF] and also called strong ergodicity on L 2 (π), implies that (P t ) t∈T is strongly ergodic on each L p (π) (1 < p < +∞), that is P t -Π p → 0 when t → +∞. Moreover, (AS2) is fulfilled under the so-called uniform ergodicity property, i.e. the strong ergodicity on L ∞ (π). These properties, established in [START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF], can be easily derived from the Riesz-Thorin interpolation theorem [START_REF] Bergh | Interpolation spaces. An introduction[END_REF] which insures, thanks to the contraction property of P t , that

P t -Π p ≤ P t -Π α p 1 P t -Π 1-α p 2 ≤ 2 min P t -Π α p 1 , P t -Π 1-α p 2 , (2.1)
where

p 1 , p 2 ∈ [1, +∞] and p ∈ [1, +∞] satisfy 1/p = α/p 1 + (1 -α)/p 2 for some α ∈ [0, 1].
Indeed, assume that Condition (AS2) holds. Then Inequality (2.1) with (p 1 = 2, p 2 = +∞) and α ∈ (0, 1) gives the strong ergodicity on L p (π) for each p ∈ (2, +∞). Notice that the value p = +∞ is obtained with α = 0, but in this case, the uniform ergodicity cannot be deduced from (AS2) and (2.1). In fact the uniform ergodicity condition is stronger than (AS2) (see [START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF]). Next Inequality (2.1) with (p 1 = 2, p 2 = 1) and α ∈ (0, 1) gives the strong ergodicity on L p (π) for each p ∈ (1, 2). The value p = 1 is obtained with α = 0, but the strong ergodicity on L 1 (π) cannot be deduced from (AS2) and (2.1). Finally, if the uniform ergodicity is assumed, then Inequality (2.1) with (p 1 = +∞, p 2 = 1) and α = 1/2 yields (AS2). Also notice that the strong ergodicity property on L p (π) holds if and only if there exists some strictly positive constants C and ε such that we have for all t ∈ T:

P t -Π p ≤ C e -εt .
(2.2) Indeed, if κ 0 := P τ -Π p < 1 for some τ ∈ T (which holds under the strong ergodicity property), then we have for all n ∈ N * : P nτ -Π p = P n τ -Π p = (P τ -Π) n p ≤ κ n 0 . Writing t = w + nτ with n ∈ N * and w ∈ [0, τ ), we obtain: P t -Π p = P w (P n τ -Π) p ≤ κ n 0 ≤ C e -εt with C := 1/κ 0 and ε := (-1/τ ) ln κ 0 . The converse implication is obvious. Thus, the strong ergodicity property on L 2 (π), i.e condition (AS2), is equivalent to require L 2 (π)-exponential ergodicity (2.2), that is the L 2 (π)-spectral gap property.

In the next subsection, Markov models with a spectral gap on L 2 (π) arising from stochastic modelling and potentially relevant to our framework are introduced. Assumption (AS2) can be also met in more abstract settings, as for instance in [START_REF] Guivarc'h | Limit theorems for random walks and products of random matrices[END_REF] where the L 2 -spectral gap property for classic Markov operators (with a state space defined as the d-dimensional torus) is proved.

Geometric ergodicity and property (AS2)

Recall that (X t ) t∈N is V -geometrically ergodic if its transition kernel P has an invariant probability measure π and is such that there are r ∈ (0, 1), a finite constant K and a π-a.e finite function V :

X → [1, +∞] such that ∀n ≥ 0, π -a.e. x ∈ X, sup |P n f (x) -π(f )|, f : X → C, |f | ≤ V ≤ K V (x) r n . (VG)
In fact, when (X t ) t∈N is ψ-irreducible (i.e. ψ(A) > 0 =⇒ P (x, A) > 0, ∀x ∈ X) and aperiodic [START_REF] Meyn | Markov chains and stochastic stability[END_REF], condition (VG) is equivalent to the standard geometric ergodicity property [START_REF] Roberts | Geometric ergodicity and hybrid markov chains[END_REF]: there are functions r : X → (0, 1) and C : X → [1, +∞) such that: for all n ∈ N, πa.e. x ∈ X,

P n (x, •) -π(•) TV := sup |P n f (x) -π(f )|, f : X → C, |f | ≤ 1 ≤ C(x) r(x) n .
There is another equivalent operational condition to geometric ergodicity for ψ-irreducible and aperiodic Markov chains (X t ) t∈N , the so-called " drift-criterion": there exist a function V : X → [1, +∞], a small set C ⊂ X and constants δ > 0, b < ∞ such that

P V ≤ (1 -δ)V + b1 C .
We refer to [START_REF] Meyn | Markov chains and stochastic stability[END_REF] for details and applications, and to [START_REF] Jones | On the Markov chain central limit theorem[END_REF] for a recent survey on the CLT for the additive functionals of (X t ) t∈N in (1.2). Now, the transition kernel P is said to be reversible with respect to π if π(dx)P (x, dy) = π(dy)P (y, dx)

or equivalently if P is self-adjoint on the space L 2 (π). It is well known that a V -geometrically ergodic Markov chain with a reversible transition kernel has the L 2 (π)-spectral gap property [START_REF] Roberts | Geometric ergodicity and hybrid markov chains[END_REF]. Moreover, for a ψ-irreducible and aperiodic Markov chain (X t ) t∈N with reversible transition kernel, (V -)geometric ergodicity is shown to be equivalent to the existence of a spectral gap in L 2 (π), and, when X 0 ∼ µ, we also have [78, Th 2.1], [START_REF] Roberts | Geometric L 2 and L 1 convergence are equivalent for reversible Markov chains[END_REF] µP

n (•) -π(•) TV ≤ 1 2 µ -π L 2 (π) r n . (R)
where r := lim n→+∞ P n -Π 2 1/n and µπ L 2 (π) := dµ/dπ -1 2 if well-defined and ∞ otherwise. Note that the reversibility condition is central to the previous discussion on the L 2 (π)-spectral gap property. Indeed, there exists a ψ-irreducible and aperiodic Markov chain which is geometrically ergodic but does not admit a spectral gap on L 2 (π) [START_REF] Häggström | Acknowledgement of priority concerning "On the central limit theorem for geometrically ergodic Markov chains[END_REF].

Such a context of geometric ergodicity and reversible kernel is relevant to the Markov Chain Monte Carlo methodology for sampling a given probability distribution, i.e. the target distribution. Indeed, the basic idea is to define a Markov chain (X t ) t∈N with the target distribution as invariant probability measure π. Then a MCMC algorithm is a scheme to draw samples from the stationary Markov chain (X t ) t∈N . But, the initial condition of the algorithm, i.e. the probability distribution of X 0 , is not π since the target distribution is inaccessible. Therefore the convergence in distribution of the Markov chain to π in regard of the probability distribution of X 0 must be guaranteed and the knowledge of the convergence rate is crucial to monitor the sampling. Thus, central limit theorem for the Markov chains and quantitative bounds as in (R) are highly expected. Geometric ergodicity of Hasting-Metropolis type algorithms has been investigated by many researchers. Two standard instances are the full dimensional and random-scan symmetric random walk Metropolis algorithm [25, 55, and references therein]. Note that the first algorithm is also referred to as a special instance of the Hasting algorithm and the second one to as a Metropolis-within-Gibbs sampler. Let π be a probability distribution on R d which is assumed to have a positive and continuous density with respect to the Lebesgue measure. The so-called proposal densities are assumed to be bounded away from 0 in some region around zero (the moves through the state space X are based on these probability distributions). These conditions assert that the corresponding transition kernel for each algorithm is ψ-irreducible, aperiodic and is reversible with respect to π. Geometric ergodicity for the Markov chain (X t ) t∈N (and so the existence of a spectral gap in L 2 (π)) is closely related to the tails of the target distribution π. For instance, in the first algorithm, it can be shown that π must have an exponential moment [55, Cor 3.4] For the second algorithm, sufficient conditions for geometric ergodicity are reported in [START_REF] Fort | On the geometric ergodicity of hybrid samplers[END_REF] when the target density decreases either subexponentially or exponentially in the tails. A very large set of examples and their respective merit are discussed in these two references. We refer to [79, and references therein] for a recent survey on the theory of Markov chains in connection with MCMC algorithms.

Uniform ergodicity and hidden Markov chains

As quoted in the introduction, a discrete-time MAP is closely related to a hidden Markov chain. Standard issues for hidden Markov chains require to be aware of the convergence rate of the hidden Markov state process (X t ) t∈N . One of them is the state estimation via filtering or smoothing. In such a context, minorization conditions on P are usually involved. The basic one is: there exists a bounded positive measure ϕ on X such that for some m ∈ N * :

∀x ∈ X, ∀A ∈ X , P m (x, A) ≥ ϕ(A). (UE)
It is well-known that this is equivalent to the uniform ergodicity property or to condition (VG) with V (x) = 1 [67, Th 16.2.1, 16.2.2]. Recall that uniform ergodicity gives the L 2 (π)spectral gap property (AS2), but the converse is not true. Another minorization condition is the so-called "Doeblin condition": there exists a probability measure ϕ such that for some m, ε < 1 and δ > 0 [START_REF] Doob | Stochastic processes[END_REF]

ϕ(A) > ε =⇒ ∀x ∈ X, P m (x, A) ≥ δ. (D 0 )
It is well known that, for ergodic and aperiodic Markov chains, (D 0 ) is equivalent to the uniform ergodicity. We refer to [14, and the references therein] for an excellent overview of the interplay between the Markov chain theory and the hidden Markov models.

Property (AS2) for continuous time Markov processes

The Markov jump processes are a basic class of continuous-time Markov models which has a wide interest in stochastic modelling. The L 2 (π)-exponential convergence has received attention a long time ago. We refer to [START_REF] Chen | From Markov chains to non-equilibrium particle systems[END_REF] for a good account of what is known on ergodic properties for such processes. In particular, the L 2 (π)-spectral gap property is shown to be equivalent to the standard exponential ergodicity for the birth-death processes:

∃β > 0 such that ∀(i, j) ∈ X 2 , ∃C i ≥ 0, |P t (i, j) -π j | ≤ C i exp(-βt) t → +∞
where (P t (i, j)) i,j∈X is the matrix semigroup of (X t ) t≥0 . This is also true for the reversible Markov jump processes. Hence, in these cases, criteria for exponential ergodicity are also valid to check the L 2 (π)-exponential convergence. Moreover, explicit bounds on the spectral gap are discussed in details in [START_REF] Chen | From Markov chains to non-equilibrium particle systems[END_REF]. For the birth-death processes, we also refer to [58, and references therein]) where explicit formulas are obtained for classical Markov queuing processes. The birth-death processes are often used as reference processes for analyzing general stochastic models. This idea was in force in the Liggetts's derivation of the L 2 -exponential convergence of supercritical nearest particle systems [START_REF] Liggett | Exponential L 2 convergence of attractive reversible nearest particle systems[END_REF]. The interacting systems of particles are also a source of examples of processes with a L 2 -spectral gap. We refer to [START_REF] Liggett | Exponential L 2 convergence of attractive reversible nearest particle systems[END_REF] for such a discussion on various classes of stochastic Ising models. In physics and specially in statistical physics, many evolution models are given by stochastic ordinary/partial equations. When the solutions are finite/infinite dimensional Markov processes, standard issues arise: existence and uniqueness of an invariant probability measure, ergodic properties which include the rate of convergence to the invariant measure with respect to some norm. Such issues may be included in the general topic of the stability of solutions of stochastic differential equations (SDEs). Thus, it is not surprising that ergodic concepts as the V -geometric ergodicity and Lyapunov-type criteria associated with, originally developed by Meyn and Tweedie [START_REF] Meyn | Markov chains and stochastic stability[END_REF] for studying the stability of discrete-time Markov models, have been found to be of value (e.g. see [32, and references therein]). Here, we are only concerned with the L 2 (π)-exponential convergence so that we only mention some results related with.

An instance of L 2 (π)-spectral gap can be found in [START_REF] Ganidis | Convergence rate of some semi-groups to their invariant probability[END_REF] where the following SDE is consid-

ered dX t = - 1 2 b(X t ) dt + dW t X 0 = x ∈ R d
where (W t ) t≥0 is the standard d-dimensional Brownian motion and b(•) is a gradient field from R d to R d (with suitable properties ensuring essentially the existence of a unique strong solution to the equation, which has a unique invariant probability measure). When b(•) is a radial function satisfying b(x) ∼ C|x| α for α > 1 when x → +∞, then the semigroup is shown to be ultracontractive and to have a L 2 (π)-spectral gap [START_REF] Ganidis | Convergence rate of some semi-groups to their invariant probability[END_REF].

Another instance of L 2 (π)-spectral gap is related to the R-valued Markov process solution to

dX t = b(X t ) dt + a(X t ) dW t (2.3)
where (W t ) t≥0 is the standard 1-dimensional Brownian motion and X 0 is a random variable independent of (W t ) t≥0 . Standard assumptions ensure that the solution of the SDE above is a positive recurrent diffusion on some interval and a (strictly) stationary ergodic time-reversible process. Under additional conditions on the scale and the speed densities of the diffusion (X t ) t≥0 [29, (A4) and reinforced (A5), Prop. 2.8], the transition semigroup of (X t ) t≥0 is shown to have the L 2 (π)-spectral gap property (explicit bounds on the spectral gap are also provided). The basic example studied in [START_REF] Genon-Catalot | Stochastic volatility models as hidden Markov models and statistical applications[END_REF] is when a(x) := cx ν and b(x

) := α(β -x) with ν ∈ [1/2, 1], α, β ∈ R.
Conditions ensuring the L 2 (π)-spectral gap property are provided in terms of these parameters. Applications to some classical models in finance are discussed. Note that statistical issues for continuous-time Markov processes as the jump or diffusion processes, are related to the time discretization or sampling schemes of these processes. This often provides discrete-time Markov chains which inherit ergodic properties of the original continuous-time process. Thus we turn to the discussion on the discrete-time case (e.g. see [START_REF] Dehay | On likelihood estimation for discretely observed Markov jump processes[END_REF] for the jump processes, [START_REF] Genon-Catalot | Stochastic volatility models as hidden Markov models and statistical applications[END_REF] and the references therein for the (hidden) diffusions).

Finally, the context of the stochastic differential equation (2.3) can be generalized to Markov H-valued processes solution to infinite dimensional SDEs, where H is a Hilbert space. A good account of these generalizations can be found in [33, and references therein].

3 The ρ-mixing property and central limit theorems

Let (X t , Y t ) t∈T be a MAP taking values in X × R d . E (x,0
) , E π,0 are the expectation with respect to the initial conditions (X 0 , Y 0 ) ∼ (δ x , δ 0 ) and (X 0 , Y 0 ) ∼ (π, δ 0 ) respectively. First, basic facts for MAPs are proposed. Second, they are used to show that, for a discrete-time MAP, the increment process (Y n -Y n-1 ) n∈N * is exponentially ρ-mixing under (AS1-AS2).

Then, a CLT is obtained under conditions (AS1-AS2) and the expected moment condition (AS3) (i.e. (AS3d)) with α = 2.

Basic facts on MAPs

Let

F (X,Y ) t := σ(X u , Y u , u ≤ t), F X t := σ(X u , u ≤ t) and F Y t := σ(Y u , u ≤ t)
be the filtration generated by the processes (X t , Y t ) t∈T , (X t ) t∈T and (Y t ) t∈T respectively.

The additivity property (1.1) for the semigroup (Q t ) t∈T reads as follows for any measurable (C-valued) function g on X × R d and any a ∈ R d :

Q t (g) a = Q t (g a ) (3.1)
where g a (x, y) := g(x, y +a) for every (x, y) ∈ X×R d . Let us introduce the following notation:

Q s (x; dx 1 × dy 1 ) := Q s (x, 0; dx 1 × dy 1 ).
Then, we have:

Lemma 3.1. For any C-valued function g on X × R d such that E[|g(X u , Y u )|] < ∞ for every u ∈ T, we have: E[g(X s+t , Y s+t ) | F (X,Y ) s ] = Q t (g Ys )(X s , 0) = Q t (g Ys )(X s ). (3.2)
or in terms of the increments of the process (Y t ) t∈T :

E[g(X s+t , Y s+t -Y s ) | F (X,Y ) s ] = Q t (g)(X s , 0) = Q t (g)(X s ) = E (Xs,0) [g(X t , Y t )]. (3.3)
Proof. The two formula are derived as follows:

E[g(X s+t , Y s+t ) | F (X,Y ) s ] = E[g(X s+t , Y s+t ) | X s , Y s ] (Markov property) = Q t (g)(X s , Y s ) = Q t (g Ys )(X s , 0) (from (3.1)) = Q t (g Ys )(X s ); E[g(X s+t , Y s+t -Y s ) | F (X,Y ) s ] = E[g(X s+t , Y s+t -Y s ) | X s , Y s ] (Markov property) = E[g -Ys (X s+t , Y s+t ) | X s , Y s ] = Q t (g 0 )(X s , 0) = Q t (g)(X s ) (from (3.2)) = E (Xs,0) [g(X t , Y t )].
Lemma 3.2. For every n ≥ 1, any C-valued function g such that for every 0

≤ u 1 ≤ • • • ≤ u n E |g(X u 1 , Y u 1 , X u 2 , Y u 2 -Y u 1 , . . . , X un , Y un -Y u n-1 )| < ∞
we have for any s ≥ 0 and t 1 , . . . , t n ≥ 0:

E[g(X s+t 1 , Y s+t 1 -Y s , . . . , X s+ n i=1 t i , Y s+ n i=1 t i -Y s+ n-1 i=1 t i ) | F (X,Y ) s ] = Q s (X s ; dx 1 × dz 1 ) n i=2 Q s (x i-1 ; dx i × dz i )g(x 1 , z 1 , . . . , x n , z n ) = ( n ⊗ i=1 Q t i )(g) X s . (3.4)
Proof. Lemma 3.1 gives the case n = 1. Let us check that Formula (3.4) is valid for n = 2. This can help the reader to follow the induction.

E[g(X s+t 1 , Y s+t 1 -Y s , X s+t 1 +t 2 , Y s+t 1 +t 2 -Y s+t 1 ) | F (X,Y ) s ] = E E g(X s+t 1 , Y s+t 1 -Y s , X s+t 1 +t 2 , Y s+t 1 +t 2 -Y s+t 1 ) | F (X,Y ) s+t 1 | F (X,Y ) s = E E g(X s+t 1 , Y s+t 1 -Y s , X s+t 1 +t 2 , Y s+t 1 +t 2 -Y s+t 1 ) | X s+t 1 , Y s+t 1 | F (X,Y ) s = E g(X s+t 1 , Y s+t 1 -Y s , x 2 , y 2 -Y s+t 1 )Q t 2 (X s+t 1 , Y s+t 1 ; dx 2 × dy 2 ) | F (X,Y ) s = E g(X s+t 1 , Y s+t 1 -Y s , x 2 , z 2 ) Q t 2 (X s+t 1 ; dx 2 × dz 2 ) | F (X,Y ) s (using (1.1)) = Q t 1 (X s ; dx 1 × dz 1 ) Q t 2 (x 1 ; dx 2 × dz 2 )g(x 1 , z 1 , x 2 , z 2 ) (using (3.3)) = ( Q t 1 ⊗ Q t 2 )(g) X s .
Let us now complete the induction. Assume that Property (3.4) is valid for n -1. Then

E[g(X s+t 1 , Y s+t 1 -Y s , . . . , X s+ n i=1 t i , Y s+ n i=1 t i -Y s+ n-1 i=1 t i ) | F (X,Y ) s ] = E E g(X s+t 1 , Y s+t 1 -Y s , . . . , X s+ n i=1 t i , Y s+ n i=1 t i -Y s+ n-1 i=1 t i ) | F (X,Y ) s+t 1 | F (X,Y ) s = E ( n ⊗ i=2 Q t i ) g(X s+t 1 , Y s+t 1 -Y s , •, • • • , •) X s+t 1 | F (X,Y ) s (induction) = Q t 1 ⊗ ( n ⊗ i=2 Q t i ))(g) X s (using (3.3)).
Corollary 3.1. Under (AS1), the following properties hold.

1. The process (Y t ) t∈T has stationary increments, i.e.

E π,0 g(Y s+t 1 -Y s , . . . , Y s+ n i=1 t i -Y s+ n-1 i=1 t i ) = π (⊗ n 1 Q t i )(g) (3.5)
does not depend on s for any function g as in Lemma 3.2.

If

E π,0 [|Y u |] < ∞ for every u ∈ T, then: ∀(s, t) ∈ T 2 , E π,0 [Y s+t ] = E π,0 [Y t ] + E π,0 [Y s ]. 3. (ξ n := Y n -Y n-1 ) n∈N * is a stationary sequence of R d -valued random variables and if h is a C-valued function such that E π,0 |h(ξ 1 , . . . , ξ n )| 2 = 1, then Q ⊗n 1 (h) ∈ L 2 (π) with Q ⊗n 1 (h) 2 ≤ 1, (3.6)
where Q ⊗n 1 denotes the n-fold kernel product

n ⊗ i=1 Q 1 .
Proof. Take the expectation of (3.4) with respect to the probability mesure π:

E π,0 g(Y s+t 1 -Y s , . . . , Y s+ n i=1 t i -Y s+ n-1 i=1 t i ) = E π,0 n ⊗ i=1 Q t i (g)(X s ) = E πPs,0 n ⊗ i=1 Q t i (g)(X 0 ) = E π,0 n ⊗ i=1 Q t i (g)(X 0 ) (invariance property of π).
The second property is deduced from the stationarity of the increments of (Y t ) t∈T . Indeed, we can write

E π,0 [Y t ] = E π,0 [Y s+t -Y s ] = E π,0 [Y s+t ] -E π,0 [Y s ]. That (ξ n ) n∈N * is a stationary sequence of random variables follows from (3.5) with s = 0, t 1 = • • • = t n = 1.
The last property follows from (3.4) and the Jensen inequality 3 . Under (AS1), the covariance has the following form

Q ⊗n 1 h 2 2 = E (•,0) h(ξ 1 , . . . , ξ n ) 2 2 ≤ E π,0 |h(ξ 1 , . . . , ξ n )| 2 = 1. Lemma 3.3. Let ξ n := Y n -Y n-1 for n ∈ N * (recall that Y 0 = 0 a.s.). Let f and h be two C- valued functions such that E π,0 |f (ξ 1 , . . . , ξ n )| 2 < ∞ and E π,0 |h(ξ n+t , . . . , ξ n+t+m )| 2 < ∞ for (t, n, m) ∈ (N * )
Cov π,0 f (ξ 1 , . . . , ξ n ); h(ξ n+t , . . . , ξ n+t+m ) = E π,0 f (ξ 1 , . . . , ξ n )(P t-1 -Π) Q ⊗m+1 1 (h) (X n ) (3.7)
where

Q ⊗m+1 1 denotes the (m + 1)-fold kernel product m+1 ⊗ i=1 Q 1 .
Proof. Apply formula (3.4) with E (x,0) to the specific function g

(x 1 , z 1 , . . . , x n+t+m , z n+t+m ) = f (z 1 , . . . , z n ) × h(z n+t , . . . , z n+t+m ) with t, n, m ≥ 1: E (x,0) f (ξ 1 , . . . , ξ n )h(ξ n+t , . . . , ξ n+t+m ) = Q ⊗n+m+t 1 (g)(x) = (X×R d ) n+t+m Q 1 (x; dx 1 × dz 1 ) n+t+m i=2 Q 1 (x i-1 ; dx i × dz i )g(x 1 , . . . , z n+t+m ) = (X×R d ) n Q 1 (x; dx 1 × dz 1 ) n i=2 Q 1 (x i-1 ; dx i × dz i )f (z 1 , . . . , z n ) × (X×R d ) t-1 n+t-1 i=n+1 Q 1 (x i-1 ; dx i × dz i ) × (X×R d ) m+1 n+t+m i=n+t Q 1 (x i-1 ; dx i × dz i )h(z n+t , . . . , z n+t+m ).
The second term reduces to

X t-1 n+t-1 i=n+1 Q 1 (x i-1 ; dx i × R d ) = X t-1 n+t-1 i=n+1 P 1 (x i-1 ; dx i ) = X P t-1 (x n ; dx n+t-1 ). The third is Q ⊗m+1 1 (h)(x n+t-1
). Then we have

E (x,0) f (ξ 1 , . . . , ξ n )h(ξ n+t , . . . , ξ n+t+m ) = (X×R d ) n Q 1 (x; dx 1 × dz 1 ) n i=2 Q 1 (x i-1 ; dx i × dz i )f (z 1 , . . . , z n )P t-1 ( Q ⊗m+1 1 (h) (x n ) = E (x,0) f (ξ 1 , . . . , ξ n )P t-1 ( Q ⊗m+1 1 h (X n ) (using (3.4) with E (x,0) ).
Then, integrating against the probability measure π(•) gives

E π,0 f (ξ 1 , . . . , ξ n )h(ξ n+t , . . . , ξ n+t+m ) = E π,0 f (ξ 1 , . . . , ξ n )P t-1 ( Q ⊗m+1 1 h (X n ) . (3.8) Since Π Q ⊗m+1 1 (h) (x) = π Q ⊗m+1 1 (h) for every x ∈ X, we obtain E π,0 f (ξ 1 , . . . , ξ n )Π( Q ⊗m+1 1 (h) (X n ) = E π,0 [f (ξ 1 , . . . , ξ n )]π Q ⊗m+1 1 (h) = E π,0 [f (ξ 1 , . . . , ξ n )]E π,0 [h(ξ n+t , . . . , ξ n+t+m )]
where the last equality follows from (3.5).

Remark 3.1. We can prove a continuous-time counterpart of Lemma 3.3. But, we restrict ourself to the discrete-time version because this is the version we need in the paper.

ρ-mixing property of (Y n -Y n-1 ) n≥1 for discrete-time stationary MAPs

Let us recall some basic facts on the ρ-mixing of a (strictly) stationary sequence of random variables (ξ n ) n∈N * (e.g. see [START_REF] Bradley | Introduction to strong mixing conditions[END_REF]). For any p ∈ N * and q ∈ N * ∪ {∞} with p ≤ q, G q p := σ(ξ p , . . . , ξ q ) denotes the σ-algebra generated by random variables ξ p , . . . , ξ q . The ρ-mixing coefficient at horizon t > 0, denoted by ρ(t), is defined by

ρ(t) := sup n∈N * sup |Corr(f ; h)| f ∈ L 2 (G n 1 ), h ∈ L 2 (G ∞ n+t ) . (3.9)
where Corr(f ; h) is the correlation coefficient of the two random variables f and g. In fact, ρ-mixing coefficient may be computed as follows from [START_REF] Bradley | Introduction to strong mixing conditions[END_REF]Prop 3.18]: for t > 0

ρ(t) = sup n∈N * sup m∈N * sup |Corr(f ; h)| f ∈ L 2 (G n 1 ), h ∈ L 2 (G n+t+m n+t ) . (3.10)
The stationary sequence (ξ n ) n∈N * is said to be ρ-mixing if lim t→+∞ ρ(t) = 0.

We know from condition (AS2) that (X n ) n∈N is ρ-mixing [START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF]. In the special case when

Y n := n k=1 ξ(X k ), it is clear that (Y n -Y n-1
) n≥1 is also ρ-mixing from [11, p. 28]. We extend this fact to general (Y n ) n∈N in the next proposition. Proposition 3.1 (ρ-mixing). Under (AS1-AS2), the stationary sequence (ξ n := Y n -Y n-1 ) n∈N * is ρ-mixing at an exponential rate: there exists ε > 0 such that

ρ(t) = O exp(-εt) .
Proof. For the sake of simplicity, assume that d = 1. First, note that the random variables f and h in (3.10) can be assumed to be of L 2 -norm 1. Thus, we just have to deal with covariances. Recall that (ξ n := Y n -Y n-1 ) n∈N * is known to be stationary under (AS1) from Corollary 3.1. The σ-algebra G n 1 and G n+t+m n+t in the mixing coefficients will be relative to the stationary sequence (ξ n ) n∈N * . Second, let us consider two L 2 -normed random variables

f (ξ 1 , . . . , ξ n ) ∈ L 2 (G n 1 ), h(ξ n+t , . . . , ξ n+t+m ) ∈ L 2 (G n+t+m n+t
). For any m ≥ 1, the map x → Q ⊗m+1 1 (h)(x) belongs to L 2 (π) and we have Q ⊗m+1 1 (h) 2 ≤ 1 from Corollary 3.1 (see (3.6)). Since P t and Π are contractions on L 2 (π), we have

(P t-1 -Π) Q ⊗m+1 1 (h) ∈ L 2 (π).
The Cauchy-Schwarz inequality and the last comments allow us to write from (3.7)

Cov(f ; h) 2 ≤ E π,0 |f (ξ 1 , . . . , ξ n )| 2 E π,0 (P t-1 -Π) Q ⊗m+1 1 (h) (X n ) 2 = E π,0 (P t-1 -Π) Q ⊗m+1 1 (h) (X 0 ) 2 (π is P n -invariant) = (P t-1 -Π) Q ⊗m+1 1 (h) 2 2 ≤ P t-1 -Π 2 2 Q ⊗m+1 1 (h) 2 2
(since

Q ⊗m+1 1 (h) ∈ L 2 (π)) ≤ P t-1 -Π 2 2 (since Q ⊗m+1 1 (h) 2 ≤ 1).
Therefore, it follows that for every t ≥ 1:

sup |Corr(f ; h)| f ∈ L 2 (G n 1 ), h ∈ L 2 (G n+t+m n+t ) ≤ P t-1 -Π 2 .
The right hand side term in the inequality above does not depend on m and n, so that we obtain from (3.10)

ρ(t) ≤ P t-1 -Π 2 .
The proof is completed by using the exponential estimate (2.2) of P t-1 -Π 2 under (AS2).

Central limit theorem for the Markov additive processes

In a first step, we consider a discrete-time

X × R d -valued MAP, (X n , Y n ) n∈N , for which the driving Markov chain (X n ) n∈N is assumed to satisfy (AS1-AS2). Recall that Condition (AS3d) for α = 2 is E π,0 |Y 1 | 2 < ∞. This condition implies that E π,0 [|Y 1 |] < ∞, and we suppose that E π,0 [Y 1 ] = 0 for convenience (if not, replace Y n by Y n -E π,0 [Y n ] = Y n -nE π,0 [Y 1 ] from Corollary 3.1).
We know from Proposition 3.1 that (Y n+1 -Y n ) n∈N is stationary and is exponentially ρmixing when (AS1-AS2) hold. Then, under the expected moment condition [START_REF] Ibragimov | A note on the central limit theorem for dependent random variables[END_REF][START_REF] Peligrad | On the central limit theorem for ρ-mixing sequences of random variables[END_REF] (e.g. see [START_REF] Bradley | Introduction to strong mixing conditions[END_REF]Th 11.4]). To the best of our knowledge, Theorem 3.1 for general MAP is new. The notation N (0, 0) stands for the Dirac distribution at 0.

E π,0 [|Y 1 | 2 ] < ∞, the CLT for (Y n ) n∈N * follows from
Σ := lim n 1 n E π,0 [ Y n Y * n ]
where the symbol * denotes the transpose operator.

That (Y n / √ n) n∈N satisfies the CLT under the condition E π,0 [|Y 1 | 2 ]
< ∞ was known in some cases. Such standard situations are recalled in the two next remarks (with d = 1 to simplify). Remark 3.2 (Martingale method). If there exists a measurable function ξ :

X → R such that Y n -Y n-1 = ξ(X n ) and E π,0 [|Y 1 | 2 ] = π(ξ 2 ) < ∞, then (Y n / √ n) n∈N converges in distribution
to the Gaussian law N (0, σ 2 ) where σ 2 = π(ξ 2 ) + 2 +∞ ℓ=1 π(ξP ℓ ξ) ∈ [0, +∞). This result follows from the Gordin-Lifsic theorem [START_REF] Gordin | On the central limit theorem for stationary Markov processes[END_REF]. Indeed, (AS2) implies that (X n ) n∈N is ergodic and that there is a solution ξ ∈ L 2 (π) to the Poisson equation: ξ -P ξ = ξ. Then, the difference martingale method of [START_REF] Gordin | On the central limit theorem for stationary Markov processes[END_REF] can be used to obtain the CLT. Remark 3.3 (Uniform ergodicity). Recall that the Markov chain (X n ) n∈N is said to be uniformly ergodic if lim n→+∞ P n -Π ∞ = 0. This property implies (AS2) (but is stronger) and is fulfilled if and only if (X n ) n∈N is ergodic, aperiodic and satisfies the Doeblin condition (D 0 ). In addition, for an aperiodic and ergodic Markov chain (X n ) n∈N , Doeblin's condition is equivalent to the uniform mixing (or φ-mixing) property, and then, the φ-mixing coefficients go to 0 at least exponentially fast (see [START_REF] Bradley | Basic properties of strong mixing conditions. a survey and some open questions[END_REF][START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF]).

Set ξ n := Y n -Y n-1 . If (X n ) n∈N is uniformly ergodic and if E π,0 [Y 2 1 ] < ∞, then the real number σ 2 = E π,0 [ξ 2 1 ] + 2 +∞ ℓ=1 E π,0 [ξ 1 ξ ℓ ] is well-defined in [0, +∞). If σ 2 > 0, then the sequence (Y n / √ n) n∈N converges in distribution to N (0, σ 2 ) [38]
. This CLT is established as follows: the stationarity and the uniform ergodicity of (X n ) n∈N extend to the sequence (ξ n ) n∈N , and the φ-mixing coefficients of (ξ n ) n∈N also go to 0 at an exponential rate (see [START_REF] Grigorescu | Limit theorems for J -X processes with a general state space[END_REF]Rk. 4,Lem. 1]). The proof is completed using [53, Th.18.5.2].

The CLT for a continuous-time MAP (X t , Y t ) t≥0 is deduced from the discrete-time statement. Theorem 3.2. Under (AS1-AS2) and (AS3) with α = 2, (Y t / √ t) t>0 satisfies a CLT.

Proof. For t ∈ [0, +∞), we set t = n + v, where n is the integer part of t and v ∈ [0, 1). We can write:

Y t √ t = (Y t -Y n ) √ t + √ n √ t Y n √ n . (3.11)
Recall that (Q t ) t≥0 is the transition semigroup of (X t , Y t ) t≥0 . It is easily checked that the MAP (X t , Y t ) t≥0 "sampled" at discrete instants, (X n , Y n ) n∈N , is a discrete-time MAP with transition kernel Q := Q 1 which satisfies (AS1-AS2) and (AS3d). Therefore, (Y n / √ n) n∈N satisfies a CLT thanks to Theorem 3.1. Finally, the sequence ((Y t -Y n )/ √ t) t>0 converges in probability to 0 from the Tchebychev inequality and condition (AS3d):

P π,0 |Y t -Y n | > √ tε = P π,0 |Y v | > √ tε (stationary increments) ≤ E π,0 |Y v | 2 tε 2 ≤ sup v∈(0,1] E π,0 |Y v | 2 tε 2 ----→ t→+∞ 0.
Therefore, (Y t / √ t) t>0 satisfies a CLT from (3.11).

Remark 3.4 (FCLT). Proposition 3.1 allows us to deduce from [8, Th. [START_REF] Dehay | On likelihood estimation for discretely observed Markov jump processes[END_REF].2] that a functional central limit theorem also holds (d = 1). That is, under the assumptions of Theorem 3.1, we have:

Y ⌊nt⌋ √ n t≥0 L -----→ n→+∞ (B t ) t≥0 (3.12)
as random elements of D[0, ∞), the Skorokhod space of cadlag functions on R + , and where (B t ) t≥0 is a Brownian motion with zero drift and some variance parameter. Let us give some comments on the FCLT relevant to our context.

(a) The case of a discrete-time MAP (X n , Y n ) n∈N with (X n ) n∈N satisfying the Doeblin condition is covered by [START_REF] Grigorescu | Limit theorems for J -X processes with a general state space[END_REF]. (Y n ) n∈N is shown to be φ-mixing and a FCLT for φ-mixing sequences is used. We extend their approach to our case of L 2 -spectral gap.

(b) Under (AS1-AS2) and the expected moment condition of order 2, Maigret [START_REF] Maigret | Théorème de limite centrale fonctionnel pour une chaîne de Markov récurrente au sens de Harris et positive[END_REF] has established a FCLT for (Y n := ξ(X n-1 , X n )) n∈N * in the specific case where (X n ) n∈N is Harris-recurrent. It is worth noticing that Condition (AS2) cannot be compared with the Harris-recurrence property.

(c) If (X t ) t≥0 is a stationary ergodic Markov process with a strongly continuous transition semigroup (P t ) t≥0 on L 2 (π), the following convergence holds for any f ∈ L 2 (π) such that π(f ) = 0 [7, Th. 2.1, Prop. 2.3] (see also [START_REF] Touati | Théorèmes de limite centrale fonctionnels pour les processus de Markov[END_REF] in the Harris-recurrent case):

1 √ n nt 0 ξ(X s )ds t≥0 L -----→ n→+∞ (B t ) t≥0
where (B t ) t≥0 is a Brownian motion with zero drift and some variance parameter. Set Y t := t 0 ξ(X s )ds. Since ξ ∈ L 2 (π), we have E π,0 |Y t | 2 ≤ π(|ξ| 2 ) for every t ∈ (0, 1], thus (AS3) with α = 2 is true. Then, the convergence result above is easily deduced from (3.12) using the discrete-time stationary MAP (X n , Y n ) n≥1 introduced in the proof of Theorem 3.2.

(d) Glynn and Whitt deal with the integral functional of a regenerative process in [START_REF] Glynn | Limit theorems for cumulative processes[END_REF][START_REF] Glynn | Necessary conditions in limit theorems for cumulative processes[END_REF].

Their results apply to a Markov process which is a specific instance of a regenerative process. Conditions for the CLT (FCLT) to hold are expressed in terms of a second moment on the increments Y T 1 := T 1 0 ξ(X s )ds of the process (Y t ) t>0 over a regeneration cycle of length T 1 (and an additional condition of negligeability in probability of sup 0≤s≤T 1 |Y s |). The fact that we only consider the Markov case makes our conditions easier to check.

Refinements of the central limit theorem for MAPs

Let (X t , Y t ) t∈T be a MAP taking values in X × R d . The canonical scalar product on R d is denoted by •, • . The Fourier operators associated with (X t , Y t ) t∈T are introduced in the next subsection and are shown to satisfy a semigroup property. In the discrete-time case, precise expansions of the characteristic function of the additive component Y t can be deduced from [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] under (AS2). They are central to the derivation of our limit theorems in this section. Limit theorems are first considered for discrete-time MAPs. A local limit theorem, a Berry-Esseen bound and a first-order Edgeworth expansion are obtained. The continuous-time case is addressed thanks to the basic reduction to the discrete-time case used for the CLT.

Fourier operators. A semigroup property

For any t ∈ T and ζ ∈ R d , we consider the linear operator S t (ζ) acting (in a first step) on the space of bounded measurable functions f : X → C as follows:

∀x ∈ X, S t (ζ)f (x) := E (x,0) e i ζ,Yt f (X t ) . (4.1)
Note that S t (0) = P t . In the discrete-time case, S 1 (ζ) corresponds to the Fourier operator which was first introduced by Nagaev [START_REF] Nagaev | Some limit theorems for stationary Markov chains[END_REF] in the special case when Y n = n k=1 ξ(X k ) (see [START_REF] Guivarc'h | Thormes limites pour une classe de chanes de Markov et applications aux diffomorphismes d'Anosov[END_REF][START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF] and the reference therein), and was extended to discrete-time MAPs in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF][START_REF] Guivarc'h | Application d'un théorème limite local à la transcience et à la récurrence de marches aléatoires[END_REF] to prove local limit theorems and renewals theorems (see also [START_REF] Fuh | Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks[END_REF]). All these works are based on the following formula (see Proposition 4.1 below):

∀ζ ∈ R d , ∀n ∈ N, S n (ζ)f (x) := E (x,0) e i ζ,Yn f (X n ) = S 1 (ζ) n f (x). (4.2)
This formula clearly reads as the semigroup property:

S m+n (ζ) = S m (ζ) • S n (ζ).
In the continuous-time, it seems that the operators S t (ζ) were first introduced in [START_REF] Hitsuda | The central limit theorem for additive functionals of Markov processes and the weak convergence to Wiener[END_REF] for investigating AFs of continuous-time Markov processes on a compact metric state space X. In [START_REF] Hitsuda | The central limit theorem for additive functionals of Markov processes and the weak convergence to Wiener[END_REF], P t was assumed to have a spectral gap on the space of all continuous C-valued functions on X, and (S t (ζ)) t>0 was thought of as a semigroup (see (SG) below) on this space.

Here, in view of (AS2), the above mentioned semigroup property has to be considered on the Lebesgue spaces L p (π) (1 ≤ p ≤ ∞). Proposition 4.1. For all t ∈ T and ζ ∈ R d , S t (ζ) defines a linear contraction on L p (π), and we have:

∀ζ ∈ R d , ∀(s, t) ∈ T 2 , S t+s (ζ) = S t (ζ) • S s (ζ). (SG)
In particular, Relation (4.2) holds for all f ∈ L p (π).

Proof. The first assertion is easy to prove. Next, for any ζ ∈ R d and f ∈ L p (π), let us set: g(x, y) := f (x) e i ζ,y with x ∈ X and y ∈ R d . Then, using the Markov property and Lemma 3.1:

(S t+s (ζ)f )(x) := E (x,0) e i ζ,Y t+s f (X t+s ) = E (x,0) E (x,0) [e i ζ,Y t+s f (X t+s ) | F (X,Y ) s ] = E (x,0) (Q t g Ys )(X s , 0) = E (x,0) [e i ζ,Ys (Q t g)(X s , 0)] = E (x,0) e i ζ,Ys )E Xs,0 [f (X t )e i ζ,Yt ] = E (x,0) e i ζ,Ys (S t (ζ)f )(X s ) = S s (ζ) S t (ζ)f (x)
the third equality results from: g Ys (x, y) = f (x) e i ζ,(y+Ys) = e i ζ,Ys g(x, y). This gives the semigroup property (SG). The last assertion is obvious.

Expansions of the characteristic function of the additive component

Here we assume that (X n , Y n ) n∈N is a discrete-time MAP taking values in X × R d (possibly derived from a continuous-time MAP) such that the driving Markov chain (X n ) n∈N is stationary and satisfies (AS2). This last property ensures that S 1 (0) has good spectral properties, and the iterates S 1 (ζ) n occurring in (4.2) are studied using the Nagaev-Guivarc'h spectral method which consists in applying the perturbation theory to the Fourier operators S 1 (ζ) for small ζ. However using the standard perturbation theorem requires strong assumptions on Y 1 . Here we shall appeal to the weak spectral method introduced in [START_REF] Hennion | Central limit theorems for iterated random Lipschitz mappings[END_REF] and based on the Keller-Liverani perturbation theorem [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF]. This method is fully developed in the Markov framework in [49,

∀n ∈ N, ∀ζ ∈ O, E π,0 e i ζ,Yn f (X n ) = λ(ζ) n L(ζ, f ) + R n (ζ, f ), (4.3)
where λ(•), L(•, f ), R n (•, f ) are C-valued functions of class C m 0 on O, with λ(0) = 1 and L(0, f ) = π(f ). Moreover, we have the following properties for ℓ = 0, . . . , m 0 :

-

sup ζ∈O |L (ℓ) (ζ, f )| < ∞ (4.4a) -∃κ ∈ (0, 1), sup ζ∈O |R (ℓ) n (ζ, f )| = O(κ n ). (4.4b) If f := 1 X , we have R n (0, 1 X ) = 0.
When Y n = n k=1 ξ(X k ), the above properties are proved in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Sect. 7.3] by using (4.2) and some operator-type derivation arguments. For a general additive component Y n , the method is the same 1 using Lemmas 4.1 and 4.2 below which slightly extend [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Lem. 4.2,7.4]. Mention that, by using the same lemmas, Proposition 4.2 can also be deduced from [START_REF] Gouëzel | Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps[END_REF] which specifies the method introduced in [START_REF] Gouëzel | Banach spaces adapted to Anosov systems[END_REF][START_REF] Hennion | Central limit theorems for iterated random Lipschitz mappings[END_REF] to prove Taylor expansions of λ(

•), L(•, f ), R n (•, f ) 2 .
The operator norm in the space L(L p , L p ′ ) of the linear bounded operators from L p (π) to

L p ′ (π) is denoted by • p,p ′ . Lemma 4.1. If 1 ≤ p ′ < p, then the map ζ → S 1 (ζ) is continuous from R d to L(L p , L p ′ ). Proof. We have for ζ ∈ R d , ζ 0 ∈ R d and f ∈ L p (π), thanks to Hölder's inequality (S 1 (ζ) -S 1 (ζ 0 ))f (x) p ′ = E (x,0) e i ζ,Y 1 f (X 1 ) -E (x,0) e i ζ 0 ,Y 1 f (X 1 ) p ′ ≤ E (x,0) e i ζ-ζ 0 ,Y 1 -1 p ′ |f (X 1 )| p ′ ≤ 2 p ′ E (x,0) min 1, | ζ -ζ 0 , Y 1 | p ′ |f (X 1 )| p ′ ,
the last inequality resulting from the classic inequality |e ia -1| ≤ 2 min 1, |a| . An integration with respect to π and the use of Hölder's inequality give

π |(S 1 (ζ) -S 1 (ζ 0 ))f | p ′ ≤ 2 p ′ E π,0 min 1, | ζ -ζ 0 , Y 1 | (pp ′ )/(p-p ′ ) (p-p ′ )/p E π,0 |f (X 1 )| p p ′ /p ≤ 2 p ′ min 1, | ζ -ζ 0 , Y 1 | p ′ (pp ′ )/(p-p ′ ) f p ′ p ,
since π is invariant. Thus, we deduce that

S 1 (ζ)-S 1 (ζ 0 ) p,p ′ ≤ 2 min 1, | ζ-ζ 0 , Y 1 | (pp ′ )/(p-p ′ )
goes to 0 when |ζζ 0 | → 0 from Lebesgue's theorem.

Lemma 4.2. Assume that (AS3d) holds for some α > m 0 (m 0 ∈ N * ), and let 1 ≤ j ≤ m 0 . If p > 1 and

p j := αp/(α + jp) ≥ 1, then ζ → S 1 (ζ) is j-times continuously differentiable from R d to L(L p , L p j ), and
sup ζ∈R d S (j) 1 (ζ) p,p j ≤ E π,0 [|Y 1 | α ] j/α .
Proof. For the sake of simplicity, we suppose that d = 1. Below we consider any ζ ∈ R, ζ 0 ∈ R and f ∈ L p (π). For 1 ≤ j ≤ m 0 , define (formally) the following linear operator:

∀x ∈ X, S (j) 1 (ζ)f (x) := E (x,0) (iY 1 ) j e iζY 1 f (X 1 ) .
1 See the beginning of the appendix. In particular, mention that λ(ζ) is the dominant eigenvalue of S1(ζ), L(ζ, •) is related to the associated eigenprojection, and κ can be chosen as κ = (e -ε + 1)/2 where ε > 0 is defined in (2.2).

2 As observed in [START_REF] Gouëzel | Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps[END_REF], the passage from the Taylor expansions to the differentiability properties can be derived from [START_REF] Campanato | Proprietà di una famiglia di spazi funzionali[END_REF].

First we have:

|S (j) 1 (ζ)f (x)| p j ≤ E (x,0) |Y 1 | jp j |f (X 1 )| p j so that, from Hölder's inequality, S (j) 1 (ζ) p,p j ≤ E π,0 |Y 1 | α j/α . Second, define ∆ := S (j-1) 1 (ζ) -S (j-1) 1 (ζ 0 ) -(ζ -ζ 0 ) S (j)
1 (ζ 0 ). Then, for j ∈ {1, . . . , m 0 -1}, we have thanks to the classic inequality

|e ia -1 -ia| ≤ 2|a| min 1, |a| |∆f (x)| p j ≤ 2 p j |ζ -ζ 0 | p j E (x,0) min 1, |(ζ -ζ 0 )Y 1 | p j |Y 1 | jp j |f (X 1 )| p j .
It follows from Hölder's inequality that the operator norm satisfies

∆ p,p j ≤ 2|ζ -ζ 0 | min 1, |(ζ -ζ 0 )Y 1 | |Y 1 | j α/j .
This proves that S (j-1) 1

(•) is differentiable from R to L(L p , L p j ),
and that its derivatives is S (j) 1 . Finally, we obtain:

(S (j) 1 (ζ) -S (j) 1 (ζ 0 ))f (x) p j ≤ 2 p j E (x,0) min 1, |(ζ -ζ 0 )Y 1 | p j |Y 1 | jp j |f (X 1 )| p j
from which we deduce that the operator norm satisfies

S (j) 1 (ζ) -S (j) 1 (ζ 0 ) p,p j ≤ 2 min 1, |(ζ -ζ 0 )Y 1 | |Y 1 | j α/j . Thus S 1 (•) is j-times continuously differentiable from R to L(L p , L p j ).
Next, let us return to our probabilistic context. Let ∇ and Hess denote the gradient and the Hessian operators respectively. In the following proposition, the d-dimensional vector ∇λ(0) and the symmetric d × d-matrix Hessλ(0) are related to the mean vector E π,0 [Y 1 ] and the asymptotic covariance matrix associated with the sequence

(Y n -n E π,0 [Y 1 ])/ √ n.
Proposition 4.3.

(i) If (AS3d) holds for some α > 1, then ∇λ(0

) = i E π,0 [Y 1 ].
(ii) If (AS3d) holds for some α > 2, then the following limit exists in the set of the nonnegative symmetric d × d-matrices:

Σ := lim n 1 n E π,0 Y n -n E π,0 [Y 1 ] Y n -n E π,0 [Y 1 ] * = -Hessλ(0).
Proof. Assume that d = 1 for the sake of simplicity (for d ≥ 2, the proof is similar by using partial derivatives). By differentiating at ζ = 0 the equality

E π,0 [e iζYn ] = λ(ζ) n L(ζ, 1 X ) + R n (ζ, 1 X ) of Proposition 4.2, we obtain: i E π,0 [Y n ] = n λ (1) (0) + L (1) (0, 1 X ) + R (1) n (0, 1 X ). Since E π,0 [Y n ] = n E π,0 [Y 1 ] (from Corollary 3.1), we deduce that λ (1) (0) = i and lim n E π,0 [Y n ]/n = i E π,0 [Y 1 ] from (4.4b).
To prove (ii), assume for convenience that E π,0 [Y 1 ] = 0. Then λ (1) (0) = 0, and differentiating twice the above equality at ζ = 0 gives:

-E π,0 [Y 2 n ] = n λ (2) (0)+ L (2) (0, 1 X )+ R (2)
n (0, 1 X ). We obtain the desired property by using again (4.4b).

Refinements of the CLT for discrete-time MAPs

In this subsection, (X n , Y n ) n∈N is a MAP taking values in X × R d , with a driving Markov chain (X n ) n∈N satisfying (AS1-AS2). The assumptions below imply that E π,0 [|Y 1 |] < ∞, and for convenience we suppose that

E π,0 [Y 1 ] = 0 (if not, replace Y n by Y n -nE π,0 [Y 1 ]).
Theorems 4.1 to 4.3 below have been established in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] for additive components of the form Y n = n k=1 ξ(X k ). To the best of our knowledge, the present extensions to general MAP are new.

A local limit theorem

The classical Markov nonlattice condition is needed to state the local limit theorem (LLT):

Nonlattice condition. There is no a ∈ R d , no closed subgroup H in R d , H = R d , and no bounded measurable function β : X → R d such that: Y 1 + β(X 1 ) -β(X 0 ) ∈ a + H P π,0 -a.s.
This condition is equivalent to the following operator-type property. For each p ∈ (1, ∞) and for all compact subset K of R d \ {0}, there exists ρ ∈ (0, 1) such that:

sup ζ∈K S 1 (ζ) n p = O(ρ n ). (4.5)
This result is established in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Sect. 5] for additive functionals. The proof for general MAPs is similar.

Since E π,0 [e i ζ,Yn ] = π(S 1 (ζ) n 1 X ) by (SG), it follows that sup ζ∈K E π,0 [e i ζ,Yn ] = O(ρ n ).
Theorem 4.1. The assumptions of Theorem 3.1 are supposed to be satisfied, so that

(Y n / √ n) n∈N *
converges in distribution to a d-dimensional Gaussian vector with covariance matrix Σ. Let us assume that Σ is a definite positive matrix. Finally, suppose that the nonlattice condition is true. Then, we have for all compactly supported continuous function g : R d → R:

lim n→+∞ √ det Σ (2πn) d 2 E π,0 [g(Y n ) ] = R d g(x)dx.
Proof. Thanks to (4.3) with f := 1 X , Theorem 4.1 can be established as in the i.i.d. case: use Proposition 4.2 to control L(•, 1 X ) and R n (ζ, 1 X ) and, as in [START_REF] Breiman | Probability. SIAM[END_REF], use the nonlattice condition and the following second-order Taylor expansion of λ(•), which follows from Theorem 3.

(ζ) = 1 -ζ, Σζ /2 + o(|ζ| 2 ).
Remark 4.1. We mention that a local limit theorem has been obtained in [START_REF] Maxwell | A local limit theorem for hidden Markov chains[END_REF] for the process (Y n := n k=1 Z k ) n∈N * associated with a stationary hidden Markov chain (X n , Z n ) n∈N . In [START_REF] Maxwell | A local limit theorem for hidden Markov chains[END_REF], (X n ) n∈N is only assumed to be an ergodic stationary Markov chain so that the additional conditions for the local limit theorem to hold are more involved than those of Theorem 4.1.

Rate of convergence in the one-dimensional CLT

Here we suppose that d = 1. Under the condition

E π,0 [|Y 1 | 2+ε ] < ∞, the asymptotic variance σ 2 of Proposition 4.3 is defined by σ 2 := lim n E π,0 [Y 2
n ]/n. Theorem 4.2. Under Conditions (AS1-AS2) and (AS3d) for some α > 3 and if σ 2 > 0, then there exists some constant B > 0 such that ∀n ≥ 1, sup

a∈R P π,0 Y n σ √ n ≤ a -Φ(a) ≤ B √ n (4.6)
where Φ(•) is the distribution function of the Gaussian distribution N (0, 1).

Proof. Here, the functions

λ(•), L(•) := L(•, 1 X ) and R n (•) := R n (•, 1 X ) in Proposition 4.
2 are three times continuously differentiable on O and satisfy the following properties:

sup u∈O |L(u) -1|/|u| < ∞ (from (4.4a) and L(0) = 1) sup u∈O |R n (u)/u| = O(κ n ) (from (4.4b) and R n (0) = 0) λ(u) = 1 -σ 2 u 2 /2 + O(u 3
) for u small enough (since λ (1) (0) = 0 and λ (2) (0) = -σ 2 from Proposition 4.3).

Then, we can borrow the proof of the Berry-Esseen theorem of the i.i.d. case (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF]).

Remark 4.2. The details of the previous proof are reported in [START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF]Th.2] for the additive functional

Y n = n k=1 ξ(X k , X k-1
) of a V -geometrically Markov chain. They are the same in our context. In fact, by writing out the arguments of [48, Th. 2], we can derive the following more precise property: the constant B in (4.6) depends on the sequence (Y n ) n∈N , but only through σ 2 and E π,0 [|Y 1 | 3+ε ]. Of course, this control is not as precise as in the i.i.d. case [START_REF] Feller | An introduction to probability theory and its applications[END_REF], but it is enough to obtain interesting statistical properties as in [START_REF] Ferré | Développement d'Edgeworth d'ordre 1 pour des M-estimateurs dans le cas de chaînes V-géométriquement ergodiques[END_REF][START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF] or in Section 5. Remark 4.3. Let us consider the specific case Y n -Y n-1 = ξ(X n ) for some real-valued measurable function ξ. Under Conditions (AS1-AS2), if the real number σ 2 defined in Remark 3.2 is positive, then we have (4.6) under the expected moment condition π(|ξ| 3 ) < ∞. This follows from [START_REF] Hervé | Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques[END_REF]Cor. 3.1] which is based on the spectral method and martingale difference arguments (see also [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Sect. 6]). Note that the moment condition π(|ξ| 3 ) < ∞ is optimal according to the i.i.d. case [START_REF] Feller | An introduction to probability theory and its applications[END_REF]. Remark 4.4. Let (X n ) n∈N be a ρ-mixing Markov chain. The additive functionals of (X n ) n∈N involved in the M -estimation of Markov models (see (5.5)) are of the form

Y n = n k=1 ξ(X k-1 , X k ). Since (X n , Y n ) n∈N is a MAP, Theorem 4.2 applies provided that ξ : X×X → R is a measurable function such that E π,0 [ξ(X 0 , X 1 )] = 0 and E π,0 [ |ξ(X 0 , X 1 )| 3+ε
] < ∞ for some ε > 0. This will be supported by the statistical result of Section 5.

Finally let us state a first-order Edgeworth expansion. Theorem 4.3. Assume that Conditions (AS1-AS2) and (AS3d) hold for some α > 3, that σ 2 is positive and the nonlattice condition is true. Then, there exists µ 3 ∈ R such that:

P π,0 Y n σ √ n ≤ a = Φ(a) + µ 3 6σ 3 √ n (1 -a 2 ) η(a) + o 1 √ n (4.7)
where η(•) is the density of the Gaussian distribution N (0, 1).

Other limit theorems can be stated under Condition (AS2) as, for instance, a multidimensional Berry-Esseen theorem in the Prohorov metric (see [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Sect. 9]), and the multidimensional renewal theorems (see [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF]). Although Proposition 4.2 extends to the case when the order of regularity m 0 is not integer, it does not allow to deal with the convergence of Y n (properly normalized) to stable laws, since we assume α > m 0 (in place of the expected condition α = m 0 ). For an additive functional Y n = n k=1 ξ(X k ), a careful examination of the proof of Lemmas 4.1 and 4.2 shows that this limitation could be overcame under a condition of the type : ξ ∈ L β (π) =⇒ P ξ ∈ L β ′ (π) with β ′ > β. Anyway mention that, under Condition (AS2) and the previous condition on ξ, convergence to stable laws is obtained in [54, Section 2.3] by using a "martingale approximation" approach. A natural question is to ask wether the last condition on ξ is necessary.

The non-stationary case

Under (AS2), we discuss the extension of the previous results to the non-stationary case. Let µ be the initial distribution of (X n ) n∈N . The careful use of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Prop. 7.3] allows us to extend Proposition 4.2 as follows. Under condition (AS3d) 4 with α > m 0 , and under the following assumption on µ (NS) µ is a bounded linear form on L r (π) with r such that 1 < r < αs/(α + m 0 s), where s > α/(αm 0 ), all the conclusions of Proposition 4.2 remain true when π is replaced by µ, namely: for some bounded open neighborhood Condition (NS) means that µ is absolutely continuous with respect to π with density φ ∈ L r ′ (π) where r ′ = r/(r -1) is the conjugate number of r. It is easily checked that r ′ > αs/ (αm 0 )sα > 1. Note that the bigger is the exponent α in Condition (AS3d), the closer to 1 is the allowed value of r ′ . Proposition 4.3 extends to the non-stationary case as follows.

O of ζ = 0 in R d , we have for f ∈ L s (π) ∀n ∈ N, ∀ζ ∈ O, E µ,0 e i ζ,Yn f (X n ) = λ(ζ) n L(ζ, f, µ) + R n (ζ, f, µ), (4.8
(i) If (AS3d) and (NS) hold with m 0 = 1, then ∇λ(0

) = i lim n E µ,0 [Y n ]/n.
(ii) If (AS3d) and (NS) hold with m 0 = 2, then the conclusions of Proposition 4.3(ii) remain true with µ in place of π.

Using the decomposition (4.8) (with f := 1 X ), we obtain as in the stationary case the following statements.

1. Under (AS2), (AS3d) with α = 2 and µ satisfying condition (NS) with m 0 = 1: the CLT, and the LLT under the additional non-lattice condition.

2. Under (AS2), (AS3d) with α > 3 and µ satisfying condition (NS) with m 0 = 3: the Berry-Esseen bound, and under the non-lattice condition, the first order Edgeworth expansion (4.7) with the additional term -b µ η(u)/(σ √ n), where b µ is the asymptotic bias: b µ = lim n E µ,0 [Y n ] (see [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] for details).

For instance, let us sketch the proof of the CLT. Equality (4.8) with f := 1 X gives In general, the previous statements 1. and 2. do not apply to the case when the initial distribution µ is a Dirac mass (which is not defined on L r (π)). However, when the state space X of the driving Markov chain is discrete, these statements are valid with any initial distribution δ x provided that π(x) > 0 (because δ x is then a continuous linear form on each L p (π) ≡ ℓ p (π)).

E µ,0 e i ζ,Yn/ √ n = λ(ζ/ √ n) n L(ζ/ √ n, 1 X , µ) + R n (ζ/ √ n, 1 X , µ). Since m 0 = 1 we have lim n L(ζ/ √ n, 1 X , µ) = 1 and lim n R n (ζ/ √ n, 1 X , µ) = 0.

The continuous-time case

In this section, we consider the case where T = (0, +∞). The process (X t ) t>0 is assumed to satisfy Conditions (AS1-AS2). Let us mention that the moment condition (AS3) reduces to

∀v ∈ (0, 1], E π,0 |Y v | α < ∞ when the semigroup (Q t ) t≥0 is strongly continuous on L 2 ((π, 0)) (so is (P t ) t>0 on L 2 (π)).
All the theorems of the previous subsection are extended to (Y t ) t>0 . Recall that Theorems 4.1 to 4.3 concern the multidimensional local limit theorem, the one-dimensional Berry-Esseen theorem, the one-dimensional first-order Edgeworth expansion respectively. For the sake of simplicity, we still assume that E π,0 [Y 1 ] = 0. Theorem 4.4. The conclusions of Theorems 4.1 to 4.3 are valid for (Y t / √ t) t>0 under the same assumptions, up to the following change: the moment condition (AS3d) is reinforced (with the same condition on α) in (AS3):

sup v∈(0,1] E π,0 |Y v | α < ∞.
Note that the extensions to the non-stationary case presented in Subsection 4.4 can be adapted to the continuous-time case.

When Y t is defined by Y t := t 0 ξ(X s ) ds, any moment condition of the type sup v∈[0,1] E π,0 |Y v | α < ∞ (α ≥ 1) is fulfilled if we have π(|ξ| α ) < ∞. Indeed: ∀v ∈ [0, 1], E π,0 |Y v | α ≤ E π,0 1 0 |ξ(X s )| α ds = 1 0 E π,0 |ξ(X s )| α ds = π(|ξ| α ).
Note that the nonlattice condition used in Theorem 4.4 is the same as in the discrete-time case (see Subsection 4.

|L (ℓ) v,f (ζ)| < ∞ ∃κ ∈ (0, 1), sup ζ∈O, v∈[0,1] |R (ℓ) n,v,f (ζ)| = O(κ n ).
Note that we have λ(0) = 1, L v,f (0) = π(f ), and R n,v,1 X (0) = 0.

Proof. From (4.1) and (SG), we obtain for any To that effect we need a careful use of the operator-type derivation procedure. This part is postponed in Appendix A on the basis of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF].

ζ ∈ R d , f ∈ L p (1 ≤ p ≤ ∞): E π,0 e i ζ,Yt f (X t ) = π S n+v (ζ)f = π S 1 (ζ) n S v (ζ)f = E π,0 e i ζ,Yn S v (ζ)f (X n ) , (4.10

A Berry-Esseen theorem for the M-estimators of ρ-mixing

Markov chains

The M -estimators are a general class of estimators in parametric statistics. This covers the special cases of maximum likelihood estimators, the least square estimators and the minimum contrast estimators. In the i.i.d. case, a modern treatment on M -estimation is reported in [START_REF] Van Der Vaart | Asymptotic statistics[END_REF]Chap. 5], and a Berry-Esseen bound for M -estimators is obtained in [START_REF] Pfanzagl | The Berry-Esseen bound for minimum contrast estimates[END_REF]. In a statistical framework, such a bound has to be uniform in the parameters. Pfanzagl's method, which is applied to Markov data in [START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF], requires a preliminary result on the rate of convergence in the CLT for additive functionals, with a precise control of the constants with respect to the functional (cf Remark 5.1). Earlier extensions of [START_REF] Pfanzagl | The Berry-Esseen bound for minimum contrast estimates[END_REF] to the Markov context are discussed in [START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF]. For ρ-mixing Markov chains, the closest work to ours is [START_REF] Rao | On the rate of convergence of estimators for Markov processes[END_REF]. Our main improvement is on the moment conditions which are now close to those of the i.i.d. case. A detailed comparison is presented at the end of the section.

Let Θ be any nonempty parameter set. For a Markov chain (X n ) n∈N with state space X and transition kernel P θ which depends on θ ∈ Θ, we introduce the uniform L 2 (π)-spectral gap (i.e. the uniform ρ-mixing) property.

(M) The Markov chain (X n ) n∈N has a uniform L 2 (π)-spectral gap with respect to the parameter set Θ if

1. for all θ ∈ Θ, (X n ) n∈N has a unique P θ -invariant distribution π θ ; 2. for all θ ∈ Θ, (X n ) n∈N is stationary (i.e. X 0 ∼ π θ ); 3. its transition kernel satisfies Condition (AS2) in a uniform way with respect to θ, namely there exist C > 0 and κ ∈ (0, 1) such that ∀θ ∈ Θ, ∀n ≥ 1,

P n θ -Π θ 2 ≤ C κ n , where Π θ (f ) := π θ (f ) 1 X for f ∈ L 2 (π)
In order to derive a Berry-Esseen bound for the M -estimators of (X n ) n∈N satisfying (M), we need a uniform Berry-Esseen bound for some specific additive functionals of the Markov chain (X n ) n∈N . In the next subsection, we propose such a uniform Berry-Esseen bound for the second component of a general parametric MAP. This result will be applied to the MAPs associated with these specific additive functionals (see Remark 5.1).

A uniform Berry-Essen bound for the second component of a parametric MAP

Here we propose a refinement of Proposition 4.3 and Theorem 4.2. Let us introduce the following condition.

(A) For every θ ∈ Θ, (X n , Y n ) n∈N is a X × R-valued MAP, Y 1 is P θ -integrable and centered (i.e. E θ [Y 1 ] = 0).
Below, the driving Markov chain (X n ) n∈N is assumed to satisfy condition (M). Thus, the notation P θ stands for the underlying probability measure, which depends on θ through the transition kernel Q θ of (Y n , X n ) n∈N and the initial (stationary) distribution (π θ , 0). E θ [•] denotes the associated expectation. Theorem 5.1. Assume that Condition (A) is true for the MAP (X n , Y n ) n∈N and that the driving Markov chain

(X n ) n∈N satisfies Condition (M). If M 2 := sup θ∈Θ E θ |Y 1 | 2+ε < ∞ with some ε > 0, then σ 2 (θ) := lim n E θ [Y 2
n ]/n is well-defined and is finite for each θ ∈ Θ, the function σ 2 (•) is bounded on Θ, and there exists a positive constant C Y such that

∀n ≥ 1, sup θ∈Θ σ 2 (θ) - E θ [Y 2 n ] n ≤ C Y n . ( 5

.1)

The constant C Y depends on the sequence (Y n ) n∈N , but only through the constant M 2 .

If the two following additional conditions hold true

∃ ε > 0, M 3 := sup θ∈Θ E θ |Y 1 | 3+ε < ∞ (5.2) σ 0 := inf θ∈Θ σ(θ) > 0, (5.3)
then there exists a positive constant B Y such that

∀θ ∈ Θ, ∀n ≥ 1, sup a∈R P θ Y n σ(θ) √ n ≤ a -Φ(a) ≤ B Y √ n . ( 5

.4)

The constant B Y depends on the sequence (Y n ) n∈N but only through σ 0 and the constant M 3 .

Recall that the proofs of Proposition 4.3 and Theorem 4.2 are based on Proposition 4.2. Here, for any fixed θ ∈ Θ, Proposition 4.2 applies and gives an expansion of E θ [e i ζYn f (X n )], but for a neighbourhood O θ of ζ = 0, some C-valued functions λ θ (•), L θ (•, f ), R θ,n (•, f ), and some κ θ ∈ (0, 1), which all may depend on θ. Consequently, in order to prove Theorem 5.1, we must establish that, under Conditions (M), (A) and the moment condition sup θ∈Θ E θ |Y 1 | m 0 +ε < ∞, all the conclusions of Proposition 4.2 are fulfilled in a uniform way with respect to θ ∈ Θ. This job has been done in [START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF]Sect. III.2] in the context of V -geometrically ergodic Markov chains. The arguments in the present setting are the same up to the following changes: replace the uniform V -geometrical ergodicity assumption of [START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF] by Assumption (M), and replace the domination condition (D m 0 ) of [START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF] by the moment condition sup θ∈Θ E θ |Y 1 | m 0 +ε < ∞. The previous assumptions allow us to extend Lemmas 4.1-4.2, and so Proposition 4.2, in a uniform way in θ ∈ Θ. Remark 5.1. In the next subsection, Theorem 5.1 will be applied as follows. Given a Markov chain (X n ) n∈N satisfying Condition (M) with respect to Θ, we consider the MAP (X n , Y n (p)) n∈N where Y n (p) depends on some parameter p ∈ P and is of the form

Y n (p) := n k=1 g(p, X k-1 , X k ).
The property of the constant C Y in Theorem 5.1 ensures that Inequality (5.1) is uniform in p and θ when M 2 := sup p∈P ,θ∈Θ E θ |Y 1 (p)| 2+ε < ∞ (of course, the asymptotic variance in (5.1) is replaced by some σ 2 (θ, p)). In the same way, the Berry-Esseen bound (5.4) is uniform in p and θ when M 3 := sup p∈P ,θ∈Θ E θ |Y 1 (p)| 3+ε < ∞ and inf p∈P ,θ∈Θ σ(θ, p) > 0.

Note that these comments extend to a general MAP (X n , Y n ) n∈N which may depend on some parameter γ via its probability distribution and its functional form, provided that the bounds M 2 , M 3 , σ 0 in Theorem 5.1 are uniform in γ. Remark 5.2. The conclusions of Theorem 5.1 are also valid when X 0 ∼ µ θ with µ θ of the form µ θ = φ θ dπ θ , provided that sup θ∈Θ φ θ r ′ < ∞, with r ′ defined as in Subsection 4.4 (case m 0 = 3).

A Berry-Esseen bound for the M-estimators of ρ-mixing Markov chains

Throughout this subsection, Θ is some general parameter space and (X n ) n≥0 is a Markov chain with state space X satisfying the uniform L 2 (π)-spectral gap condition (M). The underlying probability measure and the associated expectation are denoted by P θ and E θ [•]. Recall that (X n ) n∈N is assumed to be stationary under (M). Let us introduce the additive functional of ( Thanks to Theorem 5.1, the proof of Theorem 5.2 borrows the adaptation of Pfanzgal's method given in [START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF]. One of the main difficulties in this method is to obtain a Berry-Esseen bound for the additive functionals Y n (p) := n k=1 g(p, X k-1 , X k ) with p := (v, q, α 0 ) and

X n ) n≥0 M n (α) = 1 n n k=1 F (α, X k-1 , X k ) ( 5
g(p, X k-1 , X k ) := F (1) (α 0 , X k-1 , X k ) + v √ q σ 1 (θ) m(θ) F (2) (α 0 , X k-1 , X k ) -m(θ)
for |v| ≤ 2 √ ln q. Observe that we have from (V0-V2): sup {(v,q):|v|≤2 √ ln q}, θ∈Θ E θ g(p, X 0 , X 1 ) 3+ε < ∞.

Then, Remark 5.1 gives the desired Berry-Esseen bound for (Y n (p)) n∈N in a uniform way over the parameter (θ, p).

When the X n 's are i.i.d., Theorem 5.2 corresponds to Pfanzagl's theorem [START_REF] Pfanzagl | The Berry-Esseen bound for minimum contrast estimates[END_REF] up to the following changes: in [START_REF] Pfanzagl | The Berry-Esseen bound for minimum contrast estimates[END_REF], π θ is the common law of the X n 's; the additive functional is M n (α) = (1/n) n k=1 F (α, X k ); we simply have σ 2 1 (θ) = E θ F (1) (θ, X 0 ) 2 and σ 2 2 (θ) = E θ (F (2) (θ, X 0 )m(θ)) 2 , and finally Assumption (V0) is replaced by the weaker (and optimal) moment condition: sup θ∈Θ E θ |F (1) (θ, X 0 )| 3 + |F (2) (θ, X 0 )| 3 < ∞.

Earlier extensions of [START_REF] Pfanzagl | The Berry-Esseen bound for minimum contrast estimates[END_REF] to the Markov context are discussed in [START_REF] Hervé | A Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF]. Let us compare our result with that of [START_REF] Rao | On the rate of convergence of estimators for Markov processes[END_REF], in which the family of transition probabilities P θ is assumed to satisfy a uniform Doeblin condition with respect to θ ∈ Θ. This condition corresponds to a uniform L ∞ -spectral gap condition with respect to Θ which is stronger than our Condition (M) (see Subsection 2.1). Let us mention that the moment condition on F (1) and F (2) in [START_REF] Rao | On the rate of convergence of estimators for Markov processes[END_REF] is the following (α(θ) = θ in [START_REF] Rao | On the rate of convergence of estimators for Markov processes[END_REF]): sup x∈X,θ∈Θ E θ F (1) (θ, X 0 , X 1 )

3 + F (2) (θ, X 0 , X 1 ) 3 X 0 = x < ∞.

Because of the supremum over x ∈ X, this condition is in general much stronger than our moment condition (V0) (despite the order 3 + ε in (V0) instead of 3). To see that, neglect the role of θ and consider a functional f on X. Then the difference between the condition used in [START_REF] Rao | On the rate of convergence of estimators for Markov processes[END_REF] and (V0) is comparable to that between sup x∈X E[|f (X 1 )| 3 | X 0 = x] and E π [|f (X 1 )| 3+ε ] (or, equivalently, between the supremum norm P (|f | 3 ) ∞ and the norm f 3+ε of f in L 3+ε (π)). Consequently, Theorem 5.2 applies to the models considered in [START_REF] Rao | On the rate of convergence of estimators for Markov processes[END_REF] but requires weaker moment conditions. Remark 5.3. The conclusion of Theorem 5.2 holds true when X 0 ∼ µ θ and µ θ satisfies the condition given in Remark 5.2. In this case, if F is such that sup θ∈Θ, α∈A E θ |F (α, X 0 , X 1 ) 1+ε < ∞ for some ε > 0, then M θ (α) = E θ [F (α, X 0 , X 1 )] can also be defined by (see Subsection 4.4):

M θ (α) = lim n→∞ E θ,µ θ [M n (α)].

Conclusion

In this paper, we propose limit theorems for the second component (Y t ) t∈T of a discrete or continuous-time Markov Additive Process (MAP) (X t , Y t ) t∈T when (X t ) t∈T has a L 2 (π)spectral gap. The derivation of the CLT is based on a ρ-mixing condition strongly connected to the L 2 (π)-spectral gap property. The results related to the convergence rate in the CLT are developed from the weak spectral method of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]. Note that here the discrete and continuoustime cases are covered in a unified way. In this context, the semigroup property (SG) for the family of operators S t (ζ) t∈T defined by S t (ζ)f (x) := E (x,0) e i ζ,Yt f (X t ) (ζ ∈ R d , x ∈ X) has a central role. We mention that this semigroup property is essentially true only for MAPs. The impact of the results is expected to be high for models involving a L 2 (π)spectral gap, since the limit theorems are valid for general (discrete and continuous time) MAPs, and under optimal (or almost optimal) moment conditions. This is illustrated in Section 5 where a Berry-Esseen bound for the M -estimator associated with ρ-mixing Markov chains, is derived under the (almost) expected moment condition.

A Additional material for the proof of Proposition 4.4.

Here

Π(ζ) = 1 2iπ Γ 1 (z -S 1 (ζ)) -1 dz, N (ζ) n = 1 2iπ Γ 0 z n (z -S 1 (ζ)) -1 dz,
where these line integrals are considered respectively on some oriented circle Γ 1 centered at z = 1, and on some oriented circle Γ 0 centered at z = 0, with radius κ < 1 where κ is (for instance) (1 + exp(-ε))/2 with ε defined in (2.2).

2) Let us return to the continuous-time case. We obtain from (4.10) and (A. Therefore U (•) := (z -S 1 (•)) -1 satisfies the regularity properties stated in part 3).

5) Finally, we deduce from the property of part 3) that there exists a neighbourhood V of ζ = 0 in R d such that the map ζ → (z -S 1 (ζ)) -1 • S v (ζ) is m 0 -times continuously differentiable from V to L(L s (π), L 1 (π)) uniformly in z ∈ Γ 0 ∪ Γ 1 and furthermore we have for ℓ = 0, . . . , m 0 :

sup (z -S 1 (ζ)) -1 • S v (ζ) (ℓ) s,1 ; z ∈ Γ 0 ∪ Γ 1 , ζ ∈ O, v ∈ (0, 1] < ∞.

Theorem 3 . 1 .

 31 Under (AS1-AS2) and (AS3d) for α = 2, (Y n / √ n) n∈N converges in distribution when n → +∞ to the d-dimensional Gaussian law N (0, Σ), where Σ is the asymptotic covariance d × d-matrix

  see references therein]. In the sequel, F (ℓ) denotes the partial derivative of order ℓ of a C-valued function F defined on an open subset of R d . Conditions (AS1-AS2) are assumed to hold throughout the subsection. Proposition 4.2. Let m 0 ∈ N * . Under condition (AS3d) for some α > m 0 , there exists a bounded open neighborhood O of ζ = 0 in R d such that we have for all f ∈ L s (π) with any s > α α-m 0 :

Lemma 4 . 3 .

 43 Assume that Conditions (AS2) and (AS3d) with α = 2 hold and that E π,0 [Y 1 ] = 0. Then the function λ(•) in Equality (4.3) satisfies the following second-order Taylor expansion 3 : λ

  )with C-valued functions λ(•), L(•, f, µ), R n (•, f, µ) satisfying the same properties as in Proposition 4.2. It is worth noticing that λ(•) is the same function as in (4.3), contrary to L(•, f, µ) and R n (•, f, µ) which both depend on µ.

  Finally, the second-order Taylor expansion of Lemma 4.3 shows that lim n λ(ζ/ √ n) = exp(-ζ, Σζ /2).

  3.1) and plays the same role. Indeed, writing t = n + v where n is the integer part of t, we know thatE π,0 [e i ζ,Yt ] = π S 1 (ζ) n (S v (ζ)1 X ) . Using(4.5) and the fact that S v is a contraction on L p (π) (p ∈ (1, +∞)), it follows that sup ζ∈K E π,0 e i ζ,Yt = O(ρ n ). (4.9) We prove Proposition 4.4 below which is the continuous-time version of Proposition 4.2. Then, combining Proposition 4.4 with relation (4.9), the Fourier techniques of the i.i.d. case can be used to extend Theorems 4.1-4.3 to (Y t / √ t) t>0 Proposition 4.4. Let m 0 ∈ N * . Write time t as t = n + v where n is the integer part of t. Under condition (AS3) for some α > m 0 , there exists a bounded open neighborhood O of ζ = 0 in R d such that we have for all f ∈ L s (π) with any s > α/(αm 0 ):∀t ∈ (0, +∞), ∀ζ ∈ O, E π,0 e i ζ,Yt f (X t ) = λ(ζ) n L ζ, S v (ζ)f + R n ζ, S v (ζ)f ,where λ(•), L(•, •) and R n (•, •) are the functions of Proposition 4.2. Moreover, the C-valued functions L v,f (ζ) := L ζ, S v (ζ)f and R n,v,f (ζ) := R n ζ, S v (ζ)f are of class C m 0 on O, and we have the following properties for ℓ = 0, . . . , m 0 : sup ζ∈O, v∈[0,1]

  ) and the desired expansion then follows from Proposition 4.2. The two following (straightforward) extensions of Lemmas 4.1-4.2 are needed to establish the others assertions. Let t ∈ (0, +∞).

Lemma 4 . 4 .

 44 If 1 ≤ p ′ < p, then the map ζ → S t (ζ) is continuous from R d to L(L p , L p ′ ).

Lemma 4 . 5 .

 45 Assume that E π,0 [|Y t | α ] < ∞ for some α > m 0 (m 0 ∈ N * ), and let 1 ≤ j ≤ m 0 . If p > 1 and p j := αp/(α+ jp) ≥ 1, then ζ → S t (ζ) is j-times continuously differentiable from R d to L(L p , L p j ), and sup ζ∈R d S (j) t (ζ) p,p j ≤ E π,0 [|Y t | α ] j/α .The regularity properties (inζ) of the functions L ζ, S v (ζ)f and R n ζ, S v (ζ)f are not a direct consequence of those stated in Proposition 4.2 because of the additional term S v (ζ)f .

. 5 ) 5 . 2 . 1

 5521 where α ≡ α(θ) ∈ A is the parameter of interest, F (•, •, •) is a real-valued measurable function on A × X 2 and A is an open interval on the real line. Function F is assumed to satisfy the Theorem Assume that the Markov chain (X n ) n∈N satisfies Condition (M), that F satisfies Condition(5.6), that the M -estimator α n is defined as above, and finally that Conditions (V0-V6) are fulfilled. Set τ (θ) := σ 1 (θ)/m(θ). Then there exists a positive constant C such that ∀n ≥ ( α nα 0 ) ≤ u -Γ(u) ≤ C 1 √ n + γ n .

1 )

 1 , we study the regularity properties of the functions ζ → L ζ, S v (ζ)f and ζ → R n ζ, S v (ζ)f involved in the decomposition of Proposition 4.4. Let us recall that we have (see (4.2))∀ζ ∈ R d , ∀n ∈ N, E π,0 e i ζ,Yn f (X n ) = π S 1 (ζ) n f .and, for t in some open neighbourhood O of ζ = 0, (see [49, 7.2]) S 1 (ζ) n = λ(ζ) n Π(ζ) + N (ζ) n , where λ(ζ) is the dominant eigenvalue of S 1 (ζ), Π(ζ) is the associated rank-one eigenprojection and N (ζ) is a bounded linear operator on each Lp (π) 1 < p < ∞. Both equalities imply that E π,0 e i ζ,Yn f (X n ) = λ(ζ) n π Π(ζ)f + π N (ζ) n f . (A.1)Furthermore the eigenprojection Π(ζ) and the operators N (ζ) n are defined as in the standard perturbation theory by

1 )

 1 ∀t ∈ O, E π,0 e i ζ,Yt f (X t ) = λ(ζ) n π Π(ζ)(S v (ζ)f ) + π N (ζ) n (S v (ζ)f ) .Thus, we can write with the notations introduced in Proposition 4.4∀t ∈ O, L(ζ, S v (ζ)f ) := π Π(ζ)(S v (ζ)f ) , R n (ζ, S v (ζ)f ) := π N (ζ) n (S v (ζ)f ) .

(H 0 )

 0 if θ ∈ I and T 0 (θ) ∈ I, then ζ → (z -S 1 (ζ)) -1 ∈ C 0 (θ, T 0 (θ)) uniformly in z ∈ Γ 0 ∪ Γ 1 .and for ℓ = 1, . . . , m 0 :(H ℓ ) If θ ∈ ℓ k=0 T -1 0 (T 0 T 1 ) -k (I)∩(T 1 T 0 ) -k (I) , then ζ → (z-S 1 (ζ)) -1 ∈ C ℓ θ, (T 0 T 1 ) ℓ T 0 (θ)uniformly in z ∈ Γ 0 ∪ Γ 1 .

A direct application of Proposition

4.2 gives this expansion, but under Condition (AS3d) with α > 2.

In this non-stationary case, we only require condition (AS3d) with the stationary distribution π and the mean vector remains Eπ,0[Y1].

following moment condition sup E θ |F (α, X 0 , X 1 )| , θ ∈ Θ, α ∈ A < ∞.

(5.6)

Set M θ (α) := E θ [F (α, X 0 , X 1 )]. We assume that, for each θ ∈ Θ, there exists a unique α 0 = α 0 (θ) ∈ A, the so-called true value of the parameter of interest, such that we have M θ (α) > M θ (α 0 ), ∀α = α 0 . To estimate α 0 , we consider the M -estimator α n defined by

Also assume that, for all (x, y) ∈ X 2 , the map α → F (α, x, y) is twice continuously differentiable on A. Let F (1) and F (2) be the first and second order partial derivatives of F with respect to α. Then

(5.7)

We shall appeal to the following assumptions.

(V0) There exists some real constant ε > 0 such that sup θ∈Θ, α∈A (1) (α 0 , X 0 , X 1 )] = 0 and α 0 ≡ α 0 (θ) is the unique parameter value for which this property is true;

n (α) and Y

(2)

n (α). Then, thanks to Theorem 5.1 applied to MAPs (X n , Y

n (α)) n∈N , the conditions (V0)-(V2) enable us to define the asymptotic variances:

and we know that sup θ∈Θ σ j (θ) < ∞ for j = 1, 2. The following additional conditions are also required:

(V5) There exist η > 2 and a measurable function

(V6) There exists a sequence γ n → 0 such that

Therefore, we only need to study the regularity of the map

O for controlling that of the map ζ → E π,0 e i ζ,Yt f (X t ) on O (and, as a result, proving Proposition 4.4).

3) Recall that • p,p ′ denotes the operator norm in the space L(L p , L p ′ ) of the linear bounded operators from L p (π) to L p ′ (π). The notation W (•) ∈ C j (θ, θ ′ ) means that there exists a bounded open neighborhood V of ζ = 0 in R d such that:

Let us introduce the maps

We are going to apply the next obvious regularity property. Let 1

), and assume that we have:

4) Let us introduce the following (non-increasing) maps from [1, +∞) to R:

where ε 0 will be defined in (A.4). Let θ > 1. Lemma 4.4 and the continuous inclusions between the Lebesgue spaces show that

On the same way, Lemma 4.5 gives for j = 1, . . . , m 0 :

and the derivatives in the last property are uniformly bounded in v ∈ [0, 1] on any bounded open neighborhood of ζ = 0. Now set θ 0 := s, θ 1 := T 0 (s), and observe that the assumption on s (i.e. s > α/(αm 0 )) is equivalent to T m 0 1 (θ 0 ) = αθ 0 /(α + m 0 θ 0 ) > 1, so that there exists ε 0 > 0 such that

namely: θ 2j := (T 0 T 1 ) j-1 T 0 (θ 1 ) for j = 1, . . . , m 0 + 1, and θ 2j+1 := T 1 (T 0 T 1 ) j-1 T 0 (θ 1 ) for j = 1, . . . , m 0 . Note that θ 2m 0 +2 = 1. From (A.2)-(A. 3) (with v = 1) that condition C(m 0 ) of [49, 7.1] holds, so that the conclusions reported in [49, p.48] are true:


Bernard Sanjuan
email: b.sanjuan@brgm.fr

Ragnar Asmundsson
email: ragnar.asmundsson@tigerhd.com

Romain Millot
email: r.millot@brgm.fr

Brach Michel

Michel Brach
email: m.brach@brgm.fr

Use of two new Na/Li thermometric relationships for geothermal fluids in volcanic environments

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

O (H2O-SO4), based on empirical or semi-empirical laws derived from chemical equilibrium reactions between water and minerals in the deep reservoirs, are commonly used in geothermal exploration in order to estimate the reservoir temperatures. Unfortunately, these estimations are not always concordant because of processes which can perturb the chemical composition of the fluids during their ascent up to the surface (water mixing, fluid cooling, etc.). Given these discordances, auxiliary thermometers such as Na/Li, based on statistical relationships, were also developed. As Li is rather lowly reactive, the use of this thermometer can give more reliable temperature estimations. Presently, three different Na/Li relationships ([1], [2]) are mainly available according to the fluid salinity and the geological environment (volcanic/granitic and sedimentary rocks).

This study carried out in the framework of the European HITI project (HIgh Temperature Instruments for supercritical geothermal reservoir characterization and exploitation) with the collaboration of ISOR Iceland Geosurvey proposes two new Na/Li thermometric relationships. The first concerns the fluids derived from hightemperature seawater-basalt interaction processes existing in the oceanic ridges and rises as well as in the emerged rifts such as those of Iceland (Reykjanes, Svartsengi and Seltjarnarnes geothermal fields) and Djibouti (Asal-Ghoubbet and Obock geothermal areas). It can be expressed as follows:

Log (Na/Li in mol/l) = 920/(T°K) + 1.105 (r 2 = 0.994).

The second relationship, developed using dilute fluids collected only from Icelandic geothermal wells in the 100-325°C range and surprisingly close to that determined by Fouillac and Michard (1981) for volcanic saline fluids at temperatures higher than 200°C, is: Log (Na/Li in mol/l) = 1786/(T°K) -0.936 (r 2 = 0.976).

The uncertainty on temperature estimation is ± 25°C for both relationships. These results confirm that the Na/Li ratios not only depend on temperature but also on other parameters. The nature of the reservoir rocks and fluid seems to be the most influent one. Some case studies in the literature and thermodynamic considerations suggest that the Na/Li ratios could be controlled by chemical equilibrium reactions involving different mineral assemblages where illite and micas would be, however, always present.