Françoise Pène

PROBABILISTIC LIMIT THEOREMS VIA THE OPERATOR PERTURBATION METHOD, UNDER OPTIMAL MOMENT ASSUMPTIONS

Keywords: . 2000 Mathematics Subject Classification. 60J05-60F05 Markov chains, Markov random walks, central limit theorems, local limit theorems, Edgeworth expansion, spectral method

The Nagaev-Guivarc'h operator perturbation method is well known to provide various probabilistic limit theorems for Markov random walks. A natural conjecture is that this method should provide these limit theorems under the same moment assumptions as the optimal ones in the case of sums of independent and identically distributed random variables. In the past decades, assumptions have been weakened, without achieving fully this purpose (achieving it either with the help of an extra proof of the central limit theorem, or with an additional ε in the moment assumptions). The aim of this article is to give a positive answer to this conjecture via the Keller-Liverani theorem. We present here an approach allowing the establishment of limit theorems (including higher order ones) under optimal moment assumptions. Our method is based on Taylor expansions obtained via the perturbation operator method, combined with a new weak compacity argument without the use of any other extra tool (such as Martingale decomposition method, etc.).

Introduction

Let (X n ) n≥0 be a Markov chain with values in Ω, with transition operator P and with stationary measure µ and f : Ω × Ω × E → R be a measurable function. Let ν be the distribution of X 0 (i.e. the initial distribution of the Markov chain). We set P ν for the Markov distribution with transition operator P and initial probability measure ν. We are interested in the study of the Markov random walk (S n ) n≥1 given by 1

S n := n k=1 Y k with Y k := f (X k-1 X k , Z k ) ,
where Z i are independent and identically distributed (i.i.d.) random variables independent of (X k ) k and with common distribution P. We assume moreover throughout this article that Y 1 is centered with respect to P µ ⊗ P. Our goal is to establish probabilistic limit theorems for (S n ) n≥1 under moment assumptions known to be optimal in the case of sums of independent and identically distributed (i.i.d.) random variables. Recall that if (Y k ) k≥1 were a sequence of centered i.i.d. random variables:

• If Y 1 ∈ L 2 (µ), then the usual central limit theorem (CLT) holds true : (S n / √ n) n≥1 converges in distribution to a centered Gaussian random variable W with variance E[Y 2 1 ], with density h W .

• If Y 1 ∈ L 2 (µ) is Z-valued and satisfies some non-lattice condition, then the usual local limit theorem (LLT) holds true: 3 (µ) and satisfies some non-lattice condition, then there is a first order Edgeworth expansion:

P(S n = k) ∼ h W (k/ √ n)n -1 2 , uniformly in k ∈ Z. • If Y 1 ∈ L
P Sn √ n ≤ x = P (W ≤ x) + B 1 (x) n 1 2 h W (x) + o(n - 1
2 ), uniformly in x ∈ R, where B 1 is a polynomial function.

• If Y 1 ∈ L r+2 (µ) is Z-valued and satisfies some non-lattice condition, then there is an expansion of order r in the LLT:

P (S n = k) = h W (0)n -1 2 + ⌊r/2⌋ j=1 A j (k) n 1 2 +j + o(n - 1+r
2 ), where A j is a polynomial function.

• If Y 1 ∈ L r+2 (µ) satisfies some non-lattice condition as well as some diophantine condition of the form E[e isY 1 ] < e -C|s| -α for all s large enough (see [START_REF] Dolgopyat | An error term in the Central Limit Theorem for sums of discrete random variables[END_REF]), then there is an Edgeworth expansion of order r:

P Sn √ n ≤ x = P (W ≤ x) + h W (x) r j=1 B j (x) n j 2
+ o(n -r 2 ), uniformly in x ∈ R, for some polynomial functions B j . 1 We explain in appendix that this notion includes the discrete-time Markov additive processes considered in [START_REF] Hervé | Limit theorems for stationary processes with L2-spectral gap[END_REF].

We will establish such results for Markov random walks. We will also investigate other results such as

• convergence to a stable distribution in the multi-dimensional setting, • local limit theorem for observables with values in Z d , including the case of convergence to stable distributions, • expansions in the LLT for non Z-valued random variables.

We will state general results in the context of geometrically ergodic Markov chains and will illustrate all of them on a toy model of Knudsen gas. Let us recall that the Markov chain (X n ) n≥0 (or equivalently its transition operator P ) is said to be geometrically ergodic on some complex Banach space of functions B 1 if its transition operator P satisfies

∃ϑ ∈]0, 1[, ∥P n -E µ [•]∥ L(B 1 ) = O(ϑ n ) . ( 1
)
As a consequence of our general results, we will prove limit theorems under optimal moment assumptions. We illustrate our results on classical families of Markov random walks (for ρmixing, V -geometrically ergodic Markov chains or Lipschitz iterative Markov chains) and obtain in particular the following result.

Theorem 1.1. Let m ≥ 2 and κ > 0. Assume one of the following conditions holds true:

• either P is ρ-mixing, ν = µ and Y 1 ∈ L m (P µ ⊗ P) centered;

• or there exists ϑ ∈]0, 1[, C > 0 and an unbounded continuous function

V : Ω → [1, +∞[ such that E µ [V ] + E ν [V κ m ] < ∞ and P n (•)-Eµ[•] V ∞ ≤ Cϑ n ∥ • /V ∥ ∞ . Assume sup (x,y)∈Ω 2 E [|f (x, y, Z 1 )| m ] /(V (x) + V (y)) < ∞, Y 1 is
centered with respect to P µ ⊗ P.

• or (Ω, d) is a non-compact metric space, P (g) = E[g(F (x, θ))] with θ a random variable and 2 with F (•, θ) : Ω → Ω strictly contracting, f : Ω → R is Lipschitz continuous (we consider here Y n = f (X n )) and E d (x 0 , F (x 0 , θ)

) (r+1) + E ν [d(x 0 , •) κ ] < ∞ (for some fix x 0 ∈ Ω).
Then (S n / √ n) n≥1 converges in distribution to a centered Gaussian random variable with variance matrix n∈Z Cov µ⊗P ⊗N Y 1 , Y |n|+1 . 3 Assume moreover that f is non-lattice (either in Z d if we precise that f takes its values in Z d or in R d otherwise). Then (i) If f is Z d -valued, then (S n ) n≥1 satisfies the local limit theorem (LLT). (ii) if d = 1, κ ≥ m-1, and m ≥ 3, then (S n ) n≥1 satisfies a first order Edgeworth expansion. (iii) if d = 1, κ ≥ m -1, m ≥ r + 2 and if f is Z-valued, then (S n ) n≥1 satisfies a LLT with expansion of order r. (iv) if d = 1, κ ≥ m -1 and m ≥ r + 2, and if some diophantine condition is satisfied, then there is an Edgeworth expansion of order r and also an expansion of order r in the LLT.

A simple case in which the diophantine condition holds true is if E[e isY 1 |X 0 , X 1 ] < e -C|s| -α for all s large enough and for some α > 0 (with the additional assumption that r < α -1 + 1 2 for r-order Edgeworth expansion). Let us indicate that other examples in compact situations are given in [START_REF] Fernando | Edgeworth expansions for weakly dependent random variables[END_REF]. This result will appear as an application of general results for Markov random walks, that are consequences of Taylor expansions for eigenprojectors and for the dominating eigenvalue of 2 Actually, we state a much more general result (Theorem 7.9) under weaker assumptions on F and f . 3 In this manuscript, given two d-dimensional square integrable random variables A = (A1, ..., A d ) and B = (B1, ..., B d ), we write Cov(A, B) for the symmetric matrix Cov(A i ,B j )+Cov(A j ,B i ) 2 i,j=1,...,d

.

the operators P t obtained from the transition operator P by Fourier perturbation (P and P t acting on some complex Banach space B 1 ). The use of operator perturbation techniques to prove probabilistic limit theorem is usually called the Nagaev-Guivarc'h method in reference to the seminal works by these two mathematicians [START_REF] Nagaev | Some limit theorems for stationary Markov chains (Russian)[END_REF][START_REF] Nagaev | More exact limit theorems for homogeneous Markov chains[END_REF][START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov[END_REF] (see also [START_REF] Keller | Un théorème de la limite centrale pour une classe de transformations monotones par morceaux[END_REF] and [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF]). This method was first implemented in the case of nice bounded observables so that t → P t ∈ L(B 1 ) is smooth (continuous, C k , analytic) implying the smoothness of the eigenprojectors. The Keller-Liverani theorem [START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF] strenghen this approach making possible the study of the case of unbounded observables for which t → P t ∈ L(B 1 ) is not continuous (see also [START_REF] Ferré | Théorème de Keller-Liverani et forte ergodicité[END_REF] for a presentation of this method in french). The idea consists in considering two Banach spaces B 1 ⊂ B 2 (with continuous inclusion) and then in using the continuity of t → P t ∈ L(B 1 , B 2 ) to prove the continuity of the eigenprojectors as elements of L(B 1 , B 2 ).

In [START_REF] Hervé | Théorème local pour chaînes de Markov de probabilité de transition quasi-compacte. Applications aux chaînes V -géométriquement ergodiques et aux modèles itératifs[END_REF][START_REF] Hervé | Vitesse de convergence dans le théorème limite central pour des chaînes de Markov fortement ergodiques[END_REF], Hervé proved a local limit theorem and a Berry-Esséen estimate (i.e. an edgeworth expansion with error in O(n -1

2 )) for geometrically ergodic Markov chains (including V -geometrically ergodic) under the optimal assumption by proceeding in two steps: he first establishes the central limit theorem (CLT) using another method (using a martingale approximation, à la Gordin [START_REF] Gordin | The central limit theorem for stationary processes (Russian)[END_REF]) and then deduces from this result an expansion for the dominating eigenvalue, the continuity of the eigenprojectors being ensured by the Keller and Liverani theorem. This argument was reused in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]. This method relies on the fact that we both have the continuity of the eigenprojectors and a proof of the CLT by another argument. Note also that it can only work for limit theorems using only the first non null derivative of the dominating eigenvalue λ t of P t , that is the dominating term of λ t -1 as t goes to 0 (remind that the geometrically ergodicity implies that λ 0 = 1 is the single dominating eigenvalue of P and that it is simple with only constant eigenvectors). As soon as we need more derivatives, we have to find another way.

The idea that continuity of the eigenprojectors and the first order term of λ t -1 as t goes to 0 are enough to prove convergence in distribution as well as local limit theorems has been implemented to prove convergence to stable distribution or gaussian distribution with non standard normalization in [START_REF] Bálint | Limit theorems in the stadium billiard[END_REF][START_REF] Szász | Limit laws and recurrence for the planar Lorents process with infinite horizon[END_REF] in the context of dynamical systems (chaotic billiards) and in [START_REF] Guivarc'h | On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks[END_REF] in the context affine random walks.

In [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF], motivated by the establishment of further probabilistic limit theorems in Markovian context under the weakest possible moment assumptions, with Hervé, we extended the continuity statement of Keller and Liverani in a C r -smoothness result. This approach enabled us to prove some limit theorems under suboptimal assumptions, with an additional ε in the moment-type assumptions. In particular, we proved the first order Edgeworth expansion under the suboptimal moment assumption m > 3. The general C r -perturbation theorem of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] was later used again in [START_REF] Hervé | Limit theorems for stationary processes with L2-spectral gap[END_REF] and [START_REF] Hervé | A Berry-Esseen theorem of M-estimators for V-geometrical Markov chains[END_REF] in the context of Markov random walks and M -estimators, respectively.

In the present paper, using carefully a new weak compacity argument, we obtain the limit theorems under the optimal moment assumptions without requiring an extra probabilistic argument (such as martingale approximation). In Section 2 we present the key ideas with a focus on our new argument. We state in Sections 3 and 4 general results on quasi-compactness and Taylor expansions (in t) of the resolvant and so of eigenprojectors for general families (P t ) t of continuous linear operators. The last sections are devoted to the general context of Markov random walks. In Section 5, we establish Taylor expansions for the dominating eigenvalue. Probabilistic limit theorems are then infered in Section 6 (CLT, LLT, higher order expansions in the Berry-Esseén theorem as well as in the local limit theorem) with the use of the general results of [START_REF] Fernando | Expansions in the local and the central limit theorems for dynamical systems[END_REF]. In Section 7, we state probabilistic limit theorems in a general context of Markov chains and apply it to examples, proving in particular Theorem 1.1. Actually, Section 7 contains more general results (especially for Lipschitz iterative models). We end this article with an appendix, in which we explain how the Markov additive processes studied in e.g. [START_REF] Hervé | Limit theorems for stationary processes with L2-spectral gap[END_REF] are in the scope of the present work.

Overview of the key ideas of the proofs

The key idea of the Nagaev-Guivarc'h method [START_REF] Nagaev | Some limit theorems for stationary Markov chains (Russian)[END_REF][START_REF] Nagaev | More exact limit theorems for homogeneous Markov chains[END_REF][START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov[END_REF] consists in

• noticing that E Pµ⊗P ⊗N [e itSn ] = E µ [P n t (1)
] , and more generally that

E Pν ⊗P ⊗N g(X 0 ) e itSn h(X n ) = E ν [g P n t (h)] , with P t (h)(x) = E P (e itf (x,•,ω) h(•))(x) dP(ω) = E e itY 1 h(X 1 ) X 0 = x ,
• using the fact that the geometric ergodicity of the Markov chain (see ( 1)), i.e. the quasicompactness with single simple dominating eigenvalue 1 of P on some Banach space

B 1 : ∃ϑ ∈]0, 1[, ∥P n -E µ [•]∥ L(B 1 ) = O(ϑ n )
will imply the one of P t , for small |t| with a uniform bound:

∃ϑ 1 ∈]0, 1[, sup |t|<b ∥P n t -λ n t Π t ∥ L(B 1 ) = O(ϑ n 1 ) , (2)
with λ t ∈ C the dominating eigenvalue of P t and with Π t ∈ L(B 1 ) the corresponding eigenprojector, • in proving the smoothness of t → λ t and of t → Π t ,

• infering the probabilistic limit theorems using characteristic functions as in the case of sums of i.i.d. random variables.

In this whole paper, we will work with different Banach spaces satisfying some continuous embedding property that we introduce now. For two Banach spaces (B j , ∥ • ∥ (j) ), j ∈ {1, 2}, the notation

B 1 → B 2 will mean that B 1 ⊂ B 2 and ∥ • ∥ (2) ≤ ∥ • ∥ (1)
.

In [START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF], Keller and Liverani proved that when t → P t ∈ L(B 1 ) is not continuous but only

t → P t ∈ L(B 1 , B 2 ) is continuous with B 1 → B 2 ,
it may still be possible to implement this method to get (2) and the continuity of t → Π t ∈ L(B 1 , B 2 ). This idea has been extended in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] to prove the C r -smoothness of t → Π t ∈ L(B 1 , B 2 ) and of t → λ t ∈ C. To this end, assuming Y 1 ∈ L r+1 and exploiting actually only the fact that Y 1 ∈ L r+ε , we worked with a double chain of Banach spaces:

B 0 → B 0 → B 1 → B 1 → ... → B r → B r → L 1 such that t → P t ∈ L(B j , B j ) is continuous and such that t → P t ∈ L( B j , B j+m ) is C m with P t
acting quasi-compactly on each B j and B j . It is proved in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] that the resolvant and so the eigenprojectors are C r as elements of L(B j , B j+m ) and that t → λ t is also C r . This way of proceeding allowed us to prove limit theorems under suboptimal hypotheses (typically Y 1 ∈ L r+ε when the optimal condition in the i.i.d. case was Y 1 ∈ L r ). This was already a great improvement, but was not totally satisfactory because of the additional ε in the moment assumptions.

We present here an approach that allows to obtain the optimal moment assumptions. Before entering deaplier in the presentation of the operator method used here, we explain the key ideas making this adaptation possible (in particular a new key weak compacity argument in L p combined with tailored adaptations of the chain of Banach spaces). Assuming Y 1 ∈ L r+1 , we work with a single chain of Banach spaces:

B 0 → B 1 → B 2 → ... → B r+1 → L 1
such that the operators P t are quasi-compact on B 1 , ..., B r and such that t → P t admits a Taylor expansion with error in O(t m ) in L(B r-m , B r ), and with an error in o(t m ) in L(B r-m , B r+1 ).

Compared to [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF], we establish (see Proposition 4.1, Theorem 5.5 and Corollary 5.6):

• Taylor expansions for eigenprojectors take the first space B 0 to be the space of constant functions which is preserved just by P , we do not assume that P t acts on B 0 (gain of space at the begining of the chain), -replace C r -smoothness of t → P t and t → Π t by Taylor expansions with error in O(t r ) in L(B 0 , B r ), and with an error in o(t r ) in L(B 0 , B r+1 ) (gain of space all along the chain, gain of space at the end of the chain for the estimate in O(t r )), 4

choose the spaces B j so that B j → L r+1 j , • Taylor expansion for the dominating eigenvalue by a key weak compacity argument -The o(t r+1 )-Taylor expansion of the dominating eigenvalue will follow from an o(t r )-Taylor expansion of E µ [Π t [START_REF] Bálint | Limit theorems in the stadium billiard[END_REF]] and from o(t r+1 ) Taylor expansions of E µ [(e itf -1)] and of

E µ [(e itf -1)(Π t -Π 0 )(1)] (Fact 3.2)
. The main issue is to prove this last expansion. But, to this end, only r-order Taylor expansions of e itf -1 and (Π t -Π 0 )(1) will be needed.

-o(t r+1 )-Taylor expansion of E µ [(e itf -1)(Π t -Π 0 )(1)]: make an o(t r )-Taylor ex- pansion of (e itf -1) and study individually each term E µ [ (itf ) k k! (Π t -Π 0 )(1)], for k = 1, ..., r. -o(t r+1 )-Taylor expansion of E µ [ (itf ) k k! (Π t -Π 0 )( 1
)]: first use an O(t r+1-k )-Taylor expansion of (Π t -Π 0 )(1) ∈ B r+1-k to prove weak compacity in the reflexive space L r+1 r+1-k , second use o(t r+1-k )-Taylor expansions in B r+1 → L 1 (µ) to identify the weak limit. Conclude.

gather the terms and conclude the desired o(t r+1 )-Taylor expansion of [START_REF] Bálint | Limit theorems in the stadium billiard[END_REF]] follow directly from the corresponding Taylor expansions of respectively (e itf -1) in L 1 (µ) and Π t (1) in B r+1 → L 1 (µ).

E µ [(e itf - 1)(Π t -Π 0 )(1)]. -The o(t r+1 )-Taylor expansion of E µ [(e itf -1)] and the o(t r )-Taylor expansion of E µ [Π t

Quasi-compactness and perturbation

Let B 1 be complex Banach space. We write L(B 1 ) for the set of continuous linear operators on B 1 . For any continuous linear operator P ∈ B 1 , we write ρ(P ) for its spectral radius and ρ ess (P ) for its essential spectral radius. Recall that the operator P is said to be quasi-compact if ρ ess (P ) < ρ(P ). Theorem 3.1 (Browder [4]). Let P ∈ L(B 1 ) be quasi-compact. Let r ∈]ρ ess (P ), ρ(P )[. Then the spectrum of P outside B(0, r) consists in a finite number of eigenvalues λ 1 , ..., λ s (isolated 4 Let us indicate for completness that, in our examples, it has been proved in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] that t → Pt ∈ L(B0, Br+1) is C r . Indeed, in these situations, there is a continuum of Banach spaces and so the space between Br and Br+1 (to get o(t r )) can be spread along the chain to add spaces Bj, up to slightly moving the spaces B0, ..., Br. Nevertheless we do not need this smoothness but just the Taylor expansion with error in o(t r ).

in the spectrum of P ) and there exist positive integers m 1 , ..., m s such that

B 1 = ⊕ s j=0 E j , with P (E j ) ⊂ E j and ∥P n |E 0 ∥ = O(r n ) ,
with, for all j = 1, ..., s, E j := ker(P -λ j Id) m j and dim(E j ) < ∞. For every j = 0, ..., s, there exists a continuous linear projection Π

[j] : B 1 → E j such that s j=0 Π [j] = Id, P Π [j] = Π [j] P, Π [j] Π [ℓ] = δ j,ℓ Π [j] and ∀n ≥ 0, P n Π [j] = 1 2iπ Γ j z n (z Id -P ) -1 dz , ( 3
)
with Γ 0 an oriented circle C(0, r 0 ) containing no λ j and with r 0 < r, and with, for j = 1, ..., s,

Γ j an oriented circle C(λ i , r i ) ⊂ C such that λ j is the only spectral value of P contained in the closed disk D(λ i , r i ].
If moreover m j = 1 for every j = 1, ..., s, then

P n = s j=1 λ n j Π [j] + P n Π [0] , with ∥P n Π [0] ∥ = O(r n ) .
We consider now a quasi-compact operator P ∈ L(B) with simple peripherical spectrum and a family of quasi-compact operators (P t ) |t|<δ such that P 0 = P and admitting the same type of decomposition as P :

P n t = s j=1 λ n j,t Π [j],t + N n t , with ∥N n t ∥ = O(r n ) ,
with λ j,t contained in the open disk D(λ j , r j [. We will use the Keller-Liverani perturbation theorem recalled at the end of this section to prove that the family of operators we are considering satisfies this property. Due to Theorem 3.1, the regularity in t of the eigenelements Π [j],t of P t will follow from the regularity in t of the resolvant (z Id -P t ) -1 uniformly on z ∈ s j=0 Γ j . Such results will be stated in Proposition 4.1 thanks to Theorem 3.3 (Keller-Liverani perturbation theorem).

As explained in Section 2, we will deduce from this an higher order Taylor expansion for the dominating eigenvalues due to the following key formula. Fact 3.2 (see [START_REF] Bálint | Limit theorems in the stadium billiard[END_REF], or [START_REF] Hennion | Stable laws and products of positive random matrices[END_REF]). In this context (assuming m j = 1), if v j ∈ ker(P -λ j Id) and φ j ∈ ker(P * -λ j Id) are such that φ j (v j ) = 1 and that t →

φ j • Π [j],t (v j ) is continuous, then P t (Π [j],t (v j )) = λ j,t (Π [j],t (v j ))
and, for t small enough, φ j (Π [j],t (v j )) ̸ = 0, and so

λ j,t -λ j = φ j (P t -P )(Π [j],t (v j )) φ j Π [j],t (v j )
, where we used the fact that

φ j • P = λ j φ j . Since Π [j] = φ j (•)v j , it follows also that λ j,t -λ j = φ j ((P t -P )v j ) + φ j (P t -P ) Π [j],t (v j ) -Π [j] (Π [j],t (v j )) φ j Π [j],t (v j ) = φ j ((P t -P )v j ) + φ j (P t -P )(Id -Π [j] )(Π [j],t -Π [j] )(v j ) φ j Π [j],t (v j )
.

In Theorem 3.3 below, an auxiliary space B 2 is used to study the spectral properties of a family of continuous linear operators acting on B 1 . Theorem 3.3 (Keller-Liverani perturbation theorem, see [START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF] and [START_REF] Ferré | Théorème de Keller-Liverani et forte ergodicité[END_REF]). Let V be a neighbourhood of 0 in R d . Let (P t ) t∈V be a family of continuous linear operators on a Banach space

(B 1 , ∥ • ∥ (1) ). Let (B 2 , ∥ • ∥ (2)
) be a Banach space such that B 1 → B 2 . Assume that there exist positive real numbers C > 0, R > ρ(P 0 ) and r ∈]ρ ess (P 0 ), R[ such that (P t ) t∈V satisfies the following uniform Doeblin-Fortet type inequality

5 ∀t ∈ V, ∀f ∈ B 1 , ∀n ≥ 0 ∥P n t f ∥ (1) ≤ Cr n ∥f ∥ (1) + R n ∥f ∥ (2) . ( 4
)
Assume moreover that

P 0 ∈ L(B 2 ). Let ε ∈]0, R -r/
∥(z Id -P t ) -1 ∥ L(B 1 ) < +∞ , lim t→0 sup z∈C : r+ε<|z|<R+ε, inf j |z-λ [j] |>ε (z Id -P t ) -1 -(z Id -P 0 ) -1 L(B 1 ,B 2 ) = 0 ,
and, for all j=1,...,m,

lim t→0 λ [j],t = λ j , ∀t ∈ U, dim i:λ i =λ j k≥0 ker(P t -λ i,t Id) k = dim k≥0 ker(P t -λ [j] Id) k , lim t→0 Π [0] -Π [0],t L(B 1 ,B 2 ) = 0 and lim t→0 Π [j] - i : λ i =λ j Π [i],t L(B 1 ,B 2 ) = 0 ,
where Π [i],t are the projectors associated to P t ∈ L(B 1 ) and λ i,t as considered in Theorem 3.1.

This theorem ensures that

P n t =   m j=1 P n t Π [j],t   + N n t , with N t := P t • Π [0],t satisfies sup t ∥N n t ∥ L(B 1 ) = O ((r + ε) n ).
It will be crucial to notice that in the particular case where all the characteristic spaces k≥0 ker(P t -λ i,t Id) k consist only of eigenvectors, then this decomposition can be simplified in

P n t =   m j=1 λ n [j],t Π [j],t   + N n t .

Taylor expansions for the resolvant and eigenprojectors

The continuity in t ∈ R d of the eigenprojectors stated in Theorem 3.3 appears as a consequence of the continuity in t of the Resolvant R z,t := (z Id -P t ) -1 of P t , due to Formula (3) (taken with n = 0) providing an expression of Π [j],t as an integral of (z Id -P t ) -1 . In the present section, we establish higher order Taylor expansions for the Resolvant t → R z,t = (z Id -P t ) -1 , that will imply immediately the corresponding Taylor expansions for t → Π [j],t thanks to (3). This section is devoted to the proof of the next result providing Taylor expansions of the resolvant. This result contains estimates of orders 0 and 1 (useful to establish the convergence in distribution to stable distributions for non square integrable observables) but also higher order Taylor expansions for the resolvant. Since our result holds true in multi-dimension, we need to introduce some different notions related to the multilinear forms appearing in multi-dimensional Taylor expansions. We write ) and H (2) ∈ L(B j , B d ⊗k ℓ ), we write

B d 1 = (B
H (2) ⊗ H (1) :=   1 (m + k)! σ∈S m+k H (2) i σ(1) ,...,i σ(m) H (1) i σ(m+1) ,...,i σ(m+k)   i 1 ,...,i m+k =1,...,d
, where we write as usual S m+k for the set of permutations of {1, ..., m+k}. Note that, with these notations, H (2) ⊗ H (1) ∈ L(B i , B d ⊗m+k ℓ

) and that (H (2) ⊗ H (1) ).t ⊗(k+m) = (H (2) .t ⊗k )(H (1) .t ⊗m ). In dimension 1 (when d = 1), ⊗ as well as . both correspond to the usual product, t ⊗m simply to t m and B d ⊗k ℓ to B ℓ . Proposition 4.1. Let δ > 0, Γ ⊂ C and r be a nonnegative integer. Let (B j , ∥ • ∥ (j) ), j ∈ {0, ..., r + 1} be a chain of (r + 2) Banach spaces, increasing in the sense that B j → B j+1 for all j = 0, ..., r. Let (P t ) t be a family of linear operators acting continuously on B 1 , ..., B r+1 (for all t ∈ R d such that |t| < δ). Assume P = P 0 acts continuously on B 0 and that

K 0 := sup j=1,...,r+1 sup |t|<δ sup z∈Γ ∥(z Id -P t ) -1 ∥ L(B j ) + ∥(z Id -P ) -1 ∥ L(B 0 ) < ∞ . ( 5
)
Let (P (k) 0 ) k=0,...,r be a family of operators such that, for all k = 0, ..., r, P

(k) 0 ∈ r-k j=0 L(B j , B d ⊗k j+k
) and such that P (0) 0 = P 0 . Then (A) for all j = 0, ..., r,

(z Id -P t ) -1 -(z Id -P ) -1 L(B j ,B j+1 ) ≤ K 2 0 ∥P t -P ∥ L(B j ,B j+1 ) . ( 6
) (B) If r = 1 then, setting P ′ 0 = P (1) 0 , (z Id -P t ) -1 -(z Id -P ) -1 -(z Id -P ) -1 (P ′ 0 .t)(z Id -P ) -1 L(B 0 ,B 2 ) ≤ K 2 0 P t -P -P ′ 0 .t L(B 0 ,B 2 ) + |t| K 3 0 ∥P t -P ∥ L(B 1 ,B 2 ) ∥P ′ 0 ∥ L(B 0 ,B d 1 ) . ( 7
) (C) If 6 ∀j = 0, ..., r, P t - r-j k=0 P (k) 0 k! .t ⊗k L(B j ,Br)
≤ K 1 |t| r-j , 6 Note that Br+1 does not play any role in this result and thus we can take Br+1 = Br.

then there exists a constant K r which is given by a polynomial expression in K 0 , K 1 such that

(z Id -P t ) -1 -(z Id -P ) -1 - r j=1 R (j) z,0 .t ⊗j L(B 0 ,Br) ≤ K r |t| r ,
for all |t| < δ and all z ∈ Γ, with

R (j) z,0 := (z Id -P ) -1 ℓ≥1,k 1 ,...,k ℓ ≥1 : k 1 +...+k ℓ =j A k ℓ ⊗ • • • ⊗ A k 1 ,
and

A ℓ := P (ℓ) 0 ℓ! (z Id -P ) -1 ∈ r-ℓ j=0 L(B j , B d ⊗ℓ j+ℓ ). (D) If for all j = 0, ..., r, P t -r-j k=0 P (k) 0 k! .t ⊗k L(B j ,B r+1 ) = o(|t| r-j ), then, on L(B 0 , B r+1 ), (z Id -P t ) -1 -(z Id -P ) -1 - r j=1 R (j) z,0 .t ⊗j = o(|t| r ) ,
uniformly in |t| < δ and z ∈ Γ, with the same notations as in the previous item.

It may be worthwhile to note that

R (j) z,0 .t ⊗j = (z Id -P ) -1 ℓ≥1,k 1 ,...,k ℓ ≥1 : k 1 +...+k ℓ =j (A k ℓ .t ⊗k ℓ ) • • • (A k 1 .t ⊗k 1 ) .
Since, in our examples, P t will have the form P (e it.f •), the Taylor expansions of t → P t ∈ L(B j , B j+m ) will be proved using Taylor expansions in t of t → (e it.f •) ∈ L(B j , B j+m ) and the operators P (ℓ) 0 will have the form P (ℓ)

0 = P ((if ) ⊗ℓ •). Remark 4.2.
In practice we will prove the first part of Assumption (5) (about the control of ∥(z Id -P t ) -1 ∥ L(B j ) ) by applying Theorem 3.3 with Banach spaces B j → B j+1 for all j = 1, ..., r + 1 and thus we will need a (r + 3)-th Banach space B r+2 . For the second part of Assumption (5) (control of ∥(z Id -P ) -1 ∥ L(B 0 ) ), a useful idea (used several times in applications) will be to take the eigenspace associated to the dominating eigenvalue (in our applications, it will be the space of constant functions).

We could have stated the previous lemma in a much more general way by replacing

P (k) 0 .t ⊗k by a k,t and A k .t ⊗k by a k,t k! (z Id -P ) -1 ∈ r-ℓ j=0 L(B j , B j+ℓ ).
We have not chosen this presentation since we do not have application in mind, except maybe the case of convergence to stable distribution, but for which in practice Item (A) is enough (see Proposition 5.11 and Example 5.12).

Proof of Proposition 4.1. We will use the following key identity

(z Id -P t ) -1 -(z Id -P ) -1 = (z Id -P t ) -1 (P t -P )(z Id -P ) -1 .
Observe that Item (A) is a direct consequence of this identity. Analogously

(z Id -P t ) -1 -(z Id -P ) -1 -(z Id -P ) -1 (P ′ 0 .t)(z Id -P ) -1 = (z Id -P t ) -1 (P t -P )(z Id -P ) -1 -(z Id -P ) -1 (P ′ 0 .t)(z Id -P ) -1 = (z Id -P t ) -1 (P t -P -P ′ 0 .t)(z Id -P ) -1 + (z Id -P t ) -1 -(z Id -P ) -1 (P ′ 0 .t)(z Id -P ) -1
The first term has norm less than

K 2 0 ∥P t -P -P ′ 0 .t∥ L(B 0 ,B 2 ) in L(B 0 , B 2
) and the second one can be rewritten (z Id -P t ) -1 (P t -P )(z Id -P

) -1 (P ′ 0 .t)(z Id -P ) -1 ,
which ends the proof of (B).

To establish (C), we prove by induction on m = 1, ..., r that

R z,t -R z,0 - m-1 j=1 R (j) z,0 .t ⊗j L(B r-m ,Br) ≤ K m |t| m , (8)
for all |t| < δ and z ∈ Γ. Due to Item (A) applied with j = r -1, (8) holds true for m = 1 with

K 1 = K 1 K 2 0
. for all |t| < δ and z ∈ Γ. Let N = 2, ..., r. Assume [START_REF] Dolgopyat | An error term in the Central Limit Theorem for sums of discrete random variables[END_REF] holds true for all m = 0, ..., N -1, and let us prove it holds also true for m = N . Observe that

R z,t -R z,0 = R z,t (P t -P )R z,0 = R z,t N -1 k=1 P (k) 0 k! .t ⊗k R z,0 + O(|t| N ) = N -1 k=1 (R z,t -R z,0 ) A k .t ⊗k + R z,0 N -1 k=1 A k .t ⊗k + O(|t| N ) , ( 9
) in L(B r-N , B r ), with O(|t| N ) bounded by K 2 0 K 1 (uniformly in t, z). Recall that A k ∈ L(B r-N , B d ⊗k r-N +k ). It follows from the inductive hypothesis that, for any k = 1, ..., N -1, R z,t -R z,0 = ℓ≥1,k 1 ,...,k ℓ ≥1 : k 1 +...+k ℓ ≤N -k-1 R z,0 (A k ℓ .t ⊗k ℓ ) • • • (A k 1 .t ⊗k 1 ) + O(|t| N -k ) in L(B r-N +k , B r ) uniformly in t, z.
Using this formula in the first sum in the right hand side of (9) ends the induction and so the proof of Item (C).

It remains finally to prove Item (D). To this end, it is enough to prove by induction on

m = 0, ..., r that, on L(B r-m , B r+1 ), R z,t -R z,0 - ℓ≥1,k 1 ,...,k ℓ ≥1 : k 1 +...+k ℓ ≤m (z Id -P ) -1 (A k ℓ .t ⊗k ℓ ) • • • (A k 1 t ⊗k 1 ) = o(|t| m ) ( 10
)
uniformly in |t| < δ and z ∈ Γ. This is true for m = 0 since

R z,t -R z,0 = R z,t (P t -P )R z,0 = o(1) ,
on L(B r , B r+1 ) uniformly in t, z. Fix N = 2, ..., r. Assume [START_REF] Fernando | Expansions in the local and the central limit theorems for dynamical systems[END_REF] for all m = 0, ..., N -1 and let us prove it holds also true for m = N . Observe that

R z,t -R z,0 = R z,t (P t -P )R z,0 = R z,t N k=1 P (k) 0 k! .t ⊗k R z,0 + o |t| N = N k=1 (R z,t -R z,0 ) A k .t ⊗k + R z,0 N k=1 A k .t ⊗k + o |t| N , in L(B r-N , B r+1 ) uniformly in t, z. Recall that A k ∈ L(B r-N , B d ⊗k r-N +k ). It follows from the inductive hypothesis that R z,t -R z,0 = ℓ≥1,k 1 ,...,k ℓ ≥1 : k 1 +...+k ℓ ≤N -k R z,0 (A k ℓ .t ⊗k ℓ ) • • • (A k 1 .t ⊗k 1 ) + o(|t| N -k ) in L(B r-N +k , B r+1
) uniformly in t, z. This ends the proof of Item (D) and so of Proposition 4.1. □

To make easier the comparison with previous works, let us recall the result of [26, Appendix A] about C r -smoothness.

Proposition 4.3 ([26]). Assume there exists a double chain of Banach spaces

B 0 → B 0 → B 1 → B 1 → ... → B r → B r .
Assume (P t ) t∈U is a family of linear operators acting continuously on all these Banach spaces

(with U an open subset of R d ) such that sup j=0,...,r sup t∈U sup z∈Γ ∥(z Id -P t ) -1 ∥ L(B j ) + ∥(z Id -P t ) -1 ∥ L( B j ) < ∞ , and such that t → P t ∈ r j=0 L B j , B j is continuous on U ⊂ R d and such that t → P t ∈ r-m j=0 L( B j , B j+m ) is C m . Then t → (z Id -P t ) -1 ∈ L(B 0 , B r ) is C r
, with derivatives at 0 given by the Taylor expansion established in Proposition 4.1.

Proof. This result follows directly [26, Proposition A] applied with

I = m j=1 {B j , B j }, T 0 (B j ) = B j , T 1 ( B j ) = B j+1 (up to identify I with a subset of R). □ 5.
Expansions of Fourier eigenprojectors and eigenvalues in Markovian or dynamical contexts 5.1. General context and toy model. In this section, we will see how the general results of Section 4 can be implemented to study dynamical or markovian random walks (S n ) n≥1 defined as follows.

Hypothesis 5.1. Let (Ω, F, µ) and (E, T , P) be two probability spaces.

(I) either X n = T n where T : Ω → Ω is a µ-preserving transformation, with transfer operator P and f : Ω × E → R d is a measurable µ ⊗ P-centered function. We consider ν a probability measure on Ω absolutely continuous with density h with respect to µ. To unify notations with the markovian setting, we also set P µ = µ and f (x, y, ω)

:= f (y, ω). (II) or (X n ) n≥0 is

a Markov chain (identified with the canonical Markov chain) with values

in Ω and with stationary measure µ and f :

Ω × Ω × E → R d is a measurable function.
Let ν be the distribution of X 0 (i.e. the initial distribution of the Markov chain). We set P ν for the Markov distribution with transition operator P and initial probability measure ν. We assume that ((

x k ) k , ω) → f (x 0 , x 1 , ω) is P µ ⊗ P-centered.
We set

P t (h)(x) = E P e it.f (x,•,ω) h(•) (x) dP(ω) ,
and

S n := n k=1 Y k with Y k := f (X k-1 , X k , Z k )
where Z i are i.i.d. random variables independent of (X k ) k≥0 and with common distribution P. 7 In the dynamical setting (I), identifying H(x, ω) with H(x):

E µ [g P n t (hG)] = E µ⊗P ⊗N (g • T n )e it.Sn .hG = E ν⊗P ⊗N (g • T n )e it.Sn G . ( 11
)
7 Let us indicate for completness that Markov random walks are usually defined as a Markov chain (Xn, Sn)n≥0 satisfying

E h(Xn, Sn) Xn-1 = x, Sn-1 = y = E h(Xn, y + Sn) Xn-1 = x, Sn-1 = 0 .
and that we aprove in appendix that these two definitions are equivalent.

In the Markovian setting (II), P t (h)(x) = E e it.Y 1 h(X 1 ) X 0 = x , and so

E ν [g P n t (h)] = E Pν ⊗P ⊗N g(X 0 )e it.Sn h(X n ) . ( 12
)
Before considering applications seen in Theorem 1.1, we will state results in the general dynamical or markovian context and will illustrate them on the following toy model of Knudsen gas considered in [START_REF] Boatto | Diffusion approximation of a Knudsen gaz model : dependence of the diffusion constant upon a boundary condition[END_REF], which is one of the simplest example of uniformly geometrically ergodic Markov chains (i.e. satisfying (1) with B 1 = L ∞ the set of uniformly bounded complex-valued functions on Ω).

Example 5.2 (a Toy model of Knudsen gas). Let r be a nonnegative integer. Let (Ω, F, µ) be a probability space and α ∈]0, 1[. Consider the Markov chain (X n ) n with transition operator given by

P (h) = αh + (1 -α)E µ [h] , i.e. P (h -E µ [h]) = α(h -E µ [h]) .
This Markov chain describes the evolution of a process which, at each step, remains the same with probability α and changes to an independent copy of distribution µ with probability 1 -α.

We consider also Y n = f (X n ), with f : Ω → R d centered and admitting moments of order (r + 1) with respect both to the invariant distribution µ and to the initial distribution ν. Then P t is given by

P t (h)(x) = αe itf (x) h(x) + (1 -α)E µ [e itf h] .
More generally, we will consider the context described in the following:

Remark 5.3. Assume Hypothesis 5.1 and that P is geometrically ergodic on some Banach space

B 1 → L 1 (µ) containing the constant functions, so that ∥P n -E µ ∥ L(B 1 ) = O(ϑ n ) (for some ϑ ∈]0, 1[). Let ϑ 1 ∈]ϑ, 1[
. Assume moreover that the assumptions of Theorem 3.3 are satisfied for this choice of (P t , B 1 ) and for some B 2 . Then there exists δ 0 > 0 such that, in L(B 1 ),

∀|t| < δ 0 , P n t = λ n t Π t + N n t , with sup |t|<δ 0 ∥N n t ∥ B 1 = O(ϑ n 1 ) , ( 13
)
with t → Π t ∈ L(B 1 , B 2 ) and t → λ t ∈ C continuous and Π 0 = E µ [•]
1 and λ 0 = 1. Moreover Theorems 3.1 and 3.3 ensure that Π t and N t are given by the following formulas

Π t := 1 2iπ Γ 1 (z Id -P t ) -1 dz, N n t := 1 2iπ Γ 0 z n (z Id -P t ) -1 dz , ( 14
)
with Γ 1 the oriented circle C(1, δ) and Γ 0 the oriented circle C(0, a), with ϑ 1 < a < a + δ < 1 and that sup

|t|<δ 0 , z∈Γ 1 ∪Γ 0 ∥(z Id -P t ) -1 ∥ L(B 1 ) < ∞ , with t → R z,t = (z Id -P t ) -1 ∈ L(B 1 , B 2 ) continuous on {t ∈ R d : |t| < δ 0 } (uniformly in z ∈ Γ 1 ∪ Γ 0 ).
Example 5.4 (Knudsen gas). We consider again the Knudsen gas introduced in Example 5.2.

Note that ∀p ∈ [1, +∞], ∥P n (h) -E µ [h]∥ L p (µ) = α n ∥h -E µ [h]∥ L p (µ) ,
and if ν ̸ = µ, it is worthwhile to notice that we also have

∀γ ∈ [0, +∞[, P n (h) -E µ [h] (1 + |f |) γ ∞ = α n h -E µ [h] (1 + |f |) γ ∞ ≤ (1 + E µ [(1 + |f |) γ ]) α n h (1 + |f |) γ ∞ .
Thus, since |P n t (h)| ≤ P n (|h|), Theorem 3.3 applies with

• B i := L p i (µ) for all p 1 , p 2 such that 1 ≤ p 2 < p 1 ≤ +∞,
• and also (useful when ν ̸ = µ) with

B i = (1 + |f |) γ i L ∞ for all γ 1 , γ 2 such that 0 ≤ γ 1 < γ 2 < ∞ and E [|f | γ 2 ] < ∞,
where we write again L ∞ for the set of bounded measurable complex valued functions on Ω and B i being endowed with the norm

• (1+|f |) γ i ∞ .
In the general context of Remark 5.3, Item (C) (resp. Item (D)) of Proposition 4.1 applied to this context will provide the following expansions for Π and N n :

Π t - r-1 k=0 Π (k) 0 k! .t ⊗k L(B 0 ,Br) = O(|t| r ) and N n t - r-1 k=0 (N n ) (k) 0 k! .t ⊗k L(B 0 ,Br) ≤ O(a n )O(|t| r ) , (resp. with (O(t r ), B r ) being replaced by (o(t r ), B r+1 )) with Π (k) 0 , (N n ) (k) 0 ∈ r-k j=0 L(B j , B d ⊗k j+k ) given by Π (k) 0 k! := 1 2iπ Γ 1 ℓ≥1,k 1 ,...,k ℓ ≥1 : k 1 +...+k ℓ =k (z Id -P ) -1 A k ℓ ⊗ • • • ⊗A k 1 dz (15)
and

(N n ) (k) 0 k! := 1 2iπ Γ 0 z n ℓ≥1,k 1 ,...,k ℓ ≥1 : k 1 +...+k ℓ =k (z Id -P ) -1 A k ℓ ⊗ • • • ⊗ A k 1 dz , ( 16
)
with

A ℓ = P (ℓ) ℓ! (z Id -P ) -1 ∈ r-ℓ j=0 L(B j , B d ⊗ℓ j+ℓ ).
5.2. Square integrable observables.

Theorem 5.5 (Key result for probabilistic limit theorems). Let δ 0 > 0. Assume Hypothesis 5.1.

Let r be a nonnegative integer and Y 1 ∈ L r+1 (P µ ⊗ P). Let (B j , ∥ • ∥ (j) ), j = 0, ..., r + 1 be a chain of (r + 2) Banach spaces such that, for all j = 1, ..., r + 1, B j-1 → B j , B j → L r+1 j (µ). Assume that P t (for |t| < δ 0 ) acts continuously on B 1 , ..., B r+1 and that P 0 acts continuously on B 0 . Assume moreover that h, 1 ∈ B 0 and that g : Ω → R is such that E ν [g•] defines a continuous linear form on B r . We also assume that [START_REF] Gordin | The central limit theorem for stationary processes (Russian)[END_REF] with [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and centered case[END_REF] hold true on B 1 and that sup j=1,...,r+1

sup |t|<δ 0 sup z∈Γ 0 ∪Γ 1 ∥(z Id -P t ) -1 ∥ B j + ∥(z Id -P ) -1 ∥ B 0 < ∞ , ( 17
)
and that, for all m = 0, ..., r, P t -

m k=0 P (k) 0 k! .t ⊗k is both in O(t m ) in r-m j=0 L(B j , B j+m ) and in o(t m ) in L(B r-m , B r+1 ), with P (k) 0 (h)(x) = i k Q k (h) := E P (if (x, •, ω)) ⊗k h(•) (x) dP(ω) ∈ r-k j=0 L(B j , B d ⊗k j+k ) . Then E ν [g P n t (h)] = λ n t r ℓ=0 t ⊗ℓ ℓ! .E ν gΠ (ℓ) 0 (h) + O(t r ) + r ℓ=0 t ⊗ℓ ℓ! .E ν g (N n ) (ℓ) 0 (h) +O(a n |t| r ) , ( 18
)
with

Π (ℓ) 0 and (N n ) (ℓ)
0 given by [START_REF] Guivarc'h | Application d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF] given by [START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov[END_REF].

If moreover E ν [g•] defines a continuous linear form on B r+1 , then E ν [g P n t (h)] = λ n t r ℓ=0 t ⊗ℓ ℓ! .E ν gΠ (ℓ) 0 (h) + o(t r ) + r ℓ=0 t ⊗ℓ ℓ! .E ν g (N n ) (ℓ) 0 (h) + O(a n o(|t| r )) . (19) Moreover λ t -1 = r+1 k=2 E Pµ⊗P [(it.Y 1 ) k ] k! + r k=1 r+1-k ℓ=1 t ⊗(k+ℓ) k!ℓ! E µ [P (k) 0 (Π (ℓ) 0 (1) -E µ [Π (ℓ) 0 (1)])] 1 + r-1 ℓ=1 t ⊗ℓ ℓ! E µ [Π (ℓ) 0 (1)] + o(|t| r+1 ) . Note that, in the Markovian context, t ⊗(k+ℓ) .E µ [P (k) 0 (Π (ℓ) 0 (g) -E µ [Π (ℓ) 0 (g)])] = Cov Pµ⊗P (it.Y 1 ) k , Π (ℓ) 0 (1)(X 1 ).t ⊗ℓ . In particular, if f (x, y, z) = f (y) ∈ R, then E µ [P (k) 0 (Π (ℓ) 0 (g) -E µ [Π (ℓ) 0 (g)])] = i k Cov µ f ⊗k , Π (ℓ)
0 [START_REF] Bálint | Limit theorems in the stadium billiard[END_REF] . Before proving Theorem 5.5, we state a corollary and apply it to our Knudsen gas model. Corollary 5.6. Let δ 0 > 0, ϑ ∈]0, 1[, R > 0 and r be a nonnegative integer. Assume Hypothesis 5.1 with Y 1 ∈ L r+1 (P µ ⊗ P). Let (B j , ∥ • ∥ (j) ) j=0,...,r+2 be a chain of (r + 3) Banach spaces such that:

• for all j = 0, ..., r + 1, B j → B j+1 , and for all j = 1, ..., r + 1,

B j → L r+1 j (µ),
• for all j = 1, ..., r + 1, P is geometrically ergodic on B j :

∥P n -E µ ∥ L(B 1 ) = O(ϑ n ), • 1 ∈ B 0 , P ∈ L(B 0 ) and sup z∈Γ 0 ∪Γ 1 ∥(z Id -P ) -1 ∥ B 0 < ∞,
• for all m = 0, ..., r, P t -

m k=0 P (k) 0 k! .t ⊗k is both in O(t m ) in r-m j=0 L(B j , B j+m ) and in o(t m ) in L(B r-m , B r+1 ), with P (k) 0 (h)(x) = i k Q k (h) := E P (if (x, •, ω)) ⊗k h(•) (x) dP(ω) ∈ r-k j=0 L(B j , B d ⊗k j+k ) , • t → P t ∈ L(B r+1 , B r+2 ) is continuous at 0, P ∈ L(B r+2 ),
• for all |t| < δ 0 and all j = 1, ..., r + 1,

∀f ∈ B j , ∀n ≥ 0 ∥P n t f ∥ (j) ≤ ϑ n ∥f ∥ (j) + R n ∥f ∥ (j+1) . ( 20
)
Assume that h ∈ B 0 and that g :

Ω → R is such that E ν [g•] defines a continuous linear form on B r .
Then the assumptions of Theorem 5.5 are satisfied (except maybe the fact that E ν [g•] is a linear form on B r+1 ) and

E ν [g P n t (h)] = λ n t r ℓ=0 t ⊗ℓ ℓ! .E ν gΠ (ℓ) 0 (h) + O(t r ) + r ℓ=0 t ⊗ℓ ℓ! .E ν g (N n ) (ℓ) 0 (h) +O(a n |t| r ) , ( 21
)
with

Π (ℓ) 0 and (N n ) (ℓ)
0 given by [START_REF] Guivarc'h | Application d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF] given by [START_REF] Guivarc | Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov[END_REF] and

λ t -1 = r+1 k=2 E Pµ⊗P [(it.Y 1 ) k ] k! + r k=1 r+1-k ℓ=1 t ⊗(k+ℓ) k!ℓ! E µ [P (k) 0 (Π (ℓ) 0 (g) -E µ [Π (ℓ) 0 (g)])] 1 + r-1 ℓ=1 t ⊗ℓ ℓ! E µ [Π (ℓ) 0 (1)] + o(|t| r+1 ) .
Proof. Observe that, given ϑ 1 ∈]ϑ, 1[, up to reduce the value of δ 0 , Theorem 3.3 applied with B j , B j+1 for j = 1, ..., r + 1 ensures that ( 13), [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and centered case[END_REF] and the first part of [START_REF] Guivarc'h | On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks[END_REF]. We conclude by Theorem 5.5.

□ Example 5.7 (Knudsen gas). Let Q ∈]r, r + 1[. Since Y 1 ∈ L r+1 (P µ ⊗ P) and since |P t (H)| ≤ P (|H|), Example 5.
2 satisfies the assumptions of Corollary 5.6 for g ∈ V 1 for (18) and for

g ∈ V r+1-Q for (19) with B j = V j for j = 0, ..., r, B r+1 = V Q and B r+2 = V r+1 , with • either, if ν = µ, V j := L r+1 j (µ) (with convention r+1 0 = ∞), • or, in the general case, V j = (1 + |f |) j L ∞ .
This follows from Example 5.4 and from the two following facts

P t - m k=0 P ((it.f ) k •) k! L(V j ,V j+m ) ≤ ∥P ∥ V j+m e it.f - m k=0 (it.f ) k k! Vm ≤ ∥P ∥ V j+m |tf | m m! Vm = O(|t| m ) ,
and analogously

P t - r-j k=0 P ((it.f ) k •) k! L(V j ,V Q ) ≤ ∥P ∥ V Q e itf - r-j k=0 (itf ) k k! V Q-j ≤ ∥P ∥ V Q |tf | Q-j V Q-j = O(t Q-j ) .
Proof of Theorem 5.5. Items (C) and (D) of Proposition 4.1 apply, so that

N n t - r ℓ=0 t ⊗ℓ ℓ! .(N n ) (ℓ) 0 L(B 0 ,Br) = O(a n |t| r ), Π t - r ℓ=0 t ⊗ℓ ℓ! .Π (ℓ) 0 L(B 0 ,Br) = O(|t| r ) (22)
and

N n t - r ℓ=0 t ⊗ℓ ℓ! .(N n ) (ℓ) 0 L(B 0 ,B r+1 ) = O(a n o(|t| r )), Π t - r ℓ=0 t ⊗ℓ ℓ! .Π (ℓ) 0 L(B 0 ,B r+1 ) = o(|t| r ) , ( 23
) with Π (ℓ) 0 , (N n ) (ℓ) 0 ∈ r-ℓ j=0 L(B j , B d ⊗ℓ j+ℓ )
given by ( 15) and ( 16). This ends the proof of ( 18) and [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF].

Due to Fact 3.2 with v 0 = 1 and

φ 0 = E µ [•], λ t -1 = E µ [(P t -P )(1)] + E µ [(P t -P )(Id -Π 0 )(Π t -Π 0 )(1)] E µ [Π t (1)]
.

The first expectation of the above right hand side is in r+1 k=1

E Pµ⊗P [(it.Y 1 ) k ] k! + o(|t| r+1 ) since Y 1 is (r + 1) times integrable and E Pµ⊗P [Y 1 ] = 0. Moreover, since 1 ∈ B 0 and B r+1 → L 1 (µ), we also know that E µ [Π t (1)] = r ℓ=0 t ⊗ℓ ℓ! .E µ [Π (ℓ) 0 (1)] + o(t r ) .
It remains to study

E µ [(P t -P )(Π t -Π 0 )(1)] . Due to the dominated convergence theorem, since Y 1 is (r + 1) times integrable, e it.Y 1 - r k=0 (it.Y 1 ) k k! L r+1 r (Pµ×P) = o(t r ) .
Thus, since B 1 → L r+1 (P µ ⊗ P), in case (I):

E µ [(P t -P )(Π t -Π 0 )(1)] = E µ⊗P (e it.f -1)(Π t -Π 0 )(1) = r k=1 E µ⊗P [(it.f ) k (Π t -Π 0 )(1)] k! + o |t| r ∥(Π t -Π 0 )(1)∥ (1) . ( 24
)
and, in case (II):

E µ [(P t -P )(Π t -Π 0 )(1)] = E Pµ⊗P (e it.Y 1 -1)(Π t -Π 0 )(1)(X 1 ) = r k=1 E Pµ⊗P [(it.Y 1 ) k (Π t -Π 0 )(1)(X 1 )] k! + o |t| r ∥(Π t -Π 0 )(1)∥ (1) . ( 25
) Fix k = 1, ..., r. Note that Y k 1 ∈ L r+1 k (P µ × P). Due to Proposition 4.1, we know that the quantity Ψ r+1-k,t (1) := Π t (1) -r+1-k ℓ=0 t ⊗ℓ ℓ! .Π (ℓ) 0 (1) is in O(t r+1-k ) in B r+1-k → L r+1 r+1-k (µ) and is in o(t r+1-k ) in B r+1 → L 1 (µ). We will deduce that E µ⊗P f ⊗k .Ψ r+1-k,t (1) = o(|t| r+1-k ), resp. E Pµ⊗P Y ⊗k 1 .Ψ r+1-k,t (1)(X 1 ) = o(|t| r+1-k ) . (26) Indeed, since Ψ r+1-k,t (1)/|t| r+1-k t is bounded in L r+1 r+1-k , it
is contained in a relative compact set for the weak topology. Let h be one of its weak limits as t → 0. Recall that Ψ r+1-k,t (1)/|t| r+1-k t converges to 0 in L 1 (µ). This implies that, for any bounded measurable function H,

E µ⊗P [H.h] = lim t→0 |t| -(r+1-k) E µ⊗P [H.Ψ r+1-k,t (1)] = 0 , resp. E Pµ⊗P [H.h] = lim t→0 |t| -(r+1-k) E Pµ⊗P H ⊗k .Ψ r+1-k,t (1)(X 1 ) = 0 .
We conclude that h = 0, and so that Ψ r+1-k,t (1)/|t| r+1-k t converges weakly in L r+1 r+1-k to 0. This ends the proof of ( 26) since Y k 1 ∈ L r+1 k (P µ ⊗ P). We conclude by combining this with ( 24) and ( 25) respectively (and using the fact that

∥(Π t -Π 0 )(1)∥ (1) = O(t) since 1 ∈ B 0 ).
□

We now study the consequences on the smoothness of λ.

Proposition 5.8. Assume Assumptions of Theorem 5.5 and r ≥ 1.

Then λ t = 1-a 2 .t ⊗2 +o(|t| 2 ), with a.t ⊗2 = E µ [(t.Y 1 ) 2 ] + 2 n≥0 E µ [(t.Q 1 )(P n (t.Q 1 (1)))] = n∈Z E Pµ⊗P ⊗N [(t.Y 1 )(t.Y |n|+1 )] .
Proof of Proposition 5.8. We write Π ′ 0 for Π

0 . We study the term of order t 2 of λ t -1. Observe that E µ [P ′ 0 (1)] = 0 since Y 1 is centered. Due to [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and centered case[END_REF] and to Item (B) of Proposition 4.1,

E µ Π ′ 0 (1) = 1 2iπ Γ 1 E µ (z Id -P ) -1 P ′ 0 (z Id -P ) -1 (1) dz = 1 2iπ Γ 1 (z -1) -2 E µ P ′ 0 (1) dz = 0 , since E µ [P (h)] = E µ [h] and P (1) = 1. Thus E µ [Π ′ 0 (1)
] = 0. Therefore, it follows from Theorem 5.5 that

λ t -1 = - E Pµ⊗P ⊗N [(t.Y 1 ) 2 ] 2 + t ⊗2 .E µ P ′ 0 (Π ′ 0 (1)) + o(|t| 2 ) = - E Pµ⊗P ⊗N [(t.Y 1 ) 2 ] 2 + iE µ (t.Q 1 )(t.Π ′ 0 (1)) + o(|t| 2 ) .
It follows from [START_REF] Guivarc'h | Application d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF] combined with (z Id -P

) -1 (1) = (z -1) -1 1 that Π ′ 0 (1) = 1 2iπ Γ 1 (z Id -P ) -1 P ′ 0 (z Id -P ) -1 (1) dz = i 2iπ Γ 1 (z Id -P ) -1 Q 1 (1)(z -1) -1 dz = i 2iπ Γ 1 (z -1) -1 n≥0 z -n-1 P n (Q 1 (1)) dz = i n≥0 P n (Q 1 (1)) , ( 27
)
where we used the fact that ∥z

-n P n (Q 1 (1))∥ L(B 1 ) ≤ (1-δ) -n a n ∥Q 1 (1)∥ (1) (since E µ [Q 1 (1)] = 0
and recalling that a < 1 -δ). This ends the proof of the expression of a. □ Example 5.9 (Knudsen gas, normal case). Due to Example 5.7, Example 5.2 with r ≥ 1 satisfies the assumptions of Proposition 5.8 and so

λ t -1 = -a 2 .t ⊗2 + o(|t| 2 ), with a.t ⊗2 = n∈Z E Pµ (t.f (X 1 )).(t.f (X |n|+1 )) = n∈Z α |n| E µ [(t.f ) 2 ] = 1 + α 1 -α E µ [(t.f ) 2 ] .
If moreover r ≥ 2, then the next proposition (Proposition 5.10) also applies, with

b = E[f ⊗3 ]   1 + 6 n,m≥1 α n+m + 3 n≥1 2α n   = α 2 + 4α + 1 (1 -α) 2 E[f ⊗3 ] .
Let us compute now the term of order 3 in the Taylor expansion of t → λ t . with a 0,n,n+m = #{(p, q, r) ∈ Z 3 : {p, q, r} = {0, n, n + m}}.

Proof. It follows from Theorem 5.5 that

λ t -1 = - a.t ⊗2 2 - i 6 E Pµ⊗P [(t.Y 1 ) 3 ] + it ⊗3 2 .E µ Q 1 (Π ′′ 0 (1)) - t ⊗3 2 .E µ Q 2 (Π ′ 0 (1)) + o(|t| 3 ) .
Recall that, due to [START_REF] Ibragimov | Independent and Stationary Sequences of Random Variables[END_REF],

Π ′ 0 (1) = i n≥0 P n (Q 1 (1)
). It follows moreover from [START_REF] Guivarc'h | Application d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF] that

Π ′′ 0 (1) := 1 2iπ Γ 1 (z Id -P ) -1 (P ′′ 0 + 2P ′ 0 (z Id -P ) -1 P ′ 0 )(z -1) -1 (1) dz = 1 2iπ Γ 1 (z -1) -1 (z Id -P ) -1 (-Q 2 + 2iQ 1 (z Id -P ) -1 iQ 1 )(1) dz = - 1 2iπ Γ 1 I 1 (z) + I 2 (z) dz ,
with

I 1 (z) := (z -1) -2   E µ [Q 2 (1)] + 2E µ   Q 1   m≥0 z -m-1 P m (Q 1 (1))       I 2 (z) := (z -1) -1 n≥0 z -n-1 P n Q 2 (1) -E µ [Q 2 (1)] +2Q 1   m≥0 z -m-1 P m (Q 1 (1))   -2E µ   Q 1   m≥0 z -m-1 P m (Q 1 (1))       .
It follows that

-Π ′′ 0 (1) := -2 m≥0 (m + 1)E µ [Q 1 (P m (Q 1 (1)))] + n≥0 (P n (Q 2 (1)) -E µ [Q 2 (1)]) + 2 n≥0   P n   Q 1 m≥0 (P m Q 1 (1)) - m≥0 E µ [Q 1 (P m (Q 1 (1)))]     .
This ends the proof of the lemma. □ 5.3. Non square integrable observable. In view of establishing results of convergence to a stable random variable, we consider now a less smooth situation. If we assume that the distribution of Y 1 is in the standard domain of attraction of an α 0 -stable distribution with α 0 ∈]1, 2[ (so that P(|Y | > s) ∼ |s| -α 0 as s → +∞), then we expect that λ t -1 ∼ -c|t| α 0 . But, unlike in Theorem 5.5, we cannot use an argument of weak convergence to conclude, since we do not have convergence of

Pt-P 0 -tP ′ 0 t α 0
and thus we cannot hope the convergence of

Πt-Π 0 -tΠ ′ 0 t α 0
. The next general statement can be seen as a first step to convergence to stable random variables. We will apply it immediatly on our easy Knudsen gas model.

∥P n -E µ [•] = N n 0 ∥ L(B j ) = O (ϑ n
) and that Theorem 3.3 holds true for (P t ) t with this couple of spaces

(B 1 , B 2 ). Assume moreover that P t -P 0 is in O(t) in L(B 1 , B 2 ) and in o(1) in L(B 2 , L 1 (µ)). Let ϑ 1 > ϑ, γ ∈]0, 1[. We assume that ∥ N n 0 ∥ L(B 2 ) + sup |t|<δ ∥ N n t ∥ L(B 1 ) = O(ϑ n 1 ), that N t -N 0 L(B 1 ,B 2 ) = O(t), where N t := P t -Ψ t (•)1 with Ψ 0 (1) = γ + o(t) in C. Then λ t -1 = γE µ (P t -P )(Id -N t ) -1 (1) + o(|t| 2 ) = γ n≥0 E µ (P t -P ) N n t (1) + o(|t| 2 ) .
Example 5.12 (Knudsen gas, stable case).

Let α 0 ∈]1, 2] and p ∈]1, α 0 [ be such that ∥f ∥ L p (µ) < ∞.
Consider the Knudsen gas introduced in Example 5.2 with r = 0 and d = 1. Assume that the characteristic function φ f of f with respect to µ satisfies

φ f (t) -1 = E µ [e itf ] -1 = -|t| α 0 (1 -iβ sgn(t))L 0 (|t| -1 ) , ( 28
)
with |β| < tan(α 0 π/2), c > 0 and with L 0 slowly varying at infinity. Then

λ t -1 ∼ (φ f (t) -1)(1 -α) 2 n≥0 α n (n + 1) α 0 = (φ f (t) -1)(1 -α)E[G α 0 ] ∼ (1 -α) E e itGf -1 ∼ E e i(1-α) 1 α 0 tGf -1 . ( 29
)
as t → 0, where G is a geometric random variable with parameter 1-α (i.e. P(G = n) = (1-α)α n for all positive integer n) independent of f (up to enlarge the probability space).

The toy Knudsen gas model considered here can also be studied by induction, considering the successive "renewal times" T j at which X T j is chosen independently of X T j -1 . The increments T j -T j-1 of these times are independent with the same distribution as G and the sums Y T j-1 + ... + Y (T j )-1 = (T j -T j-1 )Y T j-1 between these times are independent with the same distribution as Gf . Furthermore the number N n of j's such that t j ≤ n is a sum of n independent Bernoulli random variables with parameter (1 -α), and so N n ∼ n(1 -α) almost surely. Thus it will not be surprising that n k=1 Y k will "behave asymptotically" as Nn j=1 Z j and as

(1-α)n j=1
Z j , where Z j are independent random variables with the same distribution as Gf . This is coherent with (29) (via the arguments of Section 6.1).

Proof of Example 5.12. We apply Proposition 5.11 with B 1 = L ∞ (µ) and B 2 = L p (µ). We have already seen in Example 5.4 that P is geometrically ergodic on B 1 and on B 2 and that P t acts continuously on both these spaces. Moreover

∥P t -P ∥ L(B 1 ,B 2 ) ≤ ∥e itf -1∥ L p (µ) = ∥tf ∥ L p (µ) = O(|t|) ,
since f ∈ L p (µ) and, setting q ∈]1, +∞[ such that 1 p + 1 q = 1 and using the dominated convergence theorem, we also obtain that

∥P t -P ∥ L(B 2 ,L 1 (µ)) ≤ ∥e itf -1∥ L q (µ) = o(1) ,
Here we take N t (h) := αe itf h (taking γ = 1 -α and Ψ

t = (1 -α)E µ [P t (•)]). Observe that N t -N 0 is in O(t) in L(B 1 , B 2 ) and in o(1) in L(B 2 , L 1 (µ)). It follows from Proposition 5.11 that λ t -1 = (1 -α)E µ (P t -P )(Id -N t ) -1 (1) + o(|t| 2 ) = (1 -α)E µ (e itf -1)(Id -αe itf •) -1 (1) + o(|t| 2 ) = (1 -α)E µ e itf -1 1 -αe itf + o(|t| 2 ) . ( 30
)
But

E µ e itf -1 1 -αe itf = n≥0 E µ (e itf -1)(αe itf ) n = n≥0 α n E µ e it(n+1)f -e itnf . ( 31
)
Recall that Karamata proved that there exists u 0 > 0 and two functions c, ε 0 such that lim s→+∞ c(s) > 0 and lim s→+∞ ε 0 (s

) = 0 such that L 0 (u) = c(u)e u u 0 ε 0 (s) s ds . Set N t = ⌊u 0 / √ t⌋ and let us control (31) as follows n≥0 α n E e i(n+1)tf -e intf = E (e itf -1)(αe itf ) Nt+1 1 -αe itf + Nt n=0 α n E e i(n+1)tf -e intf = O α u 0 / √ t 1 -α e itf -1 L 1 + φ f (t) -1 - Nt n=1 α n ((φ f ((n + 1)t) -1) -(φ f (nt) -1)) ∼ (φ f (t) -1) 1 + Nt n=1 α n ((n + 1) α 0 -|n| α 0 ) ∼ (φ f (t) -1) 1 + (α Nt+1 (N t + 1) α 0 -α) - Nt n=1 (n + 1) α 0 (α n+1 -α n ) ∼ (φ f (t) -1)(1 -α)   n≥0 (n + 1) α 0 α n   , ( 32
)
due to the dominated convergence theorem since, for all n ≥ 1, L 0 (|nt| -1 ) ∼ L 0 (|t| -1 ) as t → 0 and, for all n = 1, ..., N t ,

L 0 (|nt| -1 ) L 0 (|t| -1 ) = c(|nt| -1 ) c(|t| -1 ) e - |t| -1 |nt| -1 ε 0 (s) s ds = O n sup |s|>| √ tu 0 | -1 |ε 0 (s)| .
Combining [START_REF] Keller | Un théorème de la limite centrale pour une classe de transformations monotones par morceaux[END_REF], [START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF], and (32), we obtain the first equivalent. The others follow since 13) and ( 14) hold true (with δ 0 = δ) on B 1 and that

E[G α 0 ] = (1 -α) n≥0 α n (n + 1) α 0 , since E e itGf = E[φ f (Gt)] and since φ f (ut) -1 ∼ u α 0 (φ f (t) -
Π t = 1 2iπ Γ 1 (z Id -P t ) -1 dz (33) is continuous from ] -δ, δ[ to L(B 1 , B 2 ) and assuming moreover that Γ 1 ⊂ {z ∈ C : |z| > ϑ 2 }. Furthermore, sup |t|<δ ∥ N n t ∥ L(B 1 ) = O(ϑ n 1 ) ensures that (z Id -N t ) -1 = n≥0 z -n-1 N n t ∈ L(B 1 ) is uniformly bounded in |z| ≥ ϑ 2 (and thus in z ∈ Γ 1 ) and that (z Id -N 0 ) -1 = n≥0 z -n-1 N n 0 ∈ L(B 2 ) uniformly in |z| ≥ ϑ 2 . Due to Fact 3.2, since B 2 → L 1 (µ), λ t -1 = E µ [(P t -P )(Π t (1))] E µ [Π t (1)] = E µ [(P t -P )Π t (1)] (1 + O(t)) , as t → 0 .
Let us observe that

(z Id -P t ) -1 (h) = z Id -N t -1   h + Ψ t z Id -N t ) -1 (h) 1 -Ψ t (z Id -N t ) -1 (1) 1   . ( 34
)
Indeed g := (z Id -P t ) -1 (h) satisfies

(z Id -N t )(g) = (z Id -P t )(g) + Ψ t (g)w t = h + c t (h)1 , with c t (h) = Ψ t (g) = Ψ t (z Id -N t ) -1 (h + c t (h)1) , from which we infer that c t (h) = Ψt (z Id -Nt) -1 (h) 1-Ψt (z Id -Nt) -1 (1)
and so [START_REF] Milhaud | Etude de l'estimateur du maximum de vraisemblance dans le cas d'un processus autorégressif: convergence, normalité asymptotique, vitesse de convergence[END_REF]. Thus, applying [START_REF] Milhaud | Etude de l'estimateur du maximum de vraisemblance dans le cas d'un processus autorégressif: convergence, normalité asymptotique, vitesse de convergence[END_REF] to h = 1, we obtain that

(z Id -P t ) -1 (1) = (z Id -N t ) -1 (1)b t (z), with b t (z) := 1 1 -Ψ t (z Id -N t ) -1 (1) . ( 35
)
Note that we can recover the fact that (z Id -P ) -1 (1) = (z -1) -1 . Moreover, we will prove that sup

z∈Γ 1 |b t (z) -b 0 (z)| = O(|t|) . ( 36
)
To this end, we first notice that Ψ t = E µ (P t -N t )(•) and so sup z∈Γ 1

(Ψ t -Ψ 0 )(z Id -N t ) -1 (1) = O(t) (37)
since sup

z∈Γ 1 ∥(z Id -N t ) -1 ∥ L(B 1 ) < ∞, since B 2 → L 1 (µ) and since ∥P t -P ∥ L(B 1 ,B 2 ) + ∥ N t -N 0 ∥ L(B 1 ,B 2 ) = O(t) . ( 38
)
Second, it follows from [START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF] and

sup |t|<δ sup z∈Γ 1 ∥(z Id -N 0 ) -1 ∥ L(B 2 ) + ∥(z Id -N t ) -1 ∥ L(B 1 ) < ∞ , that (z Id -N t ) -1 -(z Id -N 0 ) -1 = (z Id -N 0 ) -1 ( N t -N 0 ) z Id -N t -1 = O(t) (39)
in L(B 1 , B 2 ) uniformly in z ∈ Γ 1 and so that

ψ 0 (z Id -N t ) -1 -(z Id -N 0 ) -1 (1) = O(t) (40)
since B 2 → L 1 (µ). Combining [START_REF] Peigné | Iterated function schemes and spectral decomposition of the associated Markov operator[END_REF] and (40), we have proved [START_REF] Nagaev | More exact limit theorems for homogeneous Markov chains[END_REF].

It follows also from [START_REF] Szász | Limit laws and recurrence for the planar Lorents process with infinite horizon[END_REF], that (z Id -

N t ) -1 (1) = (z -1 + γ) -1 1 + O(t) in B 2 . Since moreover P t -P = o(1) in L(B 2 , L 1 (µ)) and E µ [(P t -P )(1)] = φ f (t) -1 = o(t) (since Y 1 is centered), sup z∈Γ 1 E µ (P t -P )(z Id -N t ) -1 (1) = sup z∈Γ 1 E µ (P t -P )(z -1 + γ) -1 1) + o(t) = o(t) .
Combining this with (33), ( 35) and ( 36), we infer that

E µ [(P t -P )Π t (1)] = 1 2iπ Γ 1 E µ (P t -P )(z Id -N t ) -1 (1) b 0 (z) dz + O o(t) sup z∈Γ 1 |b t (z) -b 0 (z)| = γE µ (P t -P )(Id -N t ) -1 (1) + o(t 2 ) , as t → 0 ,
where we used the fact that b 0 (z) = z-1+γ z-1 . □

Probabilistic limit theorems

Let δ 0 > 0. Let (S n ) n≥1 be a sequence of X-valued random variables with X = R d or Z d defined on a probability space (M, P) such that

∀n ∈ N, ∀t ∈ R d , |t| < δ 0 , E[e it.Sn ] = λ n t Φ t + M t,n . ( 41
)
We set

X * := R d if X = R d and X * := [-π, π] d if X = Z d .
Remark 6.1. In Remark 5.3, we have seen general situations in which E[e it.Sn ] = E ν [P n t (h 0 )] (see [START_REF] Ferré | Théorème de Keller-Liverani et forte ergodicité[END_REF] and ( 12)) for some h 0 and some family of operators (P t ) t such that (41) holds true with λ and

Φ t = E ν [Π t (h 0 )] continuous in t and with sup |t|<δ 0 |M t,n | = O sup |t|<δ 0 ∥|N n t (h 0 )∥| B 2 decaying exponentially fast in n.
Recall moreover that further Taylor expansions have been studied in Theorem 5.5.

The goal of this section is to establish probabilistic limit theorems for (S n ) n≥1 . More precisely, we will study situations in which (S n ) n≥1 satisfies the same kind of limit theorems as sums of independent identically distributed random variables with characteristic function behaving at 0 as t → λ t . 6.1. Central and local limit theorems. Let W be a R d -valued random variable. Theorem 6.2 (Central Limit Theorem (CLT)). Let (A n ) n≥1 be a sequence of (normalizing) d × d matrices converging to 0. Assume that

lim t→0 Φ t = 1, that lim n→+∞ sup |t|<δ 0 |M t,n | = 0 and that lim n→+∞ λ n A * n t = E[e it.W ] for all t ∈ R d (writing A * n for the transpose matrix of A n ). Then (A n S n ) n≥1 converges in distribution to W.
Proof of Theorem 6.2. We prove the convergence of the characteristic functions. We fix t ∈ R d and write

E e it.(AnSn) -E[e it.W ] = λ n A * n t Φ A * n t + M A * n t,n -E[e it.W ] = o(1) , since lim n→+∞ Φ A * n t = 1, lim n→+∞ λ n A * n t = E[e it.W ] and since lim n→+∞ M A * n t,n ≤ lim n→+∞ sup |u|<δ 0 |M u,n | = 0 . Thus (A n S n ) n≥1 converges in distribution to W. □ The condition lim n→+∞ λ n A * n t = E[e it.W
] means that λ t behaves at 0 as the characteristic function of a distribution belonging to the domain of attraction of the stable distribution of W. In particular, if λ t -1 ∼ -a|Σt| α α as t → 0 with |s| α α = d i=1 |s i | α and with B an invertible matrix, then, setting A n = n -1 α Id and considering W such that E[e it.W ] = e -a|Bt| α α , the following estimate holds true for any t ∈ R d

λ n A * n t -E[e it.W ] = λ n t/n 1 α -e -an|Bt/n 1 α | α α ≤ n λ t/n 1 α -e -a|Bt/n 1 α | α α ≤ n o Bt/n 1 α α α = o(
E[e itW ] = e -c 0 |t| α 0 (1-iβ sgn(t)) , λ t -1 ∼ c 0 |t| α 0 (1 -iβ sgn(t))L 0 (|t| -1 )
as t → 0, with L 1 slowly varying at infinity, A n → 0 and lim n→+∞ n|A n | α 0 L 0 (A -1 n ) = 1. Indeed, proceeding as previously, we observe that, for every t ∈ R,

λ n Ant -E e it.W = λ n Ant -E[e in -1 α 0 tW ] n ≤ n λ Ant -E[e in -1 α 0 t.W ] = n|t| α 0 |c 0 (1 -iβ sgn(t))| A α 0 n L 0 (|t|/A n ) -n -1 ≤ |t| α 0 |c 0 (1 -iβ sgn(t))| |nA α 0 n L 0 (|t|/A n ) -1| ≤ |t| α 0 |c 0 (1 -iβ sgn(t))| L 0 (|t|/A n ) L 0 (A -1 n ) (1 + o(1)) -1 = o(1) ,
since L 0 is slowly varying. 8 setting Σ

• If r ≥ 1, then ( n-1 k=0 f (X k )/ √ n) n≥1 converges in distribution to a centered Gaussian random variable W with variance matrix Σ = 1+α 1-α E[f ⊗2
]. This follows from Theorem 6.2 combined with the first item of Remark 6.3, the desired second order expansion being proved in Example 5.9.

• Consider now the situation of Example 5.12, that is r = 0, d = 1 and there exists α 0 ∈ ]1, 2] and a function L 0 slowly varying at infinity such that the characteristic function φ f of f with respect to µ satisfies

φ f (t) -1 ∼ |t| α 0 (1 -iβ sgn(t))L 0 (|t| -1 ) , as t → 0, for some |β| < tan(α 0 π/2). Then (A n n-1 k=0 f (X k )) n≥1
converges in distribution to the stable random variable W with characteristic function given by E[e itW ] = e -c|t| α 0 (1-iβ sgn(t)) , for c = (1 -α) n≥0 α n (n + 1) α 0 and where A n → 0 is so that

lim n→+∞ n|A n | α 0 L 0 (A -1 n ) = 1. Indeed, due to Example 5.12, λ t -1 ∼ c|t| α 0 L 0 (|t| -1 )(1 -iβ sgn(t)
) and Theorem 6.2 applies with the second item of Remark 6.3.

Theorem 6.5 (Local Limit Theorem). Assume Assumptions of Theorem 6.2 with

A n invertible, sup |t|<δ 0 |Φ(t)| < ∞, sup |t|<δ 0 |M t,n | = o (det A n ) and |λ n A * n t | ≤ g(t) if |A * n t| < δ 0 , with g inte- grable on R d .

Assume moreover that W has density h W and integrable characteristic function, that f takes its values in

Z d and that sup t∈[-π,π] d \B(0,δ 0 ) E[e it.Sn ] = o (det A n ). Then sup k∈Z |P(S n = k) -det(A n )h W (A n k)| = o(det(A n )) .
Proof of Theorem 6.5. Observe that

P(S n = k) = E 1 {Sn-k=0} = E 1 (2π) d [-π,π] d e it.(Sn-k) dt = 1 (2π) d [-π,π] d e -itk E[e it.Sn ] dt = 1 (2π) d B(0,δ 0 ) e -itk λ n t Φ t dt + o (det A n ) ,
where we used the Fubini theorem for integrable functions and the fact that sup

|t|<δ 0 |M t,n | = o (det A n ) and that sup t∈[-π,π] d \B(0,δ 0 ) E[e it.Sn ] = o (det A n )
. Now, making the change of variable t = A * n s, we obtain

P(S n = k) = det(A n ) (2π) d (A * n ) -1 B(0,δ 0 ) e -iA * n s.k λ n A * n s Φ A * n s ds + o (det A n ) , and sup k∈Z (A * n ) -1 B(0,δ 0 ) e -iA * n s.k λ n A * n s Φ A * n s ds - R d e -is.Ank E[e is.W ] ds ≤ R d 1 (A * n ) -1 B(0,δ 0 ) λ n A * n s Φ A * n s -E[e is.W ] ds = o(1) ,
due to the dominated convergence theorem since g and s → E[e -isW ] are integrable. We end the proof by using h W (u) = To complete this remark, let us indicate that, under the assumptions of Theorem 3.3 (Keller and Liverani theorem), it has been proved in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Propositions 5.3 and 5.4] that the nonlattice property together with an additional reasonnable condition imply the exponential decay of sup t∈[-π,π] d \B(0,δ 0 ) E[e it.Sn ] as n goes to infinity. Example 6.7 (Knudsen gas, LLT). Theorem 6.5 applies in the situation considered in Example 6.4, provided f takes its values in Z d but is not supported by a sublattice of Z d .

Proof of Example 6.7. The fact that the assumptions of Theorem 6.2 hold has already been proved in Example 6.4 with the use of Remarks 6.1 and 6.3. First, we observe that, either Assume now that λ t -1 ∼ c|t| α 0 L 0 (|t| -1 )(1 -iβ sgn(t)) with α 0 ∈]1, 2] and with L 0 slowly varying at infinity. Then

A n = Id / √ n (if r ≥ 1) or A n = n 1 α 0 L(n) with L
∀t ∈ R, λ n Ant = e -nc|Ant| α 0 (1-iβ sgn(t))L 0 (|Ant| -1 )) = e -c|t| α 0 (1-iβ sgn(t)) L 0 (|Ant| -1 ) L 0 (A -1 n )
)

.

Due to Karamata's representation theorem, there exists u 0 > 0 and two functions c, ε 0 such that lim s→+∞ c(s) > 0, lim s→+∞ ε 0 (s) = 0 and such that L 0 (u) = c(u)e u u 0 ε 0 (s) s ds . Let δ 0 be such that λ u is well defined for all u satisfying |u| < δ 0 . Thus, if |A n t| < δ 0 < 1 and

|A n | < δ 0 , L 0 (|A n t| -1 ) L 0 (A -1 n ) = c(|A n t| -1 ) c(|A n | -1 ) e |Ant| -1 A -1 n ε 0 (s) s ds ≥ inf |s|>δ -1 c(s) sup |s|>δ -1 c(s) min(|t|, |t| -1 ) inf |s|>|δ -1 |ε 0 (s)| .
We can choose δ 0 < 1, such that this quantity is larger than

(µ) ≤ α∥h∥ + (1 -α)∥h∥ L 1 (µ)
, it follows by standard arguments (see e.g. [START_REF] Ionescu Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF][START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipschitziens (French)[END_REF]) that the essential spectral radius of P t is strictly smaller than 1. Consider now h ∈ L p (µ) and λ = e id ∈ C with d ∈ R and

λh = P t (h) = αe it.f h+(1-α)E µ [e it.f h]. Then E µ [|h|] ≤ E µ [|αe it.f h|]+|(1-α)E µ [e it.f h]| ≤ E µ [|h|]
and we conclude that λh = e it.f h = E µ [e it.f h] µ-a.e., thus h = e -id e it.f h is constant. So either h = 0 or e id = e it.f . But e id = e it.f would mean that t.f ∈ d + 2πZ, which would contradict the fact that f is not contained in a sublattice of Z d . Thus h = 0. Since P t has an essential spectral radius strictly smaller than 1 and does not admit any eigenvalue of modulus 1, we conclude that its spectral radius is strictly smaller than 1. So there exists

n t ≥ 1 such that |E Pµ [e itSn t ]| ≤ ∥P nt t 1∥ L p (µ) < 1. But u → E Pµ [e iuSn t ] = E µ [P nt u (1)] is continuous at t (since u → P u ∈ L(L a ′ (µ), L b ′ (µ)
) is continuous, for all a ′ > b ′ ≥ 1, using also the fact that sup u ∥P u ∥ L(L a (µ)) ≤ 1). We conclude by compacity. □

Edgeworth expansions (d = 1

). We assume d = 1 throughout this section. We recall now some general Edgeworth expansions results coming from [START_REF] Fernando | Expansions in the local and the central limit theorems for dynamical systems[END_REF]. We first introduce some assumptions.

Assumption (α ′ )[ r] (Smoothness): Assume λ, Φ and M enjoys the following Taylor expansions

λ t = 1 - σ 2 t 2 2 + r+2 k=3 α k t k + o(t r+2 ), Φ t = r k=0 B k t k + O |t| r+1 , ∃δ > 0, ∀t ∈ [-δ, δ], λ -n t M t,n - r k=0 C k,n t k ≤ K n λ -n 2 t |t| r+1 , with σ 2 > 0, sup k,n (|C k,n |) = O(n -p ) and K n = O(n p ) for all p > 0.
Remark 6.8. In the Markovian context of Section 5, Assumption (α ′ )[ r] will follow from Theorem 5.5 and Corollary 5.6 with r := r + 1 (up to assume the positivity of σ 2 , the expression of which is given by a of Proposition 5.8). Indeed the Taylor expansion of M t,n combined with the one of λ t coming from Theorem 5.5 leads to a Taylor expansion of order r + 1 of λ -n t M t,n with coefficients C k,n = O(n k a n ) and with error term in O(λ -n t n r+1 |t| r+1 a n ).

Assumption (β ′ ) (Non-arithmeticity): For any compact K of X * \ {0}, ∀p > 0, sup s∈K E e isSn = O(n -p ) ,
As already mentioned in Remark 6.6, in the Markovian context of Section 5, Assumption (β ′ ) will follow from the fact that ρ ess (P t ) < 1 (which can be established by using ( 4) if B 1 → B 2 is compact, due to a Theorem by Hennion [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipschitziens (French)[END_REF]), combined with the fact that ρ(P t ) ≤ 1 and that P t admits no eigenvalue of modulus 1 (for t in the compact K).

Assumption (γ ′ )[α ′ 1 , α 1 ]: Either X = Z, or there exist δ such that, for all p, ∀|s| > K, E e isSn = O n -p + |s| 1+α ′ 1 e -n α 1 δ|s| -α ′ 1 .
Assumption (δ ′ )[r ′ ]: X = R and for any B > 0, there exists K > 0 such that

K<|s|<Bn r ′ -1 2 E µ e isSn |s| ds = o(n -r ′ /2 ) .
Remark 6.9. Note that, both Conditions (γ ′ )(α ′ 1 , 1) and (δ ′ )[r ′ ] hold true provided there exist positive s 0 , n 0 , C > 0, such that r ′ < 2α ′ 1 -1 + 1 and such that

∀|s| > s 0 , ∀n ≥ n 0 , E[e isSn ] < e -n C|s| -α ′ 1 . ( 42
)
In the context of Section 5 with E e itSn = E ν [gP n t (h)], (42) holds true if there exist Banach spaces B 1 → B 2 containing h and the duals of which contains

E ν [g•] such that ∀n ≥ n 0 , ∀|s| > s 1 , ∥P n s ∥ L(B 1 ,B 2 ) ≤ Ce -Cn|s| -α ′ 1 .
(see [START_REF] Fernando | Expansions in the local and the central limit theorems for dynamical systems[END_REF]Lemma 4.7]). Note that this holds true if

∥P n 0 s ∥ B 1 < 1-C |s| α ′ 1
. This condition generalizes the α-Diophantine property of supp X:

E[e isY 1 ] < 1 -C |s| α ′ 1
of the i.i.d. case (see [START_REF] Dolgopyat | An error term in the Central Limit Theorem for sums of discrete random variables[END_REF]).

Example 6.10 (Knudsen gas). Consider again Example 5.2. Recall that

P (h -E µ [h]) = α(h - E µ [h]) with α ∈]0, 1[ and that Y n = f (X n ), with f : Ω → R centered.
h ∈ B 1 \ {0} such that λh = e itf h = E µ [e itf h] µ-almost surely, contradicting E µ [e itf ] < 1.
Finally, when

X = R, Assumptions (γ ′ )[α ′ 1 , 1] and (δ ′ )[⌈1 + 2α ′ 1 -1 ⌉ -1] hold true as soon as E µ [e itf ] < 1 -C 0 |t| α ′ 1
. Indeed

P 2 t (h)(x) = α 2 e i2tf (x) h(x)+α(1-α)e itf E µ [e itf h]+α(1-α)E µ [e i2tf h]+(1-α) 2 E µ [e itf ]E µ [e itf h] ,
and so, for all p ∈ [1, +∞], it follows that

∥P 2 t (h)∥ L p (µ) ≤ 1 -(1 -α) 2 ∥h∥ L p (µ) + (1 -α) 2 1 - C 0 |t| α ′ 1 E µ [e itf h] ≤ 1 -(1 -α) 2 C 0 |t| α ′ 1 ∥h∥ L p (µ) .
Set g(s) := X e -isx g(x) dλ(x) for s ∈ X * , where λ is the Lebesgue measure if X = R and where λ is the counting measure

if X = Z. If X = R, we say g ∈ F m k if g : R → R is continuous, λ-integrable and if ĝ : X * → C is k times continuously differentiable with C m k (g) := C m (g) + C k (g). < ∞ , with C m (g) := sup s∈X * | g(s)| min(1, |s| -m ) and C k (g) := ∥ g (k) ∥ ∞ . If X = Z, F m k = F 0 k is the set of functions g : Z → C satisfying the following summability condition n∈Z |n| k |g(n)| < ∞ . Note that C k (g) ≤ max 0≤j≤k X |x| j |g(x)| dλ(x). When X = R, C m (g) ≤ max 0≤j≤m ∥g (j) ∥ L 1 (R) .
Under our assumptions, we set N for the distribution function of a centered Gaussian random variable with variance σ 2 and n for the corresponding probability density function (that is n is the derivative of N). Let us recall now the general results of [START_REF] Fernando | Expansions in the local and the central limit theorems for dynamical systems[END_REF].

Theorem 6.11. [10, Theorem 1.1] Let r be a nonnegative integer, α ′ 1 ≥ 0, α 1 > 0 and q > α ′ 1 1 + r 2α 1 . Assume (α ′ )[ r], (β ′ ) and (γ ′ )[α ′ 1 , α 1 ] hold.
Then there exist polynomials R j such that, for all g ∈ F q+2 0 ,

E [g(S n )] = r j=0 1 n (j-1)/2 X (R j • n)(x/ √ n)g(x) dλ(x) + C q+2 (g) • o(n -r/2
) . Theorem 6.12. [10, Theorem 1.2] Let r be a nonnegative integer,

α ′ 1 ≥ 0, α 1 > 0. Let q > α ′ 1 1 + r+1 2α 1 . Assume (α ′ )[ r], (β ′ ) and (γ ′ )[α ′ 1 , α 1 ] hold. Then there exist polynomials Q j such that, for all g ∈ F q+2 r+1 , √ nE [g(S n )] = ⌊ r/2⌋ j=0 1 n j X g(x)Q j (x) dλ(x) + C q+2 r+1 (g) • o(n -r/2
sup x∈R P S n √ n ≤ x -N(x) -n(x) ⌊r ′ ⌋ k=1 P k (x) n k/2 = o(n -r ′ /2 ) .
Corollary 6.14. [10, Corollary 1.8] Assume (α ′ ) [START_REF] Bálint | Limit theorems in the stadium billiard[END_REF] and (β ′ ) hold with X = R. Then

sup x∈R P S n √ n ≤ x -N(x) - P 1 (x) n 1/2 n(x) = o(n -1/2 ) .
Corollary 6.15. [10, Corollary 1.9] Assume (α ′ ) [START_REF] Benda | A central limit theorem for contractive stochastic dynamical systems[END_REF], (β ′ ) and (δ ′ )[r 0 ] hold for some real number r 0 ∈ (1, 2). Then

sup x∈R P S n √ n ≤ x -N(x) - P 1 (x) √ n n(x) = o n -r 0 /2 .
Example 6.16 (Knudsen gas, Edgeworth expansions in CLT, LLT). Consider Example 5.2.

Recall that Assumptions (α ′ ), (β ′ ), (γ ′ ) and (δ ′ ) have been checked in Example 6.10. Assume f admits moments of order ( r + 2) with respect to ν = µ and that E µ [e itf ] < 1 for all t ∈ X * \ {0}.

• (Expansions of order r -1 and r in the LLT) Assume either

X = Z or |E µ [e itf ]| < 1 -C 0 |t| α ′ 1
for some α ′ 1 > 0, then the conclusions of Theorems 6.11 and 6.12 hold true with

respectively q > α ′ 1 (1 + r 2 ) and q > α ′ 1 (1 + r+1 2 ). • (Edgeworth expansion of order r) Assume X = R and |E µ [e itf ]| < 1 -C 0 |t| α ′ 1
for some

α ′ 1 < r ′ -1 2 -1
. Then the conclusion of Theorem 6.13 holds true. • (First order Edgeworth expansion) Assume X = R and r = 1. Then the conclusion of Theorem 6.14 holds true.

• (Edgeworth expansion of order r

0 ∈]1, 2[) Assume X = R, r = 2 and that |E µ [e itf ]| < 1 -C 0 |t| α ′ 1 for some α ′ 1 < ( r 0 -1 2 ) -1 .
Then the conclusion of Corollary 6.15 holds true.

Limit theorems for Markov random walks

We focus again in this section on the context of Markov random walks, that is the general Markovian setting of Section 5. Recall that (X n ) n≥0 is a Markov chain with states space Ω and with invariant distribution µ and initial distribution ν and that (Z k ) k≥1 is a sequence of independent identically distributed random variables with common distribution P and independent of the Markov chain (X n ) n≥0 . Recall that we are interested in the behaviour of

S n := n k=1 Y k , with Y k = f (X k-1 , X k , Z k ).
In a first subsection, we establish probabilistic limit theorems in the general context as a direct consequence of the results of Section 6. In the three following subsections, we apply our approach for classical families of Markov chains: the ρ-mixing Markov chains, the V -geometrically ergodic Markov chains and Lipschitz iterative model. More precisely, we prove Theorem 1.1 in these three last subsections.

7.1. General results. We set P ν for the Markov distribution with transition operator P and initial probability measure ν. We assume that ((

x k ) k≥0 , ω) → f (x 0 , x 1 , ω) is P µ ⊗ P-centered. We set P t (h)(x) = E e it.Y 1 h(X 1 )|X 0 .
We establish probabilistic limit theorems under the assumptions of Theorem 5.5 that we recall in the following statement for reader's convenience.

Theorem 7.1. Let δ 0 > 0. Let r be a positive integer and Y 1 ∈ L r+1 (P µ ⊗ P). Let (B j , ∥ • ∥ (j) ), j = 0, ..., r+1 be a chain of (r+2) Banach spaces such that 1 ∈ B 0 and that for all j = 1, ..., r+1,

B j-1 → B j , B j → L r+1 j (µ).
Assume that P t (for |t| < δ 0 ) acts continuously on B 1 , ..., B r+1 and that P 0 acts continuously on B 0 and that, for all m = 0, ..., r, P t -

m k=0 P (k) 0 k! .t ⊗k is both O(t m ) in r-m j=0 L(B j , B j+m ) and o(t m ) in L(B r-m , B r+1 ), with P (k) 0 (h)(x) = i k Q k (h)(x) := E P (if (x, •, ω)) ⊗k h(•) (x) dP(ω) ∈ r-k j=0 L(B j , B d ⊗k j+k ) .
Finally we assume that, in L(B 1 ),

∃ϑ 1 ∈]0, 1[, ∀n ∈ N * , ∀|t| < δ 0 , P n t = λ n t Π t + N n t , with sup |t|<δ 0 ∥N n t ∥ B 1 = O(ϑ n 1 ) , ( 43
)
with Π 0 = E µ [•]1, λ 0 = 1 and Π t := 1 2iπ Γ 1 (z Id -P t ) -1 dz, N n t := 1 2iπ Γ 0 z n (z Id -P t ) -1 dz , ( 44
)
with Γ 1 the oriented circle C(1, δ) and Γ 0 the oriented circle C(0, a), with ϑ 1 < a < a + δ < 1 and that 9 sup j=1,...,r+1

sup |t|<δ 0 sup z∈Γ 0 ∪Γ 1 ∥(z Id -P t ) -1 ∥ B j + ∥(z Id -P ) -1 ∥ B 0 < ∞ . ( 45
)
Assume either E ν ∈ B * 1 or, more generally, that there exists some Banach space B 0 (that can be intermediate between B 0 and B 1 ) such that and E ν ∈ B * 0 and such that t → Π t ∈ L(B 0 , B 0 ) is continuous at 0 (using e.g. Theorem 3.3) , then (i) Theorem 6.2 (CLT) applies and we conclude that

(S n / √ n) n≥1 converges in distribution to a centered Gaussian random variable with variance matrix n∈Z Cov µ⊗P ⊗N Y 1 , Y |n|+1 . (ii) Theorem 6.5 (LLT) applies if Y 1 is Z d valued and if the non-arithmetic condition (β ′ ) is satisfied. If moreover E ν ∈ B * r , then Assumption (α ′ )[ r := r -1] is satisfied.
In particular: 9 recall that, in practice, (43), (44) and the first part of (45) follow from the Keller-Liverani theorem (Theorem 3.3) applied with the Banach spaces Bj → Bj+1 up to consider an additional Banach space Br+2.

(iii) (d = 1) Theorems 6.11 and 6.12 (Expansions of order r -1 = r -2 and r = r -1 in the LLT) apply if Y 1 is Z-valued and if the non-arithmetic condition (β ′ ) is satisfied. (iii') (d = 1) Theorems 6.11 and 6.12 (Expansions of order r -1 = r -2 and r = r -1 in the LLT) apply if conditions (β ′ ) (non-arithmeticity) and (γ ′ )[α ′ 1 , α 1 ] (Diophantine-type condition) are satisfied. (iv) (d = 1) Corollary 6.14 (First order Edgeworth expansion) holds true if r ≥ 2 and if conditions (β ′ ) (non-arithmeticity) is satisfied on X = R. (v) (d = 1) Theorem 6.13 (Edgeworth expansion of order r = r -1) applies if r ≥ 2 and if the non-arithmetic condition (β ′ ) and Condition (δ ′ )[r ′ ] hold true.

7.2. ρ-mixing Markov chains. We consider here the case of Markov chains that are ρ-mixing, i.e. i.e. when the transfer operator P is geometrically ergodic on L 2 (µ), that is satisfies

∃C > 0, ∃ϑ ∈]0, 1[, ∀g ∈ L 2 (µ), ∥P n (g) -E µ [g]1∥ L 2 (µ) ≤ Cϑ n ∥g∥ L 2 (µ) .
Recall that this also implies the geometric ergodicity on each L p (µ) for p ∈]1, +∞[ (see [START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF]).

Let us observe that the study of Markov random walks driven by a ρ-mixing Markov chain (X k ) k can be simplified in an additive function of a ρ-mixing Markov chain.

10 Proposition 7.2. If (X k ) k≥0 is ρ-mixing, then the Markov chain ( X k := (X k-1 , X k , Z k )) k≥0
with invariant probability measure µ the distribution of (X 0 , X 1 , Z 0 ) with respect to P µ , which is given by

µ(A × B × C) = E µ [P (1 B )1 A ] P(C)
is also ρ-mixing, with same rate.

Proof. Let P be the transfer operator of X. Then, for all n ≥ 1,

P n (G)(x, y, z) = E [ G(X n-1 , X n , Z n )| X 0 = y] = E (P n-1 (H(•, z)))(y) dP(z) ,
where

H(x, z) = E [ G(X 0 , X 1 , z)| X 0 = x],
and so

∥ P n (G) -E µ [G]∥ 2 L 2 ( µ) = Ω E P n-1 (H(•, z))(y) -E µ [H(•, z)] dP(z) 2 dµ(y)
≤

Ω E P n-1 (H(•, z))(y) -E µ [H(•, z)] 2 dP(z) dµ(y) ≤ E P n-1 (H(•, z))(y) -E µ [H(•, z)] 2 L 2 (µ) dP(z) ≤ E C 2 ϑ 2(n-1) ∥H(•, z)∥ 2 L 2 (µ) dP(z) = E C 2 ϑ 2(n-1) ∥E [ G(X 0 , X 1 , z)| X 0 ] ∥ 2 L 2 (µ) dP(z) ≤ C 2 ϑ 2(n-1) ∥G∥ 2 L 2 ( µ) .
□ Thus, without any loss of generality, from now on, in this subsection, we replace f (x, y, z) by f (y) (up to replace the Markov chain X by the Markov chain X). Note that this replacement changes the notion of non-lattice, which can be corrected by using [START_REF] Hervé | Limit theorems for stationary processes with L2-spectral gap[END_REF]. Theorem 7.3. Assume P is geometrically ergodic on L 2 . Assume the initial measure ν is the stationary measure µ and Y k = f (X k ), with f : Ω → R d . Let r be a positive integer. Assume f is µ-centered and in L r+1 (µ). Then the assumptions of Theorem 7.1 hold true with B 0 = C.1 endowed with the infinite norm and with B j = L r+1 j (µ) for all j ∈ {1, ..., r} and B r+1 = L 2r+1 2r (µ) (note that 1 < 2r+1 2r < r+1 r ). In particular

(i)
′ ∈ [1, r = r -1]) applies if r ≥ 2, if Y 1 is non-lattice in R and if Assumption (δ ′ )[r ′ ] holds true.
Proof. Since P = Id on B 0 , 1 is the single spectral value of P |B 0 and ∥(z

Id -P ) -1 ∥ L(B 0 ) = |z -1| -1 . We know that for all p ∈]1, +∞[, ∥P n -E µ [•]1∥ L(L p ) decreases exponentially fast.
This implies in particular that P is quasi-compact on B 1 and on B 2 with a single dominating eigenvalue 1, which is simple and also that

∥P n t (g)∥ L p (µ) ≤ ∥P n (|g|)∥ L p (µ) ≤ ∥P n -E µ [•]1∥ L(L p ) ∥g∥ L p (µ) + ∥g∥ L 1 (µ) ,
implying the uniform Doeblin-Fortet inequality (4) for B j → B j+1 . Moreover, for all j = 0, ..., r and m ∈ {1, ..., r -j}, P t (g) -

m k=0 P ((if ) k g) k! t k B j+m ≤ P e itf - m k=0 (if ) k k! t k g B j+m ≤ e itf - m k=0 (if ) k k! t k Bm ∥g∥ B j ≤ o(t m )∥g∥ B j ,
due to the dominated convergence theorem. Note that this last inequality implies in particular the continuity assumption of t → P t ∈ L(B j , B j+1 ) required in Theorem 3.3. Moreover P

(k) 0 = P ((if ) k •) ∈ r-k j=0 L(B j , B j+k ).
We also apply Theorem 3.3 with B r → B r+1 and with B r+1 → L 1 (µ). The above inequalities combined with the Keller and Liverani perturbation theorem (Theorem 3.3) ensure that the assumptions of Items (C) and (D) of Proposition 4.1. Thus Corollary 5.6 with ν = µ and Proposition 5.8 apply. For (i), we apply Theorem 6.2 thanks to the previous facts and we identify the variance matrix of the limit using Proposition 5.8. Condition (P) of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] is satisfied since the L p (µ) are contained and dense in L 1 (µ). Condition ( K) of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] follows from the ρ-mixing. Thus, due to [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Proposition 5.4], the non-lattice property implies the non-arithmeticity (exponential decay of sup t∈ [-π,π] We illustrate the previous theorem with the following explicit example with a smooth transition density and an observable that does not admit moment of every order. Example 7.4. Let κ > 0. We consider the following von Mises Markov chain (X n ) n≥0 on Ω = [-1 2 , 1 2 ] endowed with the Lebesgue measure µ (we also take ν = µ). We assume that conditionally to X k , X k+1 has von Mises distribution with mean X k and with concentration κ. Let r be a positive integer and γ ∈]r + 1, r + 2[. We consider

S n = n k=1 Y k , with Y k = f (X k ) and f (y) := sgn(y)|y| -1 γ . Then Y k = f (X k ) ∈ L r+1 \ L r+2 and
(i) Theorem 6.2 (CLT) applies and so (S n / √ n) n≥1 converges in distribution to a centered Gaussian random variable with variance matrix n∈Z Cov Pµ Y 1 , Y |n|+1 . (ii) Theorem 6.5 (LLT) applies. (iii') Theorems 6.11 and 6.12 (Expansions of order r -1 = r -2 and r = r -1 in the LLT) apply (for any α 1 and α ′ 1 ). (iv) Corollary 6.14 (First order Edgeworth expansion) holds true if r ≥ 2. (v) Theorem 6.13 (Edgeworth expansion of order r = r -1) applies if r ≥ 2 (for r ′ = r = r -1).

Proof. We apply Theorem 7.3. This Markov chain has a smooth transition density. The family of Fourier-perturbed operators (P t ) t is given by ∀t ∈ R, P t (u)(x) = Ω p(x, y)e itf (y) u(y) dy , with p(x, y) = e κ cos(2π(y-x))

I 0 (κ)
, where I 0 is the modified Bessel function of order 0.

• We first observe that the Lebesgue measure µ on Ω := [-1 2 , 1 2 ] is invariant. This follows from the fact that Ω p(x, y) dx = 1.

• We observe that ∀|t| > 2 1 γ , µ(|f | k > t) = µ(|f | k > t 1 k ) = µ(| Id | < t -γ k ) = 2t -γ k . Thus Ω |f | k dµ = +∞ 0 µ(|f | k > t) dt
is finite if and only if γ > k. This ensures that f ∈ L r+1 (µ) \ L r+2 (µ).

• Now let us prove that the Markov chain (X n ) n is ρ-mixing. To this end, we establish a Doeblin-Fortet inequality and study the peripherical spectrum. For any n ∈ N and x ∈ Ω, the following inequalities hold true:

P n (u)(x) ≤ Ω n n-1 i=0 p(x i , x i+1 ) |u(x n )| dx 1 ...dx n ≤ e κ I 0 (κ) ∥u∥ L 1 (µ) ,
with the convention x 0 = x, where we used the fact that p(x n-1 , x n ) ≤ e κ I 0 (κ) and that, for all i = 0, ..., n -2, p(x i , •) are probability density functions. Thus we have proved that, for any p ∈ [1, +∞] and any n ∈ N,

∥P n (u)∥ L p (µ) ≤ e κ I 0 (κ) ∥u∥ L 1 (µ) . ( 46
)
This Doeblin-Fortet inequality combined with e.g. Hennion's theorem [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux lipschitziens (French)[END_REF] ensures the quasi-compactness of P on L p (µ) for all p ∈]1, +∞[ and the nullity of its spectral radius. then implies that h is constant and so also that λ = 1. Hence 1 is the only eigenvalue of modulus one of P and its eigenspace consists in constant functions. Finally (46) ensures the that sup n ∥P n ∥ L 2 (µ) < ∞ and so that the generalized eigenspace of P associated to 1 coincide with its eigenspace, ending the proof of the simplicity of 1 as an eigenvalue of P . Thus we ave proved that the Markov chain (X n ) n is ρ-mixing. • Now let us prove Assumptions (β ′ ), (γ ′ ) and (δ ′ ). To this end we will prove that lim t→+∞ ∥P 2 t ∥ ∞ = 0. Observe that P 2 t (u)(x) = Ω q t (x, z)e itf (z) u(z) dz, with

q t (x, z) = Ω p(x, y)p(y, z)e itf (y) dy = |y|>2 1 γ
p(x, y -γ )p(y -γ , z)e ity γ|y| -γ-1 dy .

The Riemann-Lebesgue theorem ensures that lim t→+∞ q t (x, z) = 0. By uniform continuity of p and since Ω is compact, (x, z) → q t (x, z) is also uniformly continuous (uniformly in (x, z, t)). Since Ω is compact, we conclude that q t (x, z) converges to 0 uniformly in (x, z), as t → +∞. Thus lim t→+∞ ∥P 2 t ∥ ∞ = 0, ensuring (42) and so Assumptions We conclude by applying 7.3. □ 7.3. V -geometrically ergodic Markov chains. Let V : Ω → [1, +∞[ be an unbounded measurable function. The random walk (X n ) n , or equivalently its transition operator P , is said to be V -geometrically ergodic if there exist C > 0 and ϑ ∈]0, 1[ such that

(γ ′ )[α ′ 1 , α 1 ] for any α 1 , α ′ 1 and (δ ′ )[r ′ ]
∀n ≥ 1, P n (•) -E π [•] V ∞ ≤ Cϑ n ∥ • /V ∥ ∞ .
Again the study of Markov random walks driven by a V -geometrically ergodic random walk (X k ) k≥0 can be reduced to an additive function of a V -geometrically ergodic Markov chain.

Proposition 7.5. If the Markov chain (X k ) k≥1 is V -geometrically ergodic, then the Markov chain ( X k = (X k-1 , X k , Z k )) k≥0 with invariant measure µ defined in Proposition 7.2 is V - geometrically ergodic with same rate, with V (x, y, z) = V (x) + V (y).
Proof. Let G : Ω 2 × E → C be a bounded measurable function. We have seen in the proof of Proposition 7.2 that the transfer operator P of X satisfies

∀n ≥ 2, P n ((G V )(x, y, z)) = E (P n-1 (H(•, z)))(y) dP(z) , where H(x, z) = E [ G(X 0 , X 1 , z)(V (x) + V (X 1 ))| X 0 = x] =: K(x, z)V (x). Moreover ∥K∥ ∞ ≤ E [∥G∥ ∞ (V (X 0 ) + V (X 1 ))|X 0 = x] V (x) ≤ ∥G∥ ∞ (1 + ∥P (V )/V ∥ ∞ ) . Thus P n (G V ) -E µ [G V ] V ∞ = 1 V E P n-1 (K(•, z)V ) -E µ [K(•, z)V ] dP(z) ∞ ≤ Cϑ n-1 E ∥K(•, z)∥ ∞ dP(z) ≤ Cϑ n-1 ∥G∥ ∞ (1 + ∥P (V )/V ∥ ∞ ) .
This ends the proof of the proposition since P acts continuously on the space V.L ∞ endowed with the norm ∥ • /V ∥ ∞ . □ Thus, without any loss of restriction, from now on, in this subsection, we replace f (x, y, z) by f (y) (again this complicates the notion of non-lattice, see [START_REF] Hervé | A Berry-Esseen theorem of M-estimators for V-geometrical Markov chains[END_REF] for a simple one). Theorem 7.6. Let r be a positive integer, r ′ ∈]r, r + 1] be a real number and f :

Ω → R d be a µ-centered function belonging to L r+1 (µ). Let V : Ω → [1, +∞[ be an unbounded measurable function such that E µ [V ] < ∞. Assume that P is V -geometrically ergodic and that max u∈{0,r ′ -r} sup j=0,...,r sup m=1,...,r-j V -j+m+u r+1 P |f | m+u V j r+1 ∞ < ∞ , ( 47
B j := V j r+1 .L ∞ and B r+1 := V r ′ r+1 .L ∞ endowed with the respective norms ∥ • ∥ (j) := ∥ • /V j r+1 ∥ ∞ and ∥ • ∥ (r+1) := ∥ • /V r ′ r+1 ∥ ∞ . If E ν [V ε ] < ∞ for some ε > 0, then (i)
. If moreover E ν [V r r+1 ] < ∞, then (iii) (d = 1
) Theorems 6.11 and 6.12 apply with r = r -1 if f is Z-valued and non-lattice in Z. (iii') (d = 1) Theorems 6.11 and 6.12 (Expansions of order r ′ in the LLT) apply with r

= r -1 if Assumption (γ ′ )[α ′ 1 , α 1 ] holds true for E Pν [e isSn ] and if Y 1 is non-lattice in R. (iv) (d = 1) Corollary 6.14 (First order Edgeworth expansion) holds true if r = 2 and f is non-lattice in R. (v) (d = 1) Theorem 6.13 (Edgeworth expansion of order r ′ ∈ [1, r = r -1]) applies if Y 1 is non-lattice in R and if Assumption (δ ′ )[r ′ ] holds true for E Pν [e isSn ] .
Proof. Recall that P is also V γ -geometrically ergodic for any γ ∈]0, 1] and that Theorem 3.3 applies with the Banach spaces

V γ L ∞ → L 1 (µ) (see e.g. [26, Lemma 10.1]). If V ∈ L 1 (m), then B j ⊂ L r+1 j (m) since ∥h∥ L r+1 j (m) = V j r+1 h V j r+1 L r+1 j (m) ≤ V j r+1 L r+1 j (m) h V j r+1 ∞ = ∥V ∥ j r+1 L 1 (m) ∥h∥ B j .
In particular, if V ∈ L 1 (ν), then ν defines a linear continuous form on B r+1 . The fact that B j → B j+1 follows from V -j+1 r ≤ V -j r . There exist C > 0 and ϑ ∈]0, 1[ such that, for any j ∈ {1, ..., r + 1},

∥P n (g) -E π [g]∥ (j) ≤ Cϑ n ∥g∥ (j) , ∥P n t (g)∥ (j) ≤ ∥P n (|g|)∥ (j) ≤ Cϑ n ∥g∥ (j) + ∥g∥ 1 ,
ensuring the Doeblin-Fortet estimate (4) for B j → B j+1 for j = 1, ..., r. For k = 0, ..., r, P (k) 0

:= P ((if ) k •) ∈ ∩ r-k j=0 L(B j , B j+k ) since P (k) 0 (h) (j+k) ≤ V -j+k r+1 P (|f | k |h|) ∞ ≤ V -j+k r+1 P (|f | k V j r+1 ) ∞ ∥h∥ (j)
and due to (47). It remains to check the regularity assumptions. For all j = 0, ..., r and m ∈ {1, ..., r -j},

P t (g) - m k=0 P ((if ) k g) k! t k (j+m) ≤ P e itf - m k=0 (if ) k k! t k g (j+m) ≤ V -j+m r+1 P |tf | m ∥g∥ (j) V j r+1 ∞ ≤ O(t m ) ∥g∥ (j) ,
and

P t (g) - m k=0 P ((if ) k g) k! t k (r+1) ≤ P |tf | m+r ′ -r V j r+1 V -j+m+r ′ -r r+1 ∞ ∥g∥ (j) ≤ O(t m+r ′ -r ) ∥g∥ (j) = o(t m ) ∥g∥ (j) .
Condition (P) of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] is satisfied since the Banach spaces are stables under complex modulus and Condition ( K) of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] follows from the V -geometric ergodicity. Thus, due to [26, Proposition 5.4], the non-lattice property implies the exponential decay of sup t∈ [-π,π]

d \B(0,δ 0 ) E[e it.Sn ] |
as n → +∞. We end the proof by applying Theorem 7.1 with

B 0 := V ε .L ∞ for (applying Theorem 3.3 for V ε 2 .L ∞ → B 0 = V ε .L ∞ ). □ Remark 7.7. Assume ν = µ. Observe that (47) is satisfied as soon as ∥f r+1 /V ∥ ∞ < ∞ (which also ensures that f ∈ L r+1 (µ)) with r ≥ 1, since for j + m ≤ r and u ∈ {0, r ′ -r}: V -j+m+u r+1 P |f | m+u V j r+1 ∞ ≤ V -j+m+u r+1 P V j+m+u r+1 ∞ f r+1 /V m+u r+1 ∞ < ∞ .
Moreover, for the Markov chain (X k-1 , X k , Z k ) considered in Proposition 7.5 with the reference function V , (47

) holds true if E |f (•, •, ω)| r+1 dP(ω)/ V ∞ < ∞. Indeed, setting g(x, y) = E |f (x,y,ω)| r+1 dP(ω) V (x)+V (y) , P |f | m+u V j r+1 (x) = P E |f (x, •, ω)| m+u dP(ω)(V (x) + V (•)) j r+1 (x) ≤ P E |f (x, •, ω)| r+1 dP(ω) m+u r+1 (V (x) + V (•)) j r+1 (x) ≤ P (g(x, •)) m+u r+1 (V (x) + V (•)) j+m+u r+1 (x) ≤ ∥g∥ m+u r+1 ∞ 1 + V -j+m+u r+1 P (V j+m+u r+1 ) ∞ V (x) j+m+u r+1
, since P is V j+m+u r+1 -geometrically ergodic.

7.4. Lipschitz iterative models. We consider a non-compact metric space (E, d) in which every closed ball is compact, and endow it with its Borel σ-algebra E. Let (G, G) be a measurable space, let (θ n ) n≥1 be a sequence of independent identically distributed G-valued random variables. Let F : E × G → E be a measurable function such that, for all g ∈ G, F g := F (•, g) is Lipschitz continuous with Lipschitz constant C g . We consider the random walk defined by

∀n ≥ 1, X n = F (X n-1 , θ n ) ,
the sequence (θ n ) n≥1 being independent of the initial value X 0 of the random walk. This random walk is called an iterative Lipschitz model [START_REF] Diaconis | Iterated random functions[END_REF] [7] and has transition operator For one-dimensional autoregressive chains, convergence to stable laws has been investigated in [START_REF] Guivarc'h | On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks[END_REF] for f (x) = x. Let x 0 be a fixed point in E. For x ∈ E, we set p(x) = 1 + d(x, x 0 ). We set, for all g ∈ G, M g := C g + p(F (x 0 , g)). We are interested in the asymptotic behaviour of S n = n k=1 f (X k ), with f : Ω → R d satisfying the following condition for some C 1 , s ≥ 0: ∀(x, y) ∈ E × E, |f (x) -f (y)| ≤ C 1 d(x, y) (p(x) + p(y)) s .

(48)

Recall that the case f is Lipschitz continuous (i.e. s = 0 in (48)) has been studied e.g. in [START_REF] Duflo | Random Iterative Models[END_REF] and in [START_REF] Benda | A central limit theorem for contractive stochastic dynamical systems[END_REF]. Fix α ∈]0, 1]. We set B In the sequel, we set M := M θ 1 and C := C θ 1 . Recall that it has been proved in [START_REF] Hennion | Central limit theorems for iterated random lipschitz mappings[END_REF]Th. I] that if E M α(s+1) < ∞ and E [C α max{C, 1} αs ] < 1 , then the Markov chain (X n ) n admits a unique stationary distribution µ and p ∈ L α(s+1) (µ) (see also e.g [START_REF] Diaconis | Iterated random functions[END_REF] [7]), which implies in particular that f ∈ L α (µ). Theorem 7.9. Let r be a positive integer and a real number r ′ ∈]r, r + 1]. Let ν be a probability measure on Ω. Assume that p ∈ L (s+1)(r+1) (µ) and α ∈]0, s+1 s+2 ]. Assume that f is µ-centered, satisfies (48) and that 13 E M (s+1)(r+1) + C α M (s+1)r ′ +sα < +∞ (49) and E C α max{C, 1} (s+1)r ′ +sα < 1 .

(50)

If p ∈ L ε (ν) for some ε > 0, then Let us prove this result. We consider the following notion of weighted Hölder-type spaces due to D. Guibourg [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and centered case[END_REF] and used in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] generalizing those introduced [START_REF] Page | Théorèmes de renouvellement pour les produits de matrices aléatoires[END_REF] (used also in [START_REF] Milhaud | Etude de l'estimateur du maximum de vraisemblance dans le cas d'un processus autorégressif: convergence, normalité asymptotique, vitesse de convergence[END_REF][START_REF] Peigné | Iterated function schemes and spectral decomposition of the associated Markov operator[END_REF]). For positive real numbers β, γ such that 0 < β ≤ γ and for (x, y) ∈ E 2 , we set Proof. Since α ∈]0, s+1 s+2 ], s + 1 ≤ s+1 α -1. Due to Lemmas 7.11 and 7.12, we know that, for all for j = 1, ..., r + 1, P is geometrically ergodic on B j and that (P t ) t satisfies the Doeblin Fortet assumption of Theorem 3.3 for the Banach spaces B j → L 1 (µ). Moreover 1 ∈ B 0 , and P coincide with Id on B 0 and so (z Id -P ) -1 = (z -1) -1 Id on B 0 . Due to Lemma 7.13, P t -k j=0 P We prove the following result (generalizing [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Proposition 11.6], see also [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Propositions 11.5 and 11.7] for C m -smoothness). We conclude by noticing, on the first hand (using a s+1 ≤ 1), that

(i)

  j

Proposition 5 .

 5 11 (d = 1). Let δ > 0. Assume Hypothesis 5.1 with d = 1 and that there exist two Banach spaces B 1 , B 2 such that 1 ∈ B 1 → B 2 → L 1 (µ), and such that P t (for any |t| < δ) acts continuously on B 1 . Assume that P is geometrically ergodic on B 1 and B 2 :

  d \B(0,δ 0 ) E[e it.Sn ] | as n → +∞). We conclude by applying Theorem 7.1. □ 11 non-latticity for Z d -valued observables means that there exist no triple (a, H, θ) with a ∈ Z d , H ̸ = Z d a closed subgroup in Z d and θ : Ω → Z d such that Y1 + θ(X1) -θ(X0) ∈ a + H Pµ × P-a.s. 12 non-latticity in R means that there exist no triple a, b ∈ R and θ : Ω → R such that Y1+θ(X1)-θ(X0) ∈ a+bZ Pµ × P-a.s..

Furthermore P 1 = 1 2 p

 112 and if h ∈ L p (µ) and λ ∈ C of modulus one are such that P h = λh, then |h(x)| ≤ Ω p(x, y) |h(y)| dy . But Ω |h(x)| dx = Ω (x, y) |h(y)| dydx , since µ is invariant. Since p(x, y) > 0, this implies that |h| is almost surely constant, and the relation λh(x) = Ω p(x, y)h(y) dy

  for any r ′ and also Assumption (β ′ ) (noticing that the lattice condition f (y) + θ(y) -θ(x) ∈ a + bZ (with b ̸ = 0) would imply that P 2πk

)

  then the conclusions of Items (C) and (D) of Propositions 4.1, of Corollary 5.6 (for ν = µ) and of Proposition 5.8 hold true with B 0 = C.1 endowed with the infinite norm and with, for j = 1, ..., r,

P

  (g)(x) = E[ g(F (x, θ 1 ))] .Note that this context includes the autoregressive chains on R d of the form∀n ≥ 1, X n = A n X n-1 + θ n , where (A n , θ n ) n≥1 is an i.i.d. sequence of r.v. taking values in M d (R) × R d , independent of X 0 (M d (R)denotes the set of real d × d-matrices.) with e.g. |A 1 | < 1 (| • | being a matrix norm).

Remark 7 . 8 .

 78 ,a for the set of functions g :E → C such that |f | (f ) < ∞, with |f | (0) a := ∥f /p a ∥ ∞ and m (0) α,b (f ) := sup x,y∈E, x̸ =y |f (x) -f (y)| d(x, y) α max(p(x), p(y)) b .Observe that (48) means that the coordinates of f belong to B

∆ 1 Proposition 7 . 10 .- 1

 17101 α,β,γ (x, y) := p(x) αγ p(y) αβ + p(x) αβ p(y) αγ .Then B α,β,γ denotes the space of C-valued functions g on E satisfying the following conditionm α,β,γ (g) := sup |g(x) -g(y)| d(x, y) α ∆ α,β,γ (x, y) , x, y ∈ E, x ̸ = y < +∞. Set |g| α,γ := sup x∈E |g(x)| p(x) α(γ+1) and ∥g∥ α,β,γ := m α,β,γ (g) + |g| α,γ . Then (B α,β,γ , ∥ • ∥ α,β,γ ) is a Banach space. Moreover γ ≤ γ ′ ⇒ B α,β,γ → B α,β,γ ′ , (51) and B α,β,γ → L K γ+1 (µ) if p ∈ L αK (µ). Theorem 7.9 relies on the following propositions combined with Theorem 7.Assume the Assumptions of Theorem 7.9. Then the assumptions of Corollary 5.6 are satisfied with B 0 = C.1, B j = B α,s+1, j(s+1) α for j = 1, ..., r, B r+1 = B α,s+1, r ′ (s+1) α -1 , B r+2 = L 1 (µ).

Lemma 7 . 11 . ([ 26 , 1 ]Lemma 7 . 12 . ([ 20 ,

 71126171220 in O(|t| k ) in L(B i , B i+k ) for all i = 0, ..., r -k and in O(|t| k+r ′ -r ) = o(t k ) in L(B r-k , B r+1 ), with P (j)0 := P (if ) ⊗j • . Moreover B j → p (s+1)j L ∞ → L r+1 j (µ) since p (s+1)(r+1) ∈ L 1 (µ); and t → P t ∈ L(B r+1 , B r+2) is continuous (we can use e.g.Lemma 7.13).□ Lemma B.1]) Let a ≥ α and b ∈ [0, a + αγ] and 0 ≤ β ≤ γ. Assume E M a+α(γ+1) + C α M a+α(γ+β) + C α M b+α(γ+1) < ∞ .Then there exists C ′ > 0 such that, for all g ∈ B(0) α,a,b , ∥P (g•)∥ L(B α,β,γ ,B α,max(β, b-a+α α ),γ+ a α ) ≤ C ′ ∥g∥ B (0) α,b,a, and∥P (g•)∥ L(C.1, B (0) α,b,a ) ≤ C ′ ∥g∥ B (0) α,b,a . More precisely |P (gh)| α,γ+ a α ≤ E[M a+α(γ+1) ]|g| (0) a |h| α,γ(52)andm α,max(β, b-a+α α ),γ+ a α (gh) ≤ E[C α C a+α(γ+β) 1 ]|g| (0) a m α,β,γ (h)+E[C α M b+α(γ+1) ]m (0) α,b (g)|h| α,γ , (53)withC 1 := max(C, 1) + d(F (x 0 , θ 1 ), x 0 ) ≤ M.Proof. The fact that C 1 ≤ M follows from[20, p.1945]. We start from [26, Lemma B.1] which states that|P (gh)| ≤ E[M a+α(γ+1) ]|g| (0) a |h| α,γ p a+α(γ+1) (ensuring (52)) and that|P (gh)(x) -P (gh)(y)| d(x, y) α ≤ E[C α C a+α(γ+β) |g| (0) a m α,β,γ (h)p(x) a ∆ α,β,γ (x, y) + E[C α M b+α(γ+1) ]m(0) α,b (g)|h| α,γ p(x) b p(y) α(γ+1) , if p(y) ≤ p(x). Still assuming that p(y) ≤ p(x), we conclude by noticing that p(x) a ∆ α,β,γ (x, y) = p(x) a (p(x) αβ p(y) αγ + p(y) αβ p(x) αγ ) ≤ p(x) a+αγ p(y) αβ + p(y) αβ p(x) a+αγ since β ≤ γ and p(x) b p(y) α(γ+1) ≤ p(x) a+αγ p(y) α+b-a since α(γ + 1) ≥ α + b -a. □ Theorem 11.5], [26, Propositions 11.2 and 11.4]) Assume s + 1 ≤ β ≤ γ andE M α(γ+1) + C α M α(γ+β) < +∞ and E C α max{C, 1} α(γ+β) < 1 . (54)Then P is geometrically ergodic on B α,β,γ : ∥P n -E µ ∥ L(B α,β,γ ) = O(ϑ n ) and (P t ) t satisfies the assumptions of Theorem 3.3 with B 1 = B α,β,γ and B 2 = L 1 (µ).

Lemma 7 . 13 .P

 713 If s + 1 ≤ β ≤ γ < γ ′ satisfies α(γ ′ -γ) ≤ s + 1 and E[M α(γ ′ +1) + C α M α(γ ′ +β) ] < ∞ , then t → P t ∈ L(B α,β,γ , B α,β,γ ′ ) is continuous at 0. If m is a positive integer, if m ′ ∈ [m, m + 1] and if β, γ are such that 0 ≤ β ≤ γ and E M m ′ (s+1)+α(γ+1) + C α M m ′ (s+1)+α(γ+β) + C α M m ′ (s+1)+αγ < ∞ . ⊗k k! = O t m ′ in L B α,β,γ , B α,β,γ+ m ′ (s+1) α and in L C.1, B (0) α,m ′ (s+1)-α,m ′ (s+1) = B α,0, m ′ (s+1) (if ) ⊗k • .Proof. Due to Lemma 7.11, the continuity at 0 of t → P t ∈ L(B α,β,γ , B α,β,γ ′ ) will follow from the control of ∥e it.f -1∥ B (0) α,sα,a=α(γ ′ -γ)(since sα-a+α α ≤ s + 1 ≤ β and 0 ≤ sα ≤ a + αγ = αγ ′ ).

  |e it.f -1| ≤ min(2, |t|.|f |) ≤ min 2, |t|. |f (x 0 )| + C 1 p s+1 ≤ 2 1-a s+1 |t| a s+1 (|f (x 0 )| + C 1 ) |e it.f -1| (0) a = O |t| min( a s+1,α) and, on the second hand, that|e it.f (x) -e it.f (y) | ≤ min (2, |t|.|f (x) -f (y)|) ≤ 2 1-α |t| α C α 1 d(x, y) α (p(x) + p(y)) sα , ensuring that m (0) b (e itf -1) = O |t| min( a s+1 ,α)This ends the proof of the first point. Let us prove the second point. To this end, we first observe thatP t -Due to Lemma 7.11, it is enough to prove that e it.fm k=0 (it.f ) k k! B (0) α,m ′ (s+1)-α,m ′ (s+1) = O(|t| m ′ ).We set h(u) := e ium k=0 (iu) k k! and notice that|h(t.f )| ≤ 2 |t.f | m ′ m! ≤ 2|t| m ′ |f | )m ′ .Observe that there existsC ′ 0 > 0 such that |h ′ (iu)| = i e ium-1 k=0 (iu) k k! ≤ C ′ 0 |u| m ′ -1 for all u ∈ R. Thus |h(t.f (x)) -h(t.f (y))| ≤ C ′ 0 max (|t.f (x)|, |t.f (y)|) m ′ -1 |t.(f (x) -f (y))| ≤ C ′′ 1 |t| m ′ -1 max p(x) s+1 , p(y) s+1 m ′ -1|t|d(x, y) α max(p(x), p(y)) s+1-α ≤ |t| m ′ d(x, y) α max(p(x), p(y)) m ′ (s+1)-α . □

  1 ) d for the set of d-dimensional vectors with entries in B 1 , and more generally B d ⊗k ℓ for the set of k-linear maps on (R d ) k with values in B ℓ , and we identify its elements H with H = (h i 1 ,...,i k ) i 1 ,...,i k =1,...,d . We write t ⊗m for (t i 1 ...t im ) i 1 ,...,im=1,...,d and . for the scalar product. Finally, for any H (1) ∈ L(B i , B d ⊗m

  Proposition 5.10. Assume the Assumptions of Theorem 5.5 with r ≥ 2. Then λ t = 1 -a 2 .t ⊗2ib 6 .t ⊗3 + o(|t| 3 ), with a as in Proposition 5.8 and with b.t ⊗3 =

	n,m≥0

a 0,n,n+m E Pµ⊗P ⊗N [(t.Y 1 )(t.Y 1+n )(t.Y 1+n+m )] ,

  It follows from Theorem 3.3 applied to (P t ) t that, for ϑ 2 ∈]ϑ 1 , 1[, up to reduce δ if necessary, Formulas (

1) (using the Lebesgue dominated convergence theorem and again the Karamata representation of slowly varying functions). □ Proof of Proposition 5.11. Observe that N 0 (1) = (1 -γ)1.

  We consider again the context of Remark 5.3. We recall that the fact that (41) holds with lim t→0 Φ t = 1 and lim n→+∞ sup |t|<δ 0 |M t,n | = 0 follows from Remark 6.1.

								1) ,
	as n goes to infinity.
	Remark 6.3. Further-
	more,
	(a) Under the assumptions of Theorem 5.5 for r ≥ 1 (and so of Proposition 5.8), Theo-rem 6.2 holds true with A n = Id / √ n and W a centered Gaussian random variable with
	variance
	Σ :=
	λ n t/ √	1 α	-e -1 2 n|Σ	1 2 t/ √	n| 2 2
	≤ n λ t/ √	n -e -1 2 |Σ	1 2 t/ √	n| 2 2 ≤ n o t/	√	n	2 2 = o(1) .
							n k=1 W k ) n converges in distribution
	to W, then Theorem 6.2 applies.
	This assumption ensures (see e.g. [27, Theorem 2.6.5]) the existence of α 0 ∈ [1, 2] and
	of |β| < tan(α 0 π/2) such that

n∈Z Cov Pµ⊗P ⊗N (Y 1 , Y |n|+1 ) . Indeed, Proposition 5.8 ensures 8 that λ t -1 = 1 2 |Σ 1 2 .t| 2 2 + o(|t| 2 ), whence we infer that, for every fixed t ∈ R d , n -E[e it.W ] = λ n t/n (b) If Proposition 5.11 allows to prove that λ t -1 ∼ E[e it.W 1 ] -1, where (W n ) n≥1 is a sequence of i.i.d. random variables such that (A n

  (Knudsen gas, convergence to gaussian or stable distributions). Consider the simple Knudsen gas model introduced in Example 5.2. The continuity of t → Φ t = E µ [Π t (1)] as well as the fact that sup |t|<δ E µ [N n t (1)] decays exponentially fast as n → +∞ have been proved in Example 5.4 thanks to Theorem 3.3 and Remark 6.1. Furthermore

1

2

for the nonnegative symmetric matrix the square of which is Σ. Example 6.

[START_REF] Browder | On the spectral theory of elliptic differential operators[END_REF]

  1 (2π) d R d e -is.u E[e is.W ] ds. □ Consider the context of Remark 6.3 with (det(A n )) n subexponential in n. We have already seen in Remark 6.1that sup |t|<δ 0 |Φ(t)| < ∞ and sup |t|<δ 0 |M t,n | = o (det A n ) for some δ 0 > 0. The integrability of t → sup n |λ n |A * n t|<δ 0 | follows in practice from the control of |λ t -1| (e.g. in case (a) of Remark 6.3, if a is invertible, |λ t | ≤ e -a.t ⊗2 4 as soon as |t| is small enough). Finally, the condition sup t∈[-π,π] d \B(0,δ 0 ) E[e it.Sn ] = o (det A n ) is usually a consequence of the fact that, for t ̸ ∈ 2πZ, ρ ess (P t ) < 1, ρ(P t ) ≤ 1 and that P t admits no eigenvalue of modulus 1, which implies that ∥P nt t ∥ < 1 (for some n t ) which, combined with a continuity argument of t → ∥P t ∥ on the compact [-π, π] d , leads to the existence of a positive integer n ′ such that sup t∈[-π,π] d \B(0,δ 0 ) ∥P n ′ t ∥ < 1 and implies the exponential decay of sup t∈[-π,π] d \B(0,δ 0 ) E[e it.Sn ] as n → +∞.

	Remark 6.6. A * n t 1

  slowly varying at infinity (in the situation of Example 5.12. The fact that lim n→+∞ λ n A * n t = E[e it.W ] with either W as in Example 6.4 has been proved in this example using Theorem 6.2 and Remark 6.3. Let us prove the domination of |λ n A * n t | using the expansion of λ t around t = 0. If r ≥ 1. The nonlattice assumption ensures the invertibility of the matrix Σ. Example 5.9 ensures that λ

t -1 ∼ -Σ 2 .t ⊗2 + o(|t| 2 ). Thus, there exists δ > 0, such that, for all |t| < δ, |λ t | ≤ g(t) := e -1 4 Σ.t ⊗2 , and so |λ n t/ √ n | ≤ (g(t/ √ n)) n = g(t) and g is integrable.

.

  such that |A n t| < δ 0 and |A n | < δ 0 , It remains to prove that sup t∈[-π,π] d \B(0,δ 0 ) E[e it.Sn ] | = o(det A n ).To this end, we follow the strategy explained in Remark 6.6. Let t ∈ R \ 2πZ and p ∈]1, +∞[. Since ∥P t ∥ L p

		1 2 min(|t|	α 0 2 , |t| -α 0 2 ) and so, for every
	t and n λ n Ant ≤ g(t) := e -c 2 min |t|	α 0 2 ,|t|	3α 0 2

  Assume f admits moments of order ( r + 2) with respect to ν = µ. The fact that Assumption (α ′ )[ r] holds true follows from Theorem 5.5 and Example 5.7.

Assumption (β ′ ) will hold true if, for all t ∈ X * \ {0}, E µ [e itf ] < 1. Indeed, as seen in Example 6.7, it is enough to prove that P t (for t ∈ X * ) admits no eigenvalue λ of modulus 1. If it was the case, there would exist

  ) . Let r be a positive integer and r ′ ∈ [1, r] be a real number. Assume (α ′ )[ r], (β ′ ) and (δ ′ )[r ′ ] hold. Then there exist polynomials P k such that

	Theorem 6.13. [10, Theorem 1.7]

  Theorem 6.2 (CLT) applies and we conclude that (S n / √ n) n≥1 converges in distribution to a centered Gaussian random variable with variance matrix n∈Z Cov Pµ Y 1 , Y |n|+1 .(ii) Theorem 6.5 (LLT) applies if Y 1 is Z d -valued and non-lattice in Z d .11 (iii) (d = 1) Theorems 6.11 and 6.12 (Expansions of order r -1 = r -2 and r = r -1 in the LLT) apply if Y 1 is Z-valued and non-lattice in Z. (iii') (d = 1) Theorems 6.11 and 6.12 (Expansions of order r -1 = r -2 and r = r -1 in the LLT) apply if Assumption (γ ′ )[α ′ 1 , α 1 ] holds true and if Y 1 is non-lattice in R.12 (iv) (d = 1) Corollary 6.14 (First order Edgeworth expansion) holds true if r ≥ 2 and if Y 1 is non-lattice in R. (v) (d = 1) Theorem 6.13 (Edgeworth expansion of order r

  Theorem 6.2 (CLT) applies and we conclude that (S n / √ n) n≥1 converges in distribution to a centered Gaussian random variable with variance matrix n∈Z Cov µ⊗P ⊗N Y 1 , Y |n|+1 . Theorem 6.5 (LLT) applies if Y 1 is Z d valued and non-lattice in Z d

	(ii)

  Theorem 6.2 (CLT) applies and we conclude that (S n / √ n) n≥1 converges in distribution to a centered Gaussian random variable with variance matrix n∈Z Cov µ⊗P ⊗N Y 1 , Y |n|+1 . (ii) Theorem 6.5 (LLT) applies if Y 1 is Z d valued and non-lattice in Z d . If moreover p ∈ L (s+1)r (ν), then (iii) (d = 1) Theorems 6.11 and 6.12 apply with r = r -1 if f is Z-valued and non-lattice in Z. (iii') (d = 1) Theorems 6.11 and 6.12 (Expansions of order r in the LLT) apply with r = r -1 if Assumption (γ ′ ) holds true for E ν [e isSn ] and if Y 1 is non-lattice in R. (iv) (d = 1) Corollary 6.14 (First order Edgeworth expansion) holds true if r ≥ 2 and if f is non-lattice in R. (v) (d = 1) Theorem 6.13 (Edgeworth expansion of order r ′ ∈ [1, r = r -1]) applies if r ≥ 2, if Y 1 is non-lattice in R and if Assumption (δ ′ )[r ′ ] holds true for E ν [e isSn ] .

We call it Doeblin-Fortet type inequality in reference to[START_REF] Doeblin | Sur des chaînes à liaisons complètes[END_REF].

Acknowledgements. The author wishes to thank Loïc Hervé for his interest in this work and for stimulating discussions.

Proof of Theorem 7.9. Proposition 7.10 ensures that assumptions of Theorem 7.1 (except maybe those on ν) are satisfied. Observe that the condition p ∈ L (s+1)r (ν) ensures that E ν ∈ B * r since B r → p r(s+1) .L ∞ . For Items (i)-(ii) (TCL and LLT), assuming ε < 1/3 and ε < α(2s + 2), we take B 0 := B ε 2s+2 ,s+1,2s+1 and apply Theorem 3.3 for B ε 2s+2 ,s+1,s+1 → B 0 = B ε 2s+2 ,s+1,2s+1 . This will ensure the continuity of t → Π t ∈ L(B 0 , B 0 ). The fact that B 0 := B ε 2s+2 ,s+1,2s+1 → L 1 (ν) will follow from B 0 → p ε .L ∞ and p ε ∈ L 1 (ν). Let us check that the assumptions of Theorem 3.3 are satisfied for B ε 2s+2 ,s+1,s+1 → B 0 . The continuity assumption of Theorem 3.3 is ensured by Lemma 7.13 (the integrability conditions follows from ε < 1/3). The other assumptions follow from Lemma 7.12. Indeed Conditions (49) and (50) imply (54) for β = γ = s+1 since α ∈]0, s+1 s+2 ] implies α(2s + 2) ≤ (s + 1)r ′ + sα and α(s + 2) ≤ (s + 1)(r + 1); and moreover (54) implies that (54) holds also true with α being replaced by ε/(2s + 2) < α (due to the Hölder inequality).

Finally it has been proved in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]Proposition 11.8] that the non-lattice property implies the exponential decay of max j=1,...,r+1 sup t∈ [-π,π]

In [START_REF] Hervé | Limit theorems for stationary processes with L2-spectral gap[END_REF], the authors considered Markov processes (X n , Sn )

They also considered a continuous version of this form. We concentrate here on the discrete time process. The first coordinate (X n ) n≥0 of this process is also a Markov process (driving the process ( S n ) n≥1 ).

We explain here how, starting with the process (X, S) and using an independent sequence (Z k ) k≥1 of i.i.d. random variables, we can construct a process (S n ) n≥0 with

such that (X n , S0 + S n ) n≥0 is Markov with the same transition operator as (X n , Sn ) n≥0 . Let

be a sequence of i.i.d. random variables uniformly distributed on ]0, 1[ d and independent of ( S0 ,

Then we define the coordinates

In particular

This ensures that the distribution of

n ) j=1,...,k-1 . Thus, we prove by induction that, for all k = 1, ..., d, the distribution of (Y

the one of ( S(1) n , ..., S(k) n ) given (X n-1 , X n , Sn-1 = 0). Indeed this holds true for k = 1 and, moreover assuming the result holds true at some rank k ∈ {1, ..., d -1}, it follows that

and thus the result holds also at rank k+1. Therefore the distribution of Y n given X n-1 , X n , S n-1 coincides with the one of Sn given (X n-1 , X n , Sn-1 = 0). Moreover (X n , S n ) n is a Markov process.