D Ferre
email: deborah.ferre@insa-rennes.fr

Parametric rst-order Edgeworth expansion for Markov additive functionals. Application to M -estimations

We give a spectral approach to prove a parametric rst-order Edgeworth expansion for bivariate additive functionals of strongly ergodic Markov chains. In particular, given any Vgeometrically ergodic Markov chain (X n ) n∈IN whose distribution depends on a parameter θ, we prove that {ξ p (X n-1 , X n ); p ∈ P, n ≥ 1} satises a uniform (in (θ, p)) rst-order Edgeworth expansion provided that {ξ p (•, •); p ∈ P} satises some non-lattice condition and an almost optimal moment domination condition. Furthermore, the sequence (X n ) n∈IN need not be stationary. This result is applied to M -estimations.

Introduction

Let (E, E) be any measurable space, and let (X n ) n≥0 be a Markov chain on a general state space E with transition kernel (Q θ (x, •); x ∈ E) where θ is a parameter in some set Θ. The initial distribution of the chain is denoted by µ θ . The underlying probability measure is denoted by P θ,µ θ .

Let {ξ p (•, •); p ∈ P} be a family of measurable functions from E 2 into IR, where P is any set. Let us dene the following bivariate additive functionals ∀n ≥ 1, ∀p ∈ P, S n (p) :

= n k=1 ξ p (X k-1 , X k ). (1)
We are interested in appropriate conditions on the model, on the family {ξ p (•, •); p ∈ P} and on the initial probability measure µ θ , under which a rst-order Edgeworth expansion exists (also called Esseen theorem), namely there exists a polynomial function A θ,p (•) such that sup (θ,p)∈Θ×P

sup u∈IR P θ,µ θ S n (p) σ θ,p √ n ≤ u -N (u) -η(u) n -1 2 A θ,p (u) = o(n -1 2 ), (2)
where N is the standard normal distribution function and η is its density. Note that Expansion (2) holds uniformly in (θ, p) ∈ Θ × P. As illustrated later in M -estimation, the bivariate and parametric form of (1), as well as the previous uniform control and the possible nonstationarity of µ θ , are required for statistical applications.

Edgeworth expansions in the Markov setting can be established by the two following methods:

1. The regeneration method. This standard method, introduced by [START_REF] Smith | Regenerative stochastic processes[END_REF], was used by Bolthausen [START_REF] Bolthausen | The Berry Esseen theorem for strongly mixing Harris recurrent Markov chains[END_REF] to establish the Berry-Esseen theorem for univariate additive functionals of the form S n = n k=1 ξ(X k ), by splitting S n into a sum of independent blocks. This method can be applied to the general class of Harris-recurrent chains (X n ) n≥0 which either possess an accessible atom or satisfy some minorization condition. Bolthausen work was extended to Edgeworth expansions by Malinovskii [START_REF] Malinovskii | Limit theorems for Harris Markov chains I[END_REF] and next generalized to bivariate additive functionals S n = n k=1 ξ(X k-1 , X k ) by Jensen [START_REF] Jensen | Asymptotic expansions for strongly mixing Harris recurrent Markov chains[END_REF]. Note that in [START_REF] Bolthausen | The Berry Esseen theorem for strongly mixing Harris recurrent Markov chains[END_REF][START_REF] Malinovskii | Limit theorems for Harris Markov chains I[END_REF][START_REF] Jensen | Asymptotic expansions for strongly mixing Harris recurrent Markov chains[END_REF], neither the distribution of (X n ) n≥0 nor the function ξ depends on parameters. However a recent work due to Bertail-Clémençon [START_REF] Bertail | A renewal approach to markovian U-statistics[END_REF] provides a Berry-Esseen theorem adapted to the above mentioned parametric setting, but the extension to Edgeworth expansions would generate even more diculties. Furthermore this statement only concerns univariate additive functionals and the extension of their proof to the bivariate case (1) induces dependence between the regeneration blocks and hence provides at least one more diculty to handle with. For further explanations concerning extension of Bertail-Clémençon work, see Appendix B. Moreover all these works are valid under some complex block-moment conditions. If the strong mixing coecient of (X n ) n≥0 decreases at a fast enough rate, then these blockmoment conditions are entailed by some explicit moment condition provided that the initial probability is dominated by some multiple of the stationary distribution π. When considering the particular case where the chain possesses an atom A, this simplication also holds true whenever the initial probability µ is the Dirac distribution δ x at some x ∈ A. Let us note that the resulting moment condition is then almost optimal (with respect to the independent case). However, to the best of our knowledge, this simplication cannot be extended to the case where the initial probability µ is any probability distribution, in particular where µ is the Dirac distribution δ x at some x ∈ E which does not necessarily belong to an atom. For further explanations concerning conditions on µ, see Appendix B.

2. The weak Nagaev-Guivarc'h spectral method. This method, based on the Keller-Liverani perturbation theorem [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF], enables the statement of limit theorems for additive functionals associated to strongly ergodic Markov chains (Harris recurrence is no more required). This method has been fully described in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] in the case of univariate additive functionals. It is specially ecient for ρ-mixing and V -geometrically Markov chains, as well as for iterated function systems. In those models, the extension of Berry-Esseen type results of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] to the case of bivariate additive functionals of the type (1) has already been obtained in [FHL, HLP] 1 with in addition the desired control on the parameters (θ, p). The resulting moment conditions on {ξ p ; p ∈ P} are (almost) optimal with respect to the independent case. Let us note that they are explicit for these three models and do not depend on the initial probability (unlike the ones given by the regeneration method).

In this paper, we will state Expansion (2) when concerning with the class of strongly ergodic Markov chains, and apply Fourier techniques via the perturbation operator theory of Nagaev-Guivarc'h.

Our work extends the Berry-Esseen type results of [HLP] to the rst-order Edgeworth expansion. As in the independent case, the gap from Berry-Esseen to Edgeworth type results induces at least a new diculty: the requirement of the non-arithmeticity hypothesis.

In Subsection 2.1, we consider a family of random variables (r.v.) S n (p) (not necessarily derived from Markovian models) dened on a general parametric probabilistic space (Ω, F, {P θ ; θ ∈ Θ}), and we state hypotheses called R(m) and (N-A) under which Expansion (2) holds true. These hypotheses concern the behavior of the characteristic function t → φ n,p (t) of S n (p): Hypothesis R(m) focuses on the form and the regularity of φ n,p near t = 0; whereas Hypothesis (N-A), related to the non-arithmeticity assumption, focuses on the behavior of φ n,p outside t = 0.

In Subsection 2.2, we specify the form of S n (p): from now on, S n (p) is dened by (1) where (X n ) n≥0 is assumed to be a strongly ergodic Markov chain, and we give a brief review of the weak Nagaev-Guivarc'h spectral method to check Hypothesis R(m) and (N-A) in this Markov context. In fact, as already done in [FHL, HLP], Hypothesis R(m) can be investigated thanks to an easy extension of the results of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF].

By contrast, the method developed in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] is not sucient to study Hypothesis (N-A). Indeed, the non-arithmeticity condition has to be checked uniformly in both the parameter θ of the Markovian model and the parameter p of the family {ξ p ; p ∈ P} involved in (1). The study of (N-A) in this context is original and constitutes one of the main contributions of this paper (actually, even in the independent case, this question is far from being obvious). In our Markov setting, this study is based on the operator perturbation theory, quasi-compactness arguments and Ascoli theorem. Specically, in Section 3, we give three approaches to reduce Hypothesis (N-A) to some simple non-lattice conditions in the case of general strongly ergodic Markov chains.

Section 4 is devoted to V -geometrically ergodic Markov chains. For this instance and more specically for dominated models, we reduce (N-A) using one of the three approaches presented in Subsection 3.3. Combining this result together with the sucient conditions of [HLP] to check Hypothesis R(m) and the general Edgeworth type statement of Subsection 2.1, provides Expansion (2) under assumptions close to the ones of the independent case. Statistical applications are studied in Section 5: a rst-order Edgeworth expansion for Mestimators of V -geometrically ergodic Markov chains is derived (Theorem 2) from the results of Section 4. Theorem 2, which extends Pfanzagl theorem [START_REF] Pfanzagl | Asymptotic expansions related to minimum contrast estimators[END_REF] obtained for independent and identically distributed (i.i.d.) data under some moment conditions of order 3, is valid under a natural adaptation of the statistical regularity conditions of [START_REF] Pfanzagl | Asymptotic expansions related to minimum contrast estimators[END_REF], moment domination conditions of order 3 + ε, and some simple non-lattice condition as well. To the best of our knowledge, this result is new. Notice that our moment domination conditions are not only almost optimal, but also take the same form as the ones used in [START_REF] Dehay | On likelihood estimation for discretely observed Markov jump processes[END_REF] to prove the asymptotic normality of M -estimators under V -geometrically ergodicity. This result is illustrated with AR(1) processes in Subsection 5.2.

The adaptation of Pfanzagl proof is developed in Section 6 for general statistical models under Hypotheses R(3) and (N-A). Note that this adaptation is not straightforward. Finally, the intermediate results of this section are applied in Subsection 6.4 to M -estimators of AR(d) processes.

Fourier techniques and rst-order Edgeworth expansion

In this section, we present some results based on Fourier techniques. These results appeal to the next Hypotheses R(m) and (N-A) that are well-suited for the markovian case as explained in Subsection 2.2.

Hypotheses R(m) and (N-A) and rst-order Edgeworth expansion

Let (Ω, F, {P θ ; θ ∈ Θ}) be any statistical model, where Θ is some parameter space. The underlying expectation is denoted by E θ . Consider a family {S n (p); n ∈ IN * , p ∈ P} of real r.v. dened on (Ω, F, {P θ ; θ ∈ Θ}), where P is any set. Note that the parameter p may depend on θ.

E θ [e itSn(p) ] = λ θ,p (t) n (1 + l θ,p (t)) + r θ,p,n (t), (3)
where λ θ,p (•), l θ,p (•) and r θ,p,n (•) are C -valued functions of class C m on I 0 satisfying the following properties:

λ θ,p (0) = 1, λ (1)
θ,p (0) = 0, l θ,p (0) = 0, r θ,p,n (0) = 0, and for = 0, . . . , m sup |λ

( ) θ,p (t)|; t ∈ I 0 , (θ, p) ∈ Θ × P < +∞, sup |l
( ) θ,p (t)|; t ∈ I 0 , (θ, p) ∈ Θ × P < +∞, ∃κ ∈ [0, 1), ∃G > 0, ∀n ≥ 1, sup |r ( ) θ,p,n (t)|; t ∈ I 0 , (θ, p) ∈ Θ × P ≤ G κ n .
Furthermore, the functions λ (m)

θ,p (•), l (m)
θ,p (•) and r (m) θ,p,n (•) are continuous on I 00 uniformly in (θ, p) ∈ Θ × P.

Hypothesis (N-A) (Non-arithmeticity). For any compact subset K 0 of IR * , there exists ρ ∈ [0, 1) such that

∀n ≥ 1, sup E θ [e itSn(p) ] ; t ∈ K 0 , (θ, p) ∈ Θ × P = O(ρ n ).
Note that under Hypothesis R(2), the function t → E θ [e itSn (p) ] is of class C 2 on I 0 for all (θ, p) ∈ Θ × P. Then by Fatou lemma, for all (θ, p) ∈ Θ × P, one has E θ [S n (p) 2 ] < +∞. Therefore, when considering the derivative of Equality (3), one easily obtains that for all (θ, p) ∈ Θ × P, lim E θ [S n (p)]/n = 0 when n → +∞. Note that under Hypothesis R(2), when considering the derivative of Equality (3), one easily obtains as well

∀n ≥ 1, lim n→+∞ sup (θ,p)∈Θ×P E θ [S n (p) 2 ] n < +∞, (4)
and in a similar way, under Hypothesis R(4),

∀n ≥ 1, lim n→+∞ sup (θ,p)∈Θ×P E θ [S n (p) 4 ] n 2 < +∞. (5)
Finally, under Hypothesis R(3), we obtain some of the assertions of Proposition 1 below. The other ones can be proved by borrowing the proof of [START_REF] Feller | An introduction to probability theory and its applications[END_REF].

sup u∈IR P θ S n (p) σ θ,p √ n ≤ u -G n,θ,p (u) = o(n -1 2 ).

The polynomial function

G n,θ,p is of the type G n,θ,p (u) = N (u)+ a 1 (θ, p) + a 2 (θ, p) u 2 η(u)/ √ n
where the coecients satisfy

for i = 1, 2, sup (θ,p)∈Θ×P |a i (θ, p)| < +∞. Furthermore, if E θ [|S n (p)| 3 ]
< +∞ for all n ≥ 1 and (θ, p) ∈ Θ × P, then the limit

m 3 θ,p,3 := lim n→+∞ E θ [S n (p) 3 ] n -3σ 2 θ,p b θ,p
is well-dened and bounded in θ ∈ Θ, and moreover the polynomial function G n,θ,p can be explicitly expressed by

G n,θ,p (u) := N (u) + m 3 θ,p,3 6σ 3 θ,p √ n (1 -u 2 ) η(u) - b θ,p σ θ,p √ n η(u).
Remark 1. In the i.i.d. case, Hypotheses R(3) and (N-A) are easily checked. Indeed consider

sup t∈K 0 sup (θ,p)∈Θ×P E θ [e itξp(X 1 ) ] < 1. (6)
When ( 6) is considered at (θ, p) xed, it can be easily relaxed to the usual condition: ξ p (X 1 ) is non-lattice. By contrast, it is not easy to relax the uniform condition (6). Note that this condition is only discussed in [START_REF] Pfanzagl | Asymptotic expansions related to minimum contrast estimators[END_REF] under the stronger Cramér condition:

lim sup t→+∞ sup (θ,p)∈Θ×P E θ [e itξp(X 1 ) ] < 1.
Hypotheses R(m) and (N-A) are the tailor-made assumptions to borrow the proof of the rstorder Edgeworth expansion in the i.i.d. case2 , and consequently to expand

P θ {S n (p)/(σ θ,p √ n) ≤ u}
with a polynomial function independent on n. Notice that, under less restrictive conditions, the results of [START_REF] Durbin | Approximations for densities of sucient estimators[END_REF] provide a rst-order Edgeworth-type expansion but with a polynomial function depending on n.

The main lines of the weak spectral method for Markovian models

Consider from now on the following general Markovian setting. Let (E, E) be any measurable space, and let (X n ) n≥0 be a Markov chain with state space E and transition kernel

(Q θ (x, •); x ∈ E)
where θ is a parameter in some set Θ. The initial distribution of the chain is denoted by µ θ (i.e. X 0 ∼ µ θ ). The underlying probability measure and the associated expectation are denoted by P θ,µ θ and E θ,µ θ . We assume that (X n ) n∈IN admits an invariant probability measure denoted by π θ (i.e. ∀θ ∈ Θ, π θ • Q θ = π θ ). Notice that we do not require stationarity for (X n ) n∈IN .

Let {ξ p (•, •); p ∈ P} be a family of measurable functions from E 2 into IR, where P is any set. Let us dene the following r.v.

∀n ≥ 1, ∀p ∈ P, S n (p) := n k=1 ξ p (X k-1 , X k ). (7)
This kind of (parametric and bivariate) functionals is required when concerning with Markovian M -estimators, as detailed in Section 5.

Now we are going to study Hypotheses R(m) and (N-A) using the Nagaev-Guivarc'h spectral method. For all t ∈ IR, (θ, p) ∈ Θ × P and x ∈ E, let us dene the Fourier kernel of (Q θ , ξ p ) by Q θ,p (t)(x, dy) := e itξp(x,y) Q θ (x, dy).

As usual, for all bounded measurable C -valued function f on E, we set

Q θ,p (t)f := E f (y)e itξp(•,y) Q θ (•, dy).
It is easy to see that we have from Markov property

∀t ∈ IR, ∀(θ, p) ∈ Θ × P, ∀n ≥ 1, E θ,µ θ [e itSn(p) f (X n )] = µ θ [Q θ,p (t) n f ].
In particular, we obtain

∀t ∈ IR, ∀(θ, p) ∈ Θ × P, ∀n ≥ 1, E θ,µ θ [e itSn(p) ] = µ θ [Q θ,p (t) n 1 E ], (9)
where 1 E stands for the function identically equal to 1 on E.

Equality

lim n→+∞ sup θ∈Θ Q n θ -Π θ B = 0,
where Π θ denotes the rank-one projection dened on B by Π θ f := π θ (f )1 E .

More precisely, we use the following equivalent form (ERG.2') of (ERG.2):

ERG.2'. : There exist c 0 > 0 and 0 ≤ κ 0 < 1 (independent on θ ∈ Θ) such that

∀θ ∈ Θ, ∀n ∈ IN , Q n θ -Π θ B ≤ c 0 κ n 0 .
Note that under (ERG.2'), for all θ ∈ Θ, the spectrum σ(Q θ|B ) of Q θ acting on B belongs to the set {z ∈ C ; |z| ≤ κ 0 } ∪ {1}.

Then, to derive the properties of R(m) from (9), we need some spectral perturbation method to control (uniformly in (θ, p) ∈ Θ × P) the spectrum of Q θ,p (t) acting on B whenever |t| is small enough. The usual method requires the continuity at t = 0 of the L(B)-valued function t → Q θ,p (t), but this continuity assumption involves too strong hypotheses (see [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]3] for details). An alternative method consists in using the Keller-Liverani theorem [START_REF] Keller | Stability of the spectrum for transfer operators[END_REF][START_REF] Liverani | Invariant measure and their properties. a functional analytic point of view[END_REF] (see also [START_REF] Baladi | Positive Transfer Operators and Decay of Correlations[END_REF][START_REF] Ferré | Synthesis document on Keller Liverani theorem[END_REF]). Using this method, the regularity of λ θ,p (•), l θ,p (•) and r θ,p,n (•) is studied in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] in the case of ρ-mixing Markov chains, V -geometrically ergodic Markov chains and for iterated function systems. More exactly, their results are only established for additive univariate functionals of (X n ) n∈IN * , but the extension to our parametric bivariate case (7) is quite natural. This work has already been done in [HLP] in the case of V -geometrically ergodic Markov chains (in Section 4, we will directly use their results).

By contrast, as already mentioned in Introduction, the method developed in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] is not sucient to investigate Hypothesis (N-A) in our parametric bivariate case. We can all the same easily state the following implication: thanks to (9), provided that the following condition is imposed on (µ θ ) θ∈Θ :

{µ θ ; θ ∈ Θ} is bounded in B ( 10
)
and that for all t ∈ IR, (θ, p) ∈ Θ × P, the operator Q θ,p (t) belongs to L(B), the family {S n (p) := n k=1 ξ p (X k-1 , X k ); n ∈ IN * , p ∈ P} fullls Hypothesis (N-A) whenever the following condition holds: Hypothesis (N-A)' (Operator-type non-arithmeticity). For any compact subset K 0 ⊂ IR * , there exists ρ < 1 such that

∀n ≥ 1, sup Q θ,p (t) n B ; t ∈ K 0 , (θ, p) ∈ Θ × P = O(ρ n ).
In the next section, we replace this condition by some more practical non-lattice conditions.

3 From non-lattice conditions to (N-A)'

We assume that the general Markovian assumptions of the previous Subsection 2.2 hold true. Furthermore, we also assume that there exists a Banach space B of complex measurable functions dened on E which contains the function 1 E , such that for all θ ∈ Θ, π θ ∈ B and such that for all t ∈ IR, (θ, p) ∈ Θ × P, the Fourier operators Q θ,p (t) dened in (8) belong to L(B).

Let us introduce the following non-lattice condition which will be proved (under some additional conditions) to imply the previous operator-type non-arithmetic condition (N-A)'.

Hypothesis (N-L) (Non-lattice). There exist no (θ 0 , p 0 ) ∈ Θ × P, no real a = a(θ 0 , p 0 ), no closed subgroup H = cZ Z with c = c(θ 0 , p 0 ) ∈ IR * , no π θ 0 -full Q θ 0 -absorbing set3 A = A(θ 0 , p 0 ) ∈ E, and nally no measurable bounded function α = α(θ 0 , p 0 ) :

E → IR such that ∀x ∈ A, ξ p 0 (x, y) + α(y) -α(x) ∈ a + H Q θ 0 (x, dy) -a.s.. (11)

Intermediate conditions

The link between (N-L) and (N-A)' is based on the three following operator-type properties.

The rst one concerns a control of the spectral radius of Q θ,p (t) acting on B denoted by

r(Q θ,p (t) |B ): ∀t = 0, ∀(θ, p) ∈ Θ × P, r Q θ,p (t) |B < 1. (i)
The second property consists in assuming that one has for any compact subset

K 0 ⊂ IR * r K 0 := sup r Q θ,p (t) |B ; t ∈ K 0 , (θ, p) ∈ Θ × P < 1. (ii)
Notice that, whenever (ii) holds true, for all z ∈ C , |z| > r K 0 and for all t ∈ K 0 , (θ, p) ∈ Θ×P, the resolvent operator (z -Q θ,p (t)) -1 is well-dened in L(B). Then the last property consists in assuming that there exists ρ 0 ∈ [r K 0 , 1) such that, for all ρ ∈ (ρ 0 , 1),

sup (z -Q θ,p (t)) -1 B ; t ∈ K 0 , (θ, p) ∈ Θ × P, |z| = ρ < +∞. (iii)
Below we study the following implications:

(a) (N-L) ⇒ (i) under some conditions (and even better: (N-L) ⇔ (i) under some more conditions)

(b) (i) ⇒ (ii)-(iii) under some conditions; (c) (ii)-(iii) ⇒ (N-A)'.
The main diculty is the proof of the statement (b). For this part, three methods are proposed in Subsection 3.3. Notice that the operator-type non-arithmetic condition (N-A)' obviously implies Property (i).

3.2 From the non-lattice condition (N-L) to Property (i)

The following lemma is an easy extension of [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]12] to our parametric bivariate case.

Lemma 1. Assume that the following assumptions hold true:

1. For all θ ∈ Θ, λ ∈ C such that |λ| ≥ 1, and for all f ∈ B, f = 0, we have

∀n ≥ 1, |λ| n |f | ≤ Q n θ |f | ⇒ |λ| = 1 and |f | = π θ (|f |) > 0 π θ -a.s. .
2. For all (θ, p) ∈ Θ × P, t ∈ IR * , there exists 0 ≤ γ = γ(θ, p, t) < 1 such that the elements of the spectrum of Q θ,p (t) acting on B with modulus greater than γ are isolated eigenvalues of nite multiplicity.

Assume that (N-L) holds true as well. Then (i) is fullled. Moreover Property (i) is equivalent to the following condition: there exist no t 0 ∈ IR * , no (θ 0 , p 0 ) ∈ Θ×P, no λ = λ(θ 0 , p 0 , t 0 ) ∈ C such that |λ| = 1, no π θ 0 -full Q θ 0 -absorbing set A = A(θ 0 , p 0 , t 0 ) ∈ E and nally no bounded w = w(θ 0 , p 0 , t 0 ) ∈ B such that |w| |A is non-null constant, satisfying ∀x ∈ A, e it 0 ξp 0 (x,y) w(y) = λw(x) Q θ 0 (x, dy) -a.s..

The last property of Lemma 1 will not be used later, it is only recalled here for a better understanding.

Remark 2. In fact, Property (i) is equivalent to (N-L) whenever e iψ ∈ B for all bounded real measurable function ψ on E. Notice that this assumption is obviously fullled in the V -geometrically ergodic Markovian model to be studied.

Assumption 1. of Lemma 1 is always satised for strongly ergodic models (cf. (ERG.2)) such that for all x ∈ E, the Dirac distribution δ x at x belongs to B . In particular, this assumption is satised by the V -geometrically ergodic Markovian model to be studied (other conditions are given in [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] to check Assumption 1.).

Assumption 2. is much more dicult to be checked. For now, we only mention that it is equivalent to the following condition: for all (θ, p) ∈ Θ × P, t ∈ IR * , the essential spectral radius

r ess (Q θ,p (t) |B ) of Q θ,p (t) acting on B is such that r ess (Q θ,p (t) |B ) ≤ γ < 1. Recall that Q θ,p (t) is said to be quasi-compact on B whenever r ess (Q θ,p (t) |B ) < r(Q θ,p (t) |B ).

Three methods for Condition (i) to imply (ii)-(iii)

To obtain the implication (i) ⇒ (ii)-(iii), we can use one of the following three approaches, in which the sets Θ and P are assumed to be compact.

• First approach. Using the standard operator perturbation theory, specically the upper- semi-continuity of the function "spectral radius" (see e.g. [HH01, p 19]), one can prove the following statement:

Assume that Q θ,p (t) -Q θ 0 ,p 0 (t 0 ) B → 0 when (t, θ, p) → (t 0 , θ 0 , p 0 ). Then the implications (i) ⇒ (ii) ⇒ (iii) are true.

However, as already mentioned in Subsection 2.2, the last assumption of continuity of t → Q θ,p (t) is too restrictive. That is why we will not apply this approach in this work.

• Second approach. It consists in using the perturbation Keller-Liverani theorem instead of the standard perturbation theory. The proof of the following proposition is not provided in this paper since it is an easy extension of [HP10, lem 12.3].

Proposition 2. Assume that there exists some semi normed space B such that for all t ∈ IR and (θ, p) ∈ Θ × P, Q θ,p (t) belongs to L( B) and B → B (i.e. B ⊂ B and the identity map is continuous from B into B). Furthermore assume that for all t 0 ∈ IR * and (θ 0 , p 0 ) ∈ Θ × P, there exists a neighborhood I 0 ⊂ IR of (t 0 , θ 0 , p 0 ) such that (C1) there exist c > 0, 0 ≤ κ < 1 and M > 0 such that for all (t, θ, p)

∈ I 0 , f ∈ B, n ∈ IN , one has Q θ,p (t) n f B ≤ c κ n f B + c M n f B . (C2) Q θ,p (t) -Q θ 0 ,p 0 (t 0 ) B, B → 0 when (t, θ, p) → (t 0 , θ 0 , p 0 ).
Then the implications (i) ⇒ (ii) ⇒ (iii) are true.

This second approach is applied in Subsection 6.4 to some AR(d) processes with d ≥ 2.

Note that Condition (C2) may be dicult to be checked because of the continuity with respect to θ. However, under the standard dominated model assumption, Condition (C2) can be dodged using the following approach.

• Third approach. It consists in using quasi-compactness and Ascoli-type arguments. For instance, let us give a brief account for the implication (i) ⇒ (ii). We assume by absurd that (ii) does not hold and (i) holds true, namely on the one hand there exists a compact subset

K 0 of IR * such that r K 0 := sup{r(Q θ,p (t) |B ); t ∈ K 0 , (θ, p) ∈ Θ × P} ≥ 1,
and on the other hand r K 0 ≤ 1 < +∞. Then there exist some sequences

(t k ) k∈IN ∈ K IN 0 and (θ k , p k ) k∈IN ∈ (Θ × P) IN such that lim r(Q θ k ,p k (t k )) ≥ 1 when k → +∞.
Under the quasi-compactness Assumption 2. of Lemma 1, the previous property implies the existence of

(λ k ) k∈IN ∈ C IN and (w k ) k∈IN ∈ B IN such that Q θ k ,p k (t k )w k = z k w k and |λ k | = r(Q θ k ,p k (t k )).
Finally, from compactness arguments (in particular by using Ascoli theorem), there exist some

t ∈ K 0 , ( θ, p) ∈ Θ × P, λ ∈ C , | λ| ≥ 1, and w ∈ B such that Q θ,p ( t)
w = λ w, which is in contradiction with (i). Similar arguments can be used to prove (ii) ⇒ (iii). In practice, it is easier to use Ascoli theorem when the model is dominated (i.e. ∀x ∈ E, ∀θ ∈ Θ, Q θ (x, dy) = q θ (x, y) µ(dy)) with suitable conditions on the function (q θ ) θ∈Θ and on the dominating positive measure µ.

This approach is detailed in Subsection 4.2 for V -geometrically ergodic Markov chains and then it is applied in Subsection 5.2 to AR(1) processes with Gaussian noise.

From Properties (ii)-(iii) to the operator-type non-arithmetic condition (N-A)'

Lemma 2. Assume that Properties (ii)-(iii) hold true. Then (N-A)' is fullled.

Proof of Lemma 2. Let K 0 ⊂ IR * be any compact set and let Γ denote the oriented circle dened by {z ∈ C ; |z| = ρ} where ρ ∈ (ρ 0 , 1). From Von Neumann series, we have for all t ∈ K 0 and (θ, p) ∈ Θ × P,

z ∈ C , |z| = ρ ⇒ (z -Q θ,p (t)) -1 = +∞ n=0 z -n-1 Q θ,p (t) n
and hence, we obtain

∀t ∈ K 0 , ∀(θ, p) ∈ Θ × P, ∀n ≥ 1, Q θ,p (t) n = 1 2iπ Γ z n (z -Q θ,p (t)) -1 dz.
Then (N-A)' can easily be derived thanks to (iii).

Conclusion

Subsections 3.2, 3.3 and 3.4 give a procedure to derive (N-A)' (and so (N-A)) from the nonlattice condition (N-L). In some cases, we may need some even more simple condition than (N-L) to check (N-A). However, notice that this new condition, denoted by (N-L)', is not equivalent to (N-L).

Assume that the set E is topological and let E := B(E) be the associated Borel algebra.

Hypothesis (N-L)'. For all p ∈ P, there do not exist A p (•) and C p such that we have for all

(x, y) ∈ E 2 , ξ p (x, y) = A p (y) -A p (x) + C p .
To connect (N-L)' with (N-L), we need the following hypotheses on both the model and (ξ p ) p∈P .

Hypothesis (S). There exists a positive measure µ on E satisfying Supp(µ) = E and such that we have for any B ∈ E:

[ ∃ (θ, x) ∈ Θ × E, Q θ (x, B) = 0 ] =⇒ [ µ(B) = 0 ].
Hypothesis (C). For all p ∈ P, the application ξ p is continuous from E 2 into IR.

Lemma 3. Assume that the set E is connex and that Assumptions (S) and (C) hold true. If the family (ξ p ) p∈P fullls (N-L)', then (N-L) is fullled.

Proof of Lemma 3. Assume that (N-L) is not fullled, that is we have (11) with some

(θ 0 , p 0 ) ∈ Θ×P, a ∈ IR, some closed subgroup H = cZ Z (c ∈ IR * ), some π θ 0 -full Q θ 0 -absorbing set A ∈ E,
and nally some bounded measurable function α : E → IR. For the sake of simplicity, let us omit the dependence on (θ 0 , p 0 ). For all x ∈ A, there exists

E x ∈ E such that Q(x, E x ) = 1 and ∀y ∈ E x , ξ(x, y) + α(y) -α(x) ∈ a + H. Let x 0 ∈ A. One has ∀y ∈ E x 0 , ξ(x 0 , y) + α(y) -α(x 0 ) ∈ a + H ∀x ∈ E x 0 , ξ(x 0 , x) + α(x) -α(x 0 ) ∈ a + H.
Thanks to Assumption (S), one has µ(E\E x 0 ) = 0 and µ(E\A) = 0 (recall that A is Qabsorbing), and hence µ(E\{A

∩ E x 0 }) = 0, A ∩ E x 0 ⊃ Supp(µ) = E where A ∩ E x 0 denotes the closure of A ∩ E x 0 . In particular, A ∩ E x 0 is not empty. Let x ∈ A ∩ E x 0 , then ∀y ∈ E x 0 ∩ E x , ξ(x, y) -(ξ(x 0 , y) -ξ(x 0 , x)) ∈ a + H. Let us dene A(x) := ξ(x 0 , x) and f (x, y) := ξ(x, y) + A(x) -A(y). Then for all x ∈ A ∩ E x 0 , f (x, E x 0 ∩ E x ) ⊂ a + H.
Then, by continuity arguments and since

E = Supp(µ) = E x 0 ∩ E x , one can easily show that f (x, E) ⊂ a + H. In the same way, f (A ∩ E x 0 , E) ⊂ a + H, and nally f (E, E) ⊂ a + H. Since f (E, E) is connex and a + H is discrete, f is constant on E 2 . 2
Remark 3. Let µ be a positive measure on E satisfying Supp(µ) = E. Assume that the following dominated model condition holds: for all θ ∈ Θ, there exists a non-negative measurable ap-

plication q θ (•, •) on (E ×E, E ⊗E) such that for all x ∈ E, B ∈ E, Q θ (x, B) = B q θ (x, y) dµ(y)
and for all x ∈ E and for µ-almost all y ∈ E, q θ (x, y) > 0. Then one can show that Assumption (S) holds true.

4 The case of uniform V -geometrically ergodic Markov chains

In this section, we illustrate the previous results for uniform V -geometrically ergodic Markov chains. From now on, for the sake of simplicity, we consider that E := IR d (with d ∈ IN * ), equipped with any norm • , and µ Leb d denotes the Lebesgue-measure on E. Let us assume that Θ is a compact set. We introduce the uniform (in θ ∈ Θ) V -geometrically ergodic Markovian model, which satises Properties (ERG.1)-(ERG.2) on the weighted-supremum normed space associated with V .

Model (M). For all θ ∈ Θ, there exist both a Q θ -invariant probability measure denoted by π θ and an unbounded function

V : E → [1, +∞) such that (VG1) sup θ∈Θ π θ (V ) < +∞, (VG2) lim n→+∞ sup |Q n θ f (x) -π θ (f )|/V (x); f : E → C measurable, |f | ≤ V, x ∈ E, θ ∈ Θ = 0.
Model (M) has already been considered for statistical investigation, see for instance [START_REF] Fuh | Ecient likelihood estimation in state space models[END_REF][START_REF] Dehay | On likelihood estimation for discretely observed Markov jump processes[END_REF][START_REF] Hervé | A Berry-Esseen theorem on M-estimators for V-geometrically ergodic Markov chains[END_REF]. When θ is xed and when the Markov chain is irreducible and aperiodic, (VG1) and (VG2) can be checked using the so-called drift-criterion, we refer to [MT93, p 367] for details. Notice that Condition (10) on initial the distribution is equivalent to the following one for Model (M): sup

θ∈Θ µ θ (V ) < +∞. (12)
In the next Subsections 4.1 and 4.2, we consider a family (ξ p ) p∈P of measurable functions from E 2 into IR, with P assumed to be a compact set, and we successively study Hypotheses R(m) and (N-A) for Model (M), before applying these results in Section 5 to M -estimators.

Study of Hypothesis R(m)

Let us recall the following proposition which has already been proven in [HLP, lem 1].

Proposition 3. Let us consider a Model (M). Assume on (ξ p ) p∈P that for all (θ, p) ∈ Θ × P, ξ p is centered with respect to the invariant measure family (π θ ) θ∈Θ (i.e. for all (θ, p) ∈ Θ × P, E θ,π θ [ξ p (X 0 , X 1 )] = 0), and that assume that (ξ p ) p∈P fullls the following moment domination condition for some m ∈ IN :

∃ε > 0, sup |ξ p (x, y)| m+ε V (x) + V (y) ; (x, y) ∈ E 2 , p ∈ P < +∞. (D m )
Finally assume that the initial distribution family (µ θ ) θ∈Θ satises (12). Then the family

{S n (p) := n k=1 ξ p (X k-1 , X k ); n ∈ IN * , p ∈ P} satises Hypothesis R(m).
Up to the arbitrarily small real number ε > 0, Condition (D m ) is the expected (with respect to the i.i.d. case) assumption to obtain Hypothesis R(m) in our model. Indeed, in [START_REF] Dehay | On likelihood estimation for discretely observed Markov jump processes[END_REF], Condition (D 2 ) is the key assumption to prove the asymptotic normality whereas in [HLP],

Condition (D 3 ) is the key assumption to prove Berry-Esseen bounds. Here one also needs to investigate Hypothesis (N-A).

Study of Hypothesis (N-A) for dominated Models (M)

Further assumptions are required to apply what we called the third approach in Subsection 3.3. Some of them concern the dominated model and the other ones involve the regularity of the applications (ξ p ) p∈P .

Assumption (S ). For all θ ∈ Θ, there exists an application q θ (•,

•) on E 2 such that ∀x ∈ E, Q θ (x, dy) = q θ (x, y) µ Leb d (dy).
Furthermore for all x ∈ E and for µ Leb d -almost all y ∈ E, the application θ → q θ (x, y) is continuous and there exists β > 0 such that

• for all θ 0 ∈ Θ, there exists a neighborhood V 1 = V 1 (θ 0 ) of θ 0 such that ∀x 0 ∈ E, lim x→x 0 sup θ∈V 1 E V (y) β |q θ (x, y) -q θ (x 0 , y)| µ Leb d (dy) = 0.
• for all x 0 ∈ E and θ 0 ∈ Θ, there exists a neighborhood

V 2 = V 2 (x 0 , θ 0 ) of θ 0 such that E V (y) β sup θ∈V 2 |q θ (x 0 , y)|µ Leb d (dy) < +∞.
Assumption (C ). The family (ξ p ) p∈P satises

• for all x ∈ E and for µ Leb d -almost all y ∈ E, the function p → ξ p (x, y) is continuous. • for all x 0 ∈ E and p 0 ∈ P, there exist neighborhoods V 3 = V 3 (x 0 , p 0 ) of x 0 and V 4 = V 4 (p 0 ) of p 0 , some positive numbers C, υ 1 and υ 2 such that we have

∀p ∈ V 4 , ∀x ∈ V 3 , ∀y ∈ E, |ξ p (x, y) -ξ p (x 0 , y)| ≤ C x -x 0 υ 1 V (y) υ 2 .
Theorem 1. Let us consider a Model (M), and assume that the preceding assumptions (S ) and (C ) hold true. If the non-lattice condition (N-L) of Section 3 holds true and if the family of initial distributions (µ θ ) θ∈Θ satises (12), then

{S n (p) := n k=1 ξ p (X k-1 , X k ); n ∈ IN * , p ∈ P} satises Hypothesis (N-A).
As discussed in Subsection 2.2, to check Hypothesis (N-A), we need a Banach space B composed of complex measurable functions dened on E, containing the function 1 E , such that for all θ ∈ Θ, π θ ∈ B , and such that for all t ∈ IR, (θ, p) ∈ Θ × P, the Fourier operator Q θ,p (t) belongs to L(B). From (VG2), the natural space for this job is the Banach space B 1 composed of measurable functions f :

E → C such that f B 1 := sup x∈E |f (x)| V (x) < +∞. (13)
Actually, for a technical reason arising in Lemma 5 below, we need to work with another space. Let β be given in Assumption (S ). Without loss of generality, one can suppose that β ∈ (0, 1).

Then we consider the Banach space B β composed of measurable functions f : E → C such that

f B β := sup x∈E |f (x)| V (x) β < +∞. (14)
Notice that for any Model (M), using the drift-criterion (cf. [START_REF] Meyn | Markov chains and stochastic stability[END_REF]) and Jensen inequality, we can prove that (see [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]10])

lim n→+∞ sup θ∈Θ Q n θ -Π θ B β = 0. (15)
Then, Assumption (ERG.2) of Subsection 2.2 holds true with B := B β . More precisely, we will use the equivalent form (ERG.2') of (ERG.2): there exist c β > 0 and 0

≤ κ β < 1 (independent on θ ∈ Θ) such that ∀θ ∈ Θ, ∀n ∈ IN , Q n θ -Π θ B β ≤ c β κ n β . (16)
Proof of Theorem 1. Let denote := (t, θ, p) ∈ IR × Θ × P and Q( ) := Q θ,p (t). First of all, notice that, since B β is a Banach lattice (i.e. for all (f, g)

∈ B β × B β , |f | ≤ |g| ⇒ f B β ≤ g B β )
and using (15), we can apply [RW97, cor 1.6] to prove that the essential spectral radius of Q( ) satises

∃ 0 ≤ κ < 1 such that ∀ 0 ∈ IR * × Θ × P, r ess (Q( 0 ) |B β ) ≤ κ. (17)
Next, let us sum up the gap from (N-L) to (N-A), specifying their link with all the intermediate conditions introduced in Subsection 3.1:

• thanks to the previous Inequality (17) on the essential spectral radius of Q( ), Assumption 2. of Lemma 1 holds true (Assumption 1. of Lemma 1 also holds true: see the comments after Lemma 1). Thus the conclusions of Lemma 1 are satised: (N-L) ⇒ (i);

• thanks to Lemma 2: (ii)-(iii) ⇒ (N-A)' with B := B β ;

• from Condition (12): (N-A)' ⇒ (N-A).
Next, it only remains to prove that (i) ⇒ (ii)-(iii). In fact, we show that (i) ⇒ (ii) ⇒ (iii), using quasi-compactness and Ascoli-type arguments, as announced in the third approach of Subsection 3.3. The proof of (i) ⇒ (ii) ⇒ (iii) involves the two following Lemmas 4 and 5.

Lemma 4 (Doeblin-Fortet Inequality). For any Model (M), there exist 16) and (VG1). Indeed, for all f ∈ B β , |f | ∈ B β , and hence one has

c β > 0, 0 ≤ κ β < 1, such that ∀ ∈ IR × Θ × P, ∀f ∈ B β , ∀n ∈ IN , Q( ) n f B β ≤ c β κ n β f B β + c β f B 1 . (D-F) Proof of Lemma 4. Doeblin-Fortet Inequality (D-F) is a consequence of Q θ,p (t) n (f ) B β ≤ Q n θ (|f |) B β (since B β is a Banach lattice) and (
Q n θ (|f |) -π θ (|f |) B β ≤ c β κ n β f B β
, from which we easily deduce the desired inequality.

2 Lemma 5. Let (w k ) k∈IN ∈ (B β ) IN such that w k B β = 1 for all k ≥ 1. If (w k ) k∈IN uniformly converges to w ≡ 0 on any compact subset of E, then sup θ∈Θ w k B 1 → 0 when k → +∞.
Proof of Lemma 5. Let ε > 0, ε := 1 -β, and let

K = K ε,ε be a compact subset of E such that sup x∈E\K V (x) -ε ≤ ε. Since |w k (x)| ≤ w k B β V (x) β = V (x) β , one has ∀k ∈ IN , w k 1 |E\K B 1 ≤ V β 1 |E\K B 1 = sup x∈E\K V (x) β V (x) ≤ ε. (18)
Furthermore

sup x∈K |w k (x)|/V (x) ≤ sup x∈K |w k (x)| → k 0, thus there exists k 0 ∈ IN such that ∀k ≥ k 0 , sup x∈K |w k (x)| V (x) = w k 1 |K B 1 ≤ ε. ( 19
)
By combining ( 18) and ( 19),

w k B 1 = max w k 1 |E\K B 1 , w k 1 |K B 1 ≤ ε. 2
We are now ready to complete the proof of Theorem 1.

Lemma 6. We have (i) ⇒ (ii).

Proof of Lemma 6. We assume by absurd that (ii) does not hold and (i) holds true, namely on the one hand there exists a compact subset

K 0 of IR * such that r K 0 := sup{r(Q( ) |B β ), ∈ K 0 × Θ × P} ≥ 1,
and on the other hand r K 0 ≤ 1 < +∞. Thus there exists

( k ) k∈IN ∈ (K 0 × Θ × P) IN such that lim r(Q( k ) |B β ) = r K 0 when k → +∞, and for all k ≥ 0, r(Q( k ) |B β ) > κ,
where κ is dened in Inequality (17) on the essential spectral radius of Q( ). Then for all k ≥ 0, there exists an eigenvalue λ k such that

|λ k | = r(Q( k ) |B β ). Let w k ∈ B β , w k = 0, w k β = 1, such that Q( k )w k = λ k w k . ( 20
)
By compacity argument, we can suppose lim k := = ( t, θ, p) and lim λ k := λ when k → +∞, with ∈ K 0 × Θ × P and | λ| = r K 0 ≥ 1.

a) (w k ) k converges on E to some w ∈ B β : Under the rst point of (S ) and the second one of (C ), and using Ascoli theorem, it is easy to see that

(Q( k )w k ) k≥k 0 is relatively compact in (C(K, IR), • ∞ ) for any compact subset K of E.
By diagonal extraction, we can suppose that (Q( k )w k ) k∈IN converges pointwise on E and uniformly on any compact subset of E, and so does the sequence (w k ) k∈IN thanks to Equality (20). Its limit is denoted by w ∈ B β .

b) w = 0: From Doeblin-Fortet Inequality (D-F), from Equality (20) which implies

Q( k ) n w k = λ n k w k for all n ∈ IN * , and from w k B β = 1, one obtains |λ k | n ≤ c β κ n β + w k B 1 . Suppose that w = 0. Then w k B 1 → 0 when k → +∞ thanks to Lemma 5. Since |λ k | → | λ| = r K 0 when k → +∞, one has for all n ∈ IN , r n K 0 ≤ c β κ n β , which is in contradiction with the fact that κ β < 1 ≤ r K 0 . Consequently w = 0. c) Conclusion: Let x 0 ∈ E. From Assumption (S ), we have Q( k )w k (x 0 ) := E w k (y)e it k ξp k (x 0 ,y) q θ k (x 0 , y) µ Leb d (dy).
Then, under the second point of (S ) and the rst one of (C ), and using Lebesgue dominated convergence theorem, one has Q( k )w k (x 0 ) → k Q( ) w(x 0 ). We have just proven that there exist λ ∈ C , | λ| = r K 0 ≥ 1, a non-null function w ∈ B β and nally a parameter ∈ K 0 ×Θ×P such that Q( ) w = λ w. This fact implies that λ ∈ σ(Q( ) |B β ), which is in contradiction with (i). 2

Lemma 7. We have (ii) ⇒ (iii).

Proof of Lemma 7. Let K 0 ⊂ IR * be compact. From (ii), we have r

K 0 := sup{r(Q( ) |B β ), ∈ K 0 ×Θ×P} < 1.
By absurd, we assume that there exists ρ be such that max(r K 0 , κ β ) < ρ < 1 (where κ β is dened in (D-F)) and such that sup

|z|=ρ sup ∈K 0 ×Θ×P { (z -Q( )) -1 B β } = +∞. Thus there exist ( k , z k ) k∈IN ∈ (K 0 × Θ × P) IN × C IN , |z k | = ρ, such that α k := (z k -Q( k )) -1 B β → +∞ when k → +∞, which implies by Banach-Steinhaus theorem that there exists f ∈ B β satisfying (z k -Q( k )) -1 f B β → +∞. Let w k := (z k -Q( k )) -1 f /α k and ε k := f /α k ∈ B β . Then one has Q( k )w k = z k w k -ε k . ( 21
)
By compacity argument, we can suppose that lim k→+∞ k := = ( t, θ, p) and lim k→+∞ z k := z, with ∈ K 0 × Θ × P, and | z| = ρ.

a) (w k ) k converges on E to some w ∈ B β : Again from Ascoli theorem, diagonal extraction and (21), we can suppose that (w k ) k converges pointwise on E and uniformly on any compact set of E, and we denote its limit by w ∈ B β .

b) w = 0: From (21), one can easily show

∀n ∈ IN , z n k w k = Q( k ) n w k + n-1 i=0 z i k Q( k ) n-1-i ε k . (22)
From (D-F), one has for all

k = (t k , θ k , p k ) ∈ IR × Θ × P and n ∈ IN : Q( k ) n ε k B β ≤ C n ε k B β where C n := c β (κ n β + b 1 ). Recall that |z k | = ρ.
Thus considering again Equality ( 22) and (D-F), we obtain

ρ n w k B β ≤ c β κ n β w k B β + w k B 1 + n-1 i=0 ρ i C n-i-1 ε k B β .
Suppose that w = 0, then w k B 1 → k 0 using Lemma 5. Since

w k B β = 1 and ε k B β = f B β /α k → k 0, one has for all n ∈ IN : ρ n ≤ c β κ n β ,
which is in contradiction with the fact that ρ > κ β . Thus we have just proven that w = 0. c) Conclusion: Using Lebesgue dominated convergence theorem, one has for all x ∈ E:

Q( k )w k (x) → k Q( ) w(x).
We have just proven that there exist z ∈ C , | z| = ρ, a nonnull function w ∈ B β and a parameter ∈ K 0 × Θ × P such that Q( )w = z w. This fact implies that r(Q( ) |B β ) ≥ ρ, which is in contradiction with the fact that ρ > r K 0 . Thus we have just proven by absurd that (ii) ⇒ (iii).

5 M -estimators associated with V -geometrically ergodic Markov chains. Examples

Let (X n ) n≥0 be a Markov chain with state space E := IR d and transition kernel (Q θ (x, •); x ∈ E), where θ is a parameter in some compact set Θ. The probability distribution of X 0 is denoted by µ θ . As before, the underlying probability measure and the associated expectation are denoted by P θ,µ θ and E θ,µ θ .

Let us introduce the parameter of interest α = α(θ) ∈ A where A is an open interval of IR.

To dene the so-called true value of the parameter of interest α 0 = α 0 (θ) ∈ A, we introduce the empirical mean functional

∀α ∈ A, ∀n ∈ IN * , M n (α) := 1 n n k=1 F (α, X k-1 , X k ), (23)
where

F (•, •, •) is a real-valued measurable function on A × E 2 .
For instance, -M n may be the log-likelihood of data (X 0 , . . . , X n ). We dene α 0 as follows

∀θ ∈ Θ, α 0 (θ) := arg min α∈A lim n→+∞ E θ,µ θ [M n (α)], (24)
and its M -estimator is supposed to be well-dened by

∀n ∈ IN * , αn := arg min α∈A M n (α). ( 25
)
Our goal is to provide an asymptotic expansion of P θ,µ θ { √ n( αn -α 0 )/σ(θ) ≤ u} uniformly in θ ∈ Θ and u ∈ IR, where σ is some suitable (asymptotic) standard deviation. As in the i.i.d. case (see for example [START_REF] Pfanzagl | Asymptotic expansions related to minimum contrast estimators[END_REF]), we assume throughout this section that the following hypotheses on (α n ) n∈IN * hold true:

HYP. 1. ∀n ≥ 1, (∂M n /∂α) (α n ) = 0, HYP. 2. ∀d > 0, sup θ∈Θ P θ,µ θ |α n -α 0 | ≥ d = o(n -1 2 ) .
Notice that the uniform consistency property (HYP.2) has already been studied in a Markovian context, see for example [Bil61, Rou65, Rao72, Gän72, DY07].

Throughout the sequel, we assume that (X n ) n∈IN belongs to the class of Models (M) (namely (X n ) n∈IN is V -geometrically ergodic uniformly in θ) and that the family of initial distributions (µ θ ) θ∈Θ satises (12). In particular, this last condition will be satised if µ θ ≡ π θ (see (VG1)), or if µ θ ≡ δ x , where δ x is the Dirac distribution at any x ∈ E. Then, under some further conditions on the model and on the function F , we prove4 that there exists a polynomial function A θ (•) such that

sup θ∈Θ sup u∈IR P θ,µ θ √ n σ(θ) (α n -α 0 ) ≤ u -N (u) -η(u) n -1 2 A θ (u) = o(n -1 2 ). (26)
Notice that the true value of the parameter of interest (see ( 24)) can also be dened by

∀θ ∈ Θ, ∀α ∈ A, α = α 0 , E θ,π θ [F (α, X 0 , X 1 )] > E θ,π θ [F (α 0 , X 0 , X 1 )].
Asymptotic expansions for M -estimators in the Markovian case have already been studied in several papers. Indeed maximum likelihood estimators are fully studied in [START_REF] Dahlhaus | Ecient parameter estimation for self similar processes[END_REF] and [START_REF] Lieberman | Valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process[END_REF] in the specic case of stationary Gaussian processes. Some M -estimators for general non-stationary models are also studied in [START_REF] Götze | Asymptotic distribution of statistics in time series[END_REF] and [START_REF] Fuh | Ecient likelihood estimation in state space models[END_REF], but each author needs some additional Cramér-type hypothesis. Here we only need the much weaker non-arithmeticity condition. Furthermore our moment conditions on F and its derivatives are almost optimal with respect to the i.i.d. case, see the comments after Theorem 2.

Edgeworth expansion for M -estimators for dominated models (M)

In addition to the previous assumptions (namely (X n ) n∈IN belongs the class of Models (M) and (α n ) n∈IN * satises (HYP.1) and (HYP.2)), we assume that (X n ) n∈IN is dominated, i.e. that Condition (S ) holds true (see its denition in Subsection 4.2). Furthermore, we assume that for all x ∈ E and for µ Leb d -almost all y ∈ E, we have q θ (x, y) > 0. Let us introduce the assumptions concerning the real-valued measurable function F involved in (23). Assume that the map α → F (α, •, •) is 3-time-dierentiable on A and let F (j) := ∂ j F/∂α j denote the derivatives for j = 1, 2, 3. Assume that F (1) , F (2) , F (3) satisfy the following moment domination condition (D 3 ):

∃ε > 0 such that ∀j = 1, 2, 3, sup |F (j) (α, x, y)| 3+ε V (x) + V (y) ; (x, y) ∈ E 2 , α ∈ A < +∞. ( 27
)
We introduce for j = 1, 2, 3

∀θ ∈ Θ, m j (θ) := E θ,π θ F (j) (α 0 , X 0 , X 1 ) , σ j (θ) 2 := lim n→+∞ E θ,π θ n M (j) n (α 0 ) -m j (θ) 2 ,
where M n (•) is given in (23) and M (j) n := ∂ j M n /∂α j , and where π θ is the Q θ -invariant probability measure given in (M). Then, from (27) and using Proposition 1 and Proposition 3, the functions σ j (•) for j = 1, 2, 3 are well-dened and bounded in θ ∈ Θ.

We consider the following additional assumptions:

C. 1. m 1 ≡ 0 and inf θ∈Θ m 2 (θ) > 0. C. 2. inf θ∈Θ σ j (θ) > 0 for j = 1, 2.
C. 3. There exists a measurable function W : E → [0, +∞) of the type W = C V η for some η ∈ (0, 1/2) and C > 0 such that

∀(α, α ) ∈ A 2 , ∀(x, y) ∈ E 2 , |F (3) (α , x, y) -F (3) (α, x, y)| ≤ |α -α| (W (x) + W (y)).
Let us introduce some assumptions similar to (C)-(C ) (see denitions in Subsections 3.5 and 4.2) concerning the regularity of (F (j) ) j=1,2,3 . The function F is supposed to satisfy C. 4. For all j = 1, 2, 3 and α ∈ A, F (j) (α, •, •) is continuous from E 2 into IR.

C. 5. For all x 0 ∈ E and α ∈ A, there exist neighborhoods V 3 = V 3 (x 0 , α) of x 0 and V 4 = V 4 (α) of α, positive real numbers C, υ 1 and υ 2 such that for all α ∈ V 4 , x ∈ V 3 and y ∈ E:

∀j = 1, 2, 3, |F (j) (α , x, y) -F (j) (α , x 0 , y)| ≤ C x -x 0 υ 1 V (y) υ 2 .
Theorem 2. Assume that all the preceding assumptions hold true, that Condition (N-L)' (see denition page 12) is satised by the following functions

(a) ∀α ∈ A, ∀j = 1, 2, F (j) (α, x, y) (b) ∀α ∈ A, ∀υ ∈ IR, F (1) (α, x, y) + υF (2) (α, x, y) + (υ 2 /2) F (3) (α, x, y)
and that the initial probability measure satises (12), namely sup θ∈Θ µ θ (V ) < +∞. Then for

j = 1, 2, 3, ∀θ ∈ Θ, m j (θ) = lim n→+∞ E θ,µ θ [M (j) n (α 0 )] , σ j (θ) 2 = lim n→+∞ E θ,µ θ n M (j) n (α 0 ) -m j (θ) 2
and there exists a polynomial function denoted by A θ such that (α n ) n∈IN * satises Expansion (26) with σ := σ 1 /m 2 . Furthermore, the coecients of A θ are bounded, and

A θ (u) := - 1 6 m 3,1 (θ) 3 σ 1 (θ) 3 + b 1 (θ) σ 1 (θ) + 1 6 m 3,1 (θ) 3 σ 1 (θ) 3 - σ 12 (θ) σ 1 (θ)m 2 (θ) + σ 1 (θ) 2m 2 (θ) 2 m 3 (θ) u 2 ,
where

         b 1 (θ) := lim n→+∞ E θ,µ θ [n M (1) n (α 0 ) ] σ 12 (θ) := lim n→+∞ E θ,π θ n M (1) n (α 0 ) M (2) n (α 0 ) -m 2 (θ) = lim n→+∞ E θ,µ θ n M (1) n (α 0 ) M (2) n (α 0 ) -m 2 (θ) m 3,1 (θ) 3 := lim n→+∞ E θ,π θ n 2 M (1) n (α 0 ) 3 = lim E θ,µ θ n 2 M (1) n (α 0 ) 3 -3σ 2 1 (θ)b 1 (θ).
When comparing with Pfanzagl results [START_REF] Pfanzagl | Asymptotic expansions related to minimum contrast estimators[END_REF] in the i.i.d. case, Expansion (26) proven in Theorem 2 is a natural substitute of the i.i.d. expansion, with an additional term due to the asymptotic bias (namely b 1 (•)). To the best of our knowledge, Theorem 2 is the most precise statement concerning the rst-order Edgeworth expansion of real-valued Mestimators associated with V -geometrically ergodic Markov chains: in fact, the dominated model condition (S ) on the model is classical in Markovian statistics, the condition (D 3 ) on the derivatives of F is the expected one (up to the real number ε > 0), Conditions (C.1)-(C.5) are the Markovian substitutes of the so-called regularity conditions of the i.i.d. case and nally the non-lattice-type conditions (a)-(b) in Theorem 2 are quite general and easy to check.

The proof of Theorem 2 is postponed to Section 6.

An application to autoregressive models is given in the next subsection.

A simple illustration of Theorem 2: a Gaussian linear model

Let us consider a Gaussian linear process (X k ) k≥0 dened by

X 0 := 0 ∀k ∈ IN * , X k := θX k-1 + Z k (28)
where

• θ is a parameter which belongs to a compact set Θ ⊂ (-1, 1)

• (Z k ) k∈IN * are i.i.d. real valued r.v. whose common distribution is supposed to be the Gaussian distribution N (m, σ 2 ) with probability density f Z with respect to µ Leb .

This simple model allows us to easily illustrate the performance of Theorem 2. For the sake of simplicity, we have considered a Gaussian noise and the Dirac distribution δ 0 , but another probability density f Z > 0 and another initial distribution µ θ could be chosen.

The sequence (X k ) k≥0 is a Markov chain with state space E := IR. Its transition kernel Q θ is given for all Borel set B ∈ B(IR) by

Q θ (x, B) = IR 1 B (θx + z) f Z (z) dz.
First of all, note that

∀j > k, X j = θ j-k X k + j-k l=1 θ j-k-l Z k+l . (29)
Then for all k ≥ 1, the r.v. X k has distribution N m(1 -θ k )/(1 -θ), σ 2 (1 -θ 2k )/(1 -θ 2 ) , and the Markov chain (X k ) k≥0 converges in distribution to π θ := N m/(1 -θ), σ 2 /(1 -θ 2 ) , so that

E θ,π θ [X 0 ] = m 1 -θ and E θ,π θ X 2 0 = σ 2 1 -θ 2 + m 2 (1 -θ) 2 . ( 30
)
For any γ > 8, let us dene the following function

V ∀x ∈ IR, V (x) := 1 + |x| γ . (31)
Since θ belongs to a compact set Θ ⊂ (-1, 1), sup θ∈Θ E θ,δ 0 [|Z 1 | γ ] < +∞ and f Z > 0, it is easily checked that (X k ) k≥0 is µ Leb -irreducible, aperiodic and fullls the drift-criterion [START_REF] Meyn | Markov chains and stochastic stability[END_REF] uniformly in θ ∈ Θ. Therefore, the Markov chain (X k ) k≥0 is an instance of Model (M) with this function V .

Assume that θ is unknown and has to be estimated (α 0 (θ) ≡ θ here). The parameters m and σ > 0 are supposed to be known. The maximum likelihood estimator (MLE) ( θn ) n∈IN * involves the following function

F ∀(x, y) ∈ IR 2 , ∀α ∈ Θ, F (α, x, y) := -ln f Z (y -α x),
and the empirical mean functional (cf. (23))

∀n ∈ IN * , ∀α ∈ Θ, M n (α) := - 1 n n k=1 ln f Z (X k -αX k-1 ). (32)
Let us check that Theorem 2 can be applied to this Gaussian linear model. First we easily obtain M

(1) n ( θn ) = 0 (that is (HYP.1)) from the denition of the estimator, and we obtain as well the following closed-form expression for the MLE

∀n ∈ IN * , θn = n k=1 (X k-1 X k -mX k-1 ) n k=1 X 2 k-1 . ( 33
)
Then uniform consistency Property (HYP.2) holds true. Indeed, write

θn -θ = ∆ 1n -θ∆ 2n -m∆ 3n ∆ 2n + E θ,π θ [X 2 0 ]
, where

• ∆ 1n := (1/n) n k=1 X k-1 X k -θE θ,π θ [X 2 0 ] -mE θ,π θ [X 0 ] • ∆ 2n := (1/n) n k=1 X 2 k-1 -E θ,π θ [X 2 0 ] • ∆ 3n := (1/n) n k=1 (X k-1 -E θ,π θ [X 0 ]).
Since the families (ξ p ) p∈P fulll Condition (D 4 ) when ξ p (x, y) := xy, ξ p (x, y) := x 2 or ξ p (x, y) := x (because γ > 8), we obtain from Property (5)

∀i = 1, 2, 3, lim n→+∞ sup θ∈Θ E θ,δ 0 ∆ 4 in n 2 < +∞.
One concludes using Markov inequality and since inf θ∈Θ E θ,π θ [X 2 0 ] > 0. Next, the moment domination conditions (27) are obviously fullled (since γ > 6). Concerning Conditions (C.1) and (C.2), we obtain using the invariant probability (π θ ) θ∈Θ :

• m 1 (θ) = E θ,π θ X 0 f (1) Z (X 1 -θX 0 )/f Z (X 1 -θX 0 ) = E θ,π θ [X 0 ] E θ,π θ f (1) Z (Z 1 )/f Z (Z 1 ) , m 1 (θ) ≡ 0, and m 2 (θ) = E θ,π θ X 2 0 /σ 2 , so that inf θ∈Θ m 2 (θ) > 0. • σ 1 (θ) 2 = lim n E θ,π θ (1/n) n k=1 X k-1 (Z k -m)/σ 2 2 = lim n E θ,π θ (1/n) n k=1 X k-1 (Z k - m)/σ 2 /σ 2 . Thus σ 1 (θ) = m 2 (θ) satises inf θ∈Θ σ 1 (θ) > 0. σ 2 (θ) 2 = lim n→+∞ E θ,π θ (1/n) n k=1 X 2 k-1 /σ 2 -n m 2 (θ)
2 and after some tedious computations using (29), we obtain inf θ∈Θ σ 2 (θ) > 0.

Conditions (C.3), (C.4) and (C.5) are obviously fullled as well. Finally Assumption (S ) on the transition kernels (Q θ ) θ∈Θ with q θ (x,

•) := f Z (• -θx) is easily checked.
Thus, we apply Theorem 2 to our Gaussian linear model to derive that

sup u∈IR sup θ∈Θ P θ,δ 0 √ n σ(θ) ( θn -θ) ≤ u -N (u) -η(u) n -1 2 A θ (u) = o(n -1 2 ).
To illustrate our results, x θ := 1/2 for example and let us simulate by Monte Carlo methods the behavior of the MLE. Then we can compare the empirical distribution function at u of √ n ( θn -1/2)/σ(1/2) with the approximations N (u) and N (u) + η(u)A 1/2 (u)/ √ n of the distribution function P 1/2,δ 0 { √ n ( θn -1/2)/σ(1/2) ≤ u}, where the asymptotic standard deviation σ(1/2) and the polynomial function A 1/2 (u) are dened by

σ 1 2 = 1 4 3 + 4 m 2 σ 2 , A 1 2 (u) = c(1 + 5u 2 ) with c := 8 18 2 3 + 6 m 2 σ 2 4 3 + 4 m 2 σ 2 3 2
.

The performance of this Edgeworth expansion is illustrated in the case where m = 0 and σ = 1. Using a Scilab program, 5000 independent samples of the estimators ( θn ) n=1,..,50 have been obtained. For u := -1 and u := 1, we represent the empirical distribution function of √ n ( θn -1/2)/σ(1/2), the Gaussian approximation and the rst-order Edgeworth one on the same graphs (see page 24). As expected, observe that the quality of the normal and the rst-order Edgeworth approximations N (u) and N (u) + η(u)A 1/2 (u)/ √ n of P 1/2,δ 0 { √ n ( θn -1/2)/σ(1/2) ≤ u} increases when n grows, but the approximation by the rst-order expansion is quickly the best one. The rst-order Edgeworth expansion greatly improves the rate of convergence in the approximation of the distribution function of the estimator, in comparison with the Berry-Esseen type results.

→ P 1/2,δ 0 √ n( θn -1/2)/σ(1/2) ≤ u ,
• the Edgeworth expansion n → N (u) + η(u)A 1/2 (u)/ √ n • and the Gaussian approximation n → N (u).

Pfanzagl method to prove Theorem 2

In Subsections 6.1 and 6.2, we adapt Pfanzagl results to some general setting. More specifically, some probabilistic Edgeworth expansions are explicitly required in Subsection 6.1, whereas the general Assumptions R(m) and (N-A) are involved in Subsection 6.2. Thanks to this work, Theorem 2 is easily proved in Subsection 6.3.

In Subsections 6.1 and 6.2, we denote by (Ω, F, {P θ ; θ ∈ Θ}) a general statistical model, where Θ is some parameter space (not necessarily compact in these subsections). The underlying expectation is denoted by E θ . We assume that the following general statistical assumptions hold true: let (M n (α)) n∈IN * be any general sequence of real observations where α ≡ α(θ) ∈ A is the parameter of interest and A is an open interval on the real line. Assume that for all n ≥ 1, the map α → M n (α) is 3-time-dierentiable on A and that the derivatives dene r.v. on (Ω, F). We denote them by M (j) n := ∂ j M n /∂α j for j = 1, 2, 3. We consider some α 0 ∈ A and assume that the A-valued r.v. αn is specied by (HYP.1) and fullls the uniform consistency Property (HYP.2) that we recall below:

HYP. 1. ∀n ≥ 1, (∂M n /∂α) (α n ) = 0, HYP. 2. ∀d > 0, sup θ∈Θ P θ |α n -α 0 | ≥ d = o(n -1/2 ) .
Note that, in Subsections 6.1 and 6.2, (M n (α)) n∈IN * is not necessarily associated with a function F as in (23).

The revisited Pfanzagl method

We appeal to the following conditions:

A. 1. For all n ≥ 1, there exists a positive r.v. W n , independent on α, such that

∀j = 2, 3, ∀(α, α ) ∈ A 2 , |M (j) n (α ) -M (j) n (α)| ≤ |α -α| W n .
Furthermore there exists l: Θ → (0, +∞) bounded on Θ such that

sup θ∈Θ P θ W n ≥ l(θ) = o(n -1/2 ).
A. 2. For j = 1, 2, there exist σ j (•) > 0 satisfying sup θ∈Θ σ j (θ) < +∞, inf θ∈Θ σ 1 (θ) > 0, m 2 (•) satisfying inf θ∈Θ m 2 (θ) > 0, and polynomial functions denoted by B θ (•) and C θ (•), such that

sup θ∈Θ sup u∈IR P θ √ n σ 1 (θ) M (1) n (α 0 ) ≤ u -N (u) -η(u)n -1 2 B θ (u) = o(n -1 2 ), sup θ∈Θ sup u∈IR P θ √ n σ 2 (θ) M (2) n (α 0 ) -m 2 (θ) ≤ u -N (u) -η(u)n -1 2 C θ (u) = o(n -1 2 ).
Furthermore the coecients of B θ (•) and C θ (•) are assumed to be bounded.

Let us dene σ(θ) := σ 1 (θ)/m 2 (θ). Notice that σ(•) satises sup θ∈Θ σ(θ) < +∞.

A. 3. For all n ≥ 1, u ∈ IR such that |u| ≤ 2 √ ln n, there exist σ n,u (•) 2 > 0, m 3 (•) bounded on Θ, D θ (•) and E θ (•) some polynomial functions such that sup θ∈Θ sup |u|≤2 √ ln n σ n,u (θ) -1 -σ 1 (θ) -1 + D θ (u) n -1 2 = o(n -1 2 ), sup θ∈Θ sup v∈IR sup |u|≤2 √ ln n P θ √ n σ n,u (θ) M n (θ, u) ≤ v -N (v) -η(v)E θ (v)n -1 2 = o(n -1 2 ),
where M n (θ, u) denotes

M n (θ, u) := M (1) n (α 0 )+ σ(θ) √ n u M (2) n (α 0 )-m 2 (θ) + σ(θ) √ n 2 u 2 2 M (3) n (α 0 )-m 3 (θ) . (34)
Furthermore, the coecients of D θ (•) and E θ (•) are assumed to be bounded.

Theorem 3. Under Conditions (A.1), (A.2) and (A.3), there exists a polynomial function A θ such that one has with

σ := σ 1 /m 2 sup θ∈Θ sup u∈IR P θ √ n σ(θ) (α n -α 0 ) ≤ u -N (u) -η(u)n -1 2 A θ (u) = o(n -1 2 ). (35)
Furthermore

∀θ ∈ Θ, ∀u ∈ IR, A θ (u) := D θ (u) σ 1 (θ) u + σ(θ) 2 2σ 1 (θ) m 3 (θ)u 2 -E θ (-u). (36)
The proof of Theorem 3 is postponed to Appendix A. It consists in adapting the Pfanzagl method [Pfa73] introduced for i.i.d. observations. Just mention that the Pfanzagl method is not exactly the one developed in Appendix A, but for convenience this discussion is omitted.

6.2 An alternative statement using Hypotheses R(m) and (N-A)

Below we appeal to the following assumptions involving Hypotheses R(m) and (N-A) of Subsection 2.1:

B. 1. For all n ≥ 1, there exists a positive r.v. W n , independent on α, such that

∀j = 2, 3, ∀(α, α ) ∈ A 2 , |M (j) n (α ) -M (j) n (α)| ≤ |α -α| W n .
Furthermore there exists l:

Θ → (-1, +∞) bounded on Θ such that n(W n -l(θ)); n ≥ 1, θ ∈ Θ fullls Hypothesis R(2). B.2. The family n M (1) n (α 0 ); n ≥ 1, θ ∈ Θ fullls Hypotheses R(3) and (N-A). Furthermore there exists m 2 (•) on Θ satisfying inf θ∈Θ m 2 (θ) > 0 such that n M (2) n (α 0 ) -m 2 (θ) ; n ≥ 1, θ ∈ Θ fullls both Hypotheses R(3) and (N-A).
Thanks to the last Assumption (B.2) and Proposition 1, we can dene the asymptotic variances

σ 1 (θ) 2 := lim n→+∞ E θ n M (1) n (α 0 ) 2 , σ 2 (θ) 2 := lim n→+∞ E θ n M (2) n (α 0 ) -m 2 (θ) 2 .
Furthermore we assume that the following conditions on these asymptotic variances hold true

B. 3. inf θ∈Θ σ 1 (θ) > 0, B. 4. inf θ∈Θ σ 2 (θ) > 0.
The following additional conditions are also required:

B. 5. There exists m 3 (•), bounded on Θ, such that n M (3) n (α 0 ) -m 3 (θ) ; n ≥ 1, θ ∈ Θ fullls Hypothesis R(2), and n M n (θ, u); n ≥ 1, θ ∈ Θ, |u| ≤ 2 √
ln n fullls both Hypotheses R(3) and (N-A) as well, where M n (θ, u) is dened by (34).

Theorem 4. Under Assumptions (B.1) to (B.5), there exists a polynomial function A θ independent on n such that one has (35) with σ := σ 1 /m 2 . The polynomial function A θ is of the type A θ (u) = a 1 (θ) + a 2 (θ)u 2 where, for i = 1, 2, sup θ∈Θ |a i (θ)| < +∞. Furthermore if we suppose that the additional moment condition holds true:

∀j = 1, 2, 3, ∀θ ∈ Θ, ∀n ∈ IN * , E θ M (j) n (α 0 ) 3 < +∞,
then one has more precisely

a 1 (θ) := - 1 6 m 3,1 (θ) 3 σ 1 (θ) 3 + b 1 (θ) σ 1 (θ) , a 2 (θ) := 1 6 m 3,1 (θ) 3 σ 1 (θ) 3 - σ 12 (θ) σ 1 (θ)m 2 (θ) + σ 1 (θ) 2m 2 (θ) 2 m 3 (θ) with          b 1 (θ) := lim n→+∞ E θ n M (1) n (α 0 ) σ 12 (θ) := lim n→+∞ E θ n M (1) n (α 0 ) M (2) n (α 0 ) -m 2 (θ) m 3,1 (θ) 3 := lim n→+∞ E θ n 2 M (1) n (α 0 ) 3 -3σ 2 1 (θ)b 1 (θ).
Proof of Theorem 4. It is sucient to show that the assumptions of Theorem 4 imply the previous ones of Theorem 3.

• From (B.1), (A.1) holds true with l := l + 1. Indeed let S n (θ) := n(W n -l(θ)).

Thanks to (4) and Markov inequality, one easily obtains

sup θ∈Θ P θ {W n ≥ l(θ)} ≤ (1/n) sup θ∈Θ (E θ [S n (θ) 2 ]/n) = O(n -1 ).
• (A.2) is directly implied by (B.2), (B.3), (B.4) using Proposition 1, where

B θ (u) := m 3,1 (θ) 3 6σ 1 (θ) 3 (1 -u 2 ) - b 1 (θ) σ 1 (θ) .
Similar expression holds for C θ (u).

• In a more intricate way, to prove (A.3) under (B.2), (B.3), (B.5), let us dene

S n (θ, p, v) := n M (1) n (α 0 ) + ς p (v, θ) M (2) n (α 0 ) -m 2 (θ) + ς p (v, θ) 2 2 M (3) n (α 0 ) -m 3 (θ)
where ς p (v, θ) := vσ(θ)/ √ p, so that S n (θ, n, u) = n M n (u, θ) (cf. (34)). From (B.5) and using Proposition 1, we can dene

σ p,v (θ) 2 := lim n→+∞ E θ [S n (θ, p, v) 2 ] n ,
and from (B.3) and using Proposition 1 again, we obtain (A.3) with

D θ (u) := - σ 12 (θ) σ 1 (θ) 3 σ(θ)u, and E θ (u) := m 31 (θ) 3 6σ 1 (θ) 3 (1-u 2 )- b 1 (θ) σ 1 (θ) . 2
6.3 Proof of Theorem 2 of Subsection 5.1

Let us go back to our Markovian model (M) and prove that the assumptions of Theorem 4 hold true whenever the assumptions of Theorem 2 are satised.

• Let us dene W n := (1/n) n k=1 (W (X k-1 ) + W (X k )) and l(θ) := 2 E θ,π θ [W (X 1 )],
where W is dened in (C.3). Then, using Proposition 3, the Lipschitz condition (B.1) for j = 3 is true. Indeed the family {ξ θ ; θ ∈ Θ} obviously fullls the moment domination condition (D 2 ) with ξ θ (x, y) := W (x) + W (y) -2 l(θ). In the same way, the remaining part of (B.1) (for j = 2) is checked under (27) (which means that the family {F (3) (α, •, •); α ∈ A} fullls (D 3 ) and a fortiori (D 2 )).

• Firstly, we deduce from Proposition 3 that the part of (B.2) concerning Hypothesis R(3) is true under (27). Secondly, thanks to Assumption (S ), we deduce from Lemma 3 and Theorem 1 that the part of (B.2) concerning Hypothesis (N-A) is true under Condition (a) of Theorem 2 (see Remark 3 concerning the assumptions of Lemma 3).

• The conditions (B.3) and (B.4) are exactly (C.2).

• We deduce from Proposition 3 that the rst point of (B.5) follows from (27). For the second point of (B.5), recall Denition (34) of M n (θ, u), and notice that inf θ∈Θ m 2 (θ) > 0 from (C.1) and sup θ∈Θ σ 1 (θ) < +∞, which imply that sup{σ(θ

)u/ √ n; n ≥ 1, θ ∈ Θ, |u| ≤ 2 √ ln n} < +∞. Thus the family    3 j=1 1 (j -1)! σ(θ) √ n u j-1 F (j) (α 0 , •, •) -m j (θ) ; n ≥ 1, θ ∈ Θ, |u| ≤ 2 √ ln n   
obviously fullls (D 3 ), and we conclude from Proposition 3 that n M n (θ, u); n ≥ 1, θ ∈ Θ, |u| ≤ 2 √ ln n fullls Hypothesis R(3). Finally, thanks to Assumption (S ), we easily check from Lemma 3 and Theorem 1 that the part of (B.5) concerning Hypothesis (N-A) is true under Condition (b) of Theorem 2. 2

Illustration of Theorem 4 in the case of some AR(d) processes

In this subsection, we apply Theorem 4 to linear autoregressive processes of order d, d ≥ 2. In substance, such a model fullls all the assumptions of Theorem 2, except the dominated model Condition (S ). Consequently, the non-arithmeticity conditions involved in the assumptions B.2 and B.5 of Theorem 4 cannot be checked using Theorem 1 any more. Here, by reinforcing the assumptions on the density of the noise, we apply the second approach of Subsection 3.3 to study these non-arithmeticity conditions.

Let us consider the following autoregressive process of order d ≥ 1 on E := IR d :

∀n ≥ d, Y n := g 1 (θ)Y n-1 + • • • + g d (θ)Y n-d + Z n (37)
where the probability distribution of (Y 0 , . . . , Y d-1 ) is denoted by µ θ and

• θ ∈ IR is a parameter,

• (g 1 , . . . , g d ) are given real continuous functions,

• and (Z k ) k∈IN * are i.i.d. real-valued r.v. independent on (Y 0 , .., Y d-1 ) with common distribution which admits some probability density f Z with respect to µ Leb .

The parameter θ of the observed AR(d) process is assumed to be in a non-empty compact set Θ ⊂ IR such that for all θ ∈ Θ the solutions of the equation

z d -g 1 (θ)z d-1 -• • • -g d-1 (θ)z -g d (θ) = 0 (38)
lie in D(0, 1) := {z ∈ C ; |z| < 1}.

Introduce the column vector X n := (Y n , . . . , Y n-d+1 ) for n ≥ d -1. Then the process (X n ) n≥d-1 is a Markov chain with the following rst-order autoregressive representation

∀n ≥ d X n = A(θ) X n-1 + (Z n , 0, . . . , 0) (39)
where

A(θ) :=          g 1 (θ) . . . g d-1 (θ) g d (θ) 1 0 . . . 0 0 0 0 1 . . . 0 0 0 . . . . . . . . . . . . 0 0 . . . 1 0 0 0 0 . . . 0 1 0          .
Assuming that the solutions of the equation (38) lie in D(0, 1) is equivalent to assume that the eigenvalues of A(θ) have moduli strictly less than unity, so that A(θ) < 1 for all θ ∈ Θ and sup θ∈Θ A(θ) < 1.

The initial distribution of (X n ) n≥d-1 is µ θ and its transition kernel Q θ is given for all Borel set B ∈ B(IR d ) by

Q θ (x, B) = IR 1 B A(θ)x + (z, 0, • • • , 0) f Z (z) dz.
By Fatou lemma, lim sup

x →∞ sup θ∈Θ Q θ V (x) V (x) ≤ sup θ∈Θ A(θ) γ < 1.
Next, pick ∈ (sup θ∈Θ A(θ) γ , 1). There exists s > 0 such that for all x > s and θ ∈ Θ,

Q θ V (x) ≤ V (x). Set S := {x ∈ IR d ; x ≤ s}. Note that ∀θ ∈ Θ, ∀x ∈ S, Q θ V (x) ≤ ς := sup θ∈Θ IR 1 + A(θ) s + |z| γ f Z (z)dz < +∞, so that ∀θ ∈ Θ, ∀x ∈ IR d , Q θ V (x) ≤ V (x) + ς.
Finally, since Condition (A) holds true, it is easily checked that (X k ) k≥0 is µ Leb d -irreducible, aperiodic and fullls the drift-criterion [START_REF] Meyn | Markov chains and stochastic stability[END_REF] uniformly in θ ∈ Θ.

2

Set e 1 := (1, 0, . . . , 0) ∈ IR d . Then, let us consider the MLE ( θn ) n∈IN * of the parameter θ (α 0 (θ) ≡ θ). We have ∀n ≥ d, e 1 , X n = g(θ), X n-1 + Z n .

Maximum likelihood estimation requires to deal with the following function

F ∀(x, y) ∈ IR d × IR d , ∀α ∈ Θ, F (α, x, y) := -ln f Z e 1 , y -g(α), x ,
and the empirical mean functional

∀n ∈ IN * , ∀α ∈ Θ, M n (α) := - 1 n n k=1 ln f Z e 1 , X k -g(α), X k-1 .
Proposition 4. Assume that the previous assumptions on the model hold true and that the MLE ( θn ) n∈IN * of the parameter θ associated with (X n ) n≥d-1 satises the uniform consistency Property (HYP.2). In addition, assume that the initial probability measure satises sup θ∈Θ µ θ (V ) < +∞ (that is Property (12)) and that the following conditions hold true:

1. ∀α ∈ A, ∀j = 1, 2, F (j) (α, x, y) fullls (N-L)' as dened page 12 2. ∀α ∈ A, ∀υ ∈ IR, F (1) (α, x, y) + υF (2) (α, x, y) + (υ 2 /2)F (3) (α, x, y) fullls (N-L)'.

Then all the conclusions of Theorem 2 are true.

Proof of Proposition 4. It is easily checked that the family {F (j) (α, x, y); α ∈ A, j = 1, 2, 3} fullls the moment domination condition (D 3 ) (i.e. ( 27)) mainly thanks to (E) and γ > 3 × 3.

Next, we claim that Conditions (C.1) to (C.5) of Theorem 2 are satised. Indeed, concerning Conditions (C.1) and (C.2), and recalling that (π θ ) θ∈Θ denotes the invariant probability of the Markov chain (X n ) n≥d-1 , we have

• m 1 (θ) = E θ,π θ [ g (1) (θ), X 0 ]E θ,π θ [f (1) Z (Z 1 )/f Z (Z 1 )] ≡ 0; m 2 (θ) = E θ,π θ [ g (1) (θ), X 0 2 ] E θ,π θ f (1) Z (Z 1 ) 2 /f Z (Z 1 ) 2 ≥ g (1) 1 (θ) 2 E θ,π θ [Z 2 1 ]
, which implies that inf θ∈Θ m 2 (θ) > 0.

• σ 1 (θ) 2 = m 2 (θ), hence one has inf θ∈Θ σ 1 (θ) > 0 ; σ 2 (θ) 2 ≥ C where C > 0 depends on the variances of Z 1 and Z 2 1 .

Conditions (C.4) and (C.5) are obviously satised. Concerning (C.3), use the fact that the family {F (4) (α, x, y); α ∈ A} fullls (D 2 ) (this statement holds true mainly thanks to Assumption (E) and γ > 2 × 4).

By using the previous facts and proceeding as in the proof of Theorem 2 (see Subsection 6.3), one can see that all the assumptions of Theorem 4 but those concerning Hypothesis (N-A) are fullled. Consequently, to deduce Proposition 4 from Theorem 4, it only remains to establish that the characteristic functions of the following families (ξ p ) p∈P (involved in Assumptions B.2 and B.5 of Theorem 4) satisfy Hypothesis (N-A):

(a) {F (j) (α, x, y); α ∈ A} with j = 1, 2, (b) {F (1) (α, x, y) + υF (2) (α, x, y) + (υ 2 /2)F (3) (α, x, y); α ∈ A, υ ∈ IR}.

To that eect, we make use of the second approach of Subsection 3.3. Below, (i), (ii) and (iii) refer to the conditions introduced in Subsection 3.1:

Fact1. Families (a)-(b) satisfy Condition (N-L). Indeed, thanks to Conditions (C) and (S) (use the fact that f Z > 0 to check that (S) holds true), we deduce from Lemma 3 that Fact1 follows from Assumptions 1. and 2. of this Proposition 4.

Fact2. The Fourier operators of Families (a)-(b) satisfy Condition (i). Indeed Assumptions 1. and 2. of Lemma 1 are fullled (see the comments after Lemma 1 concerning Assumption 1. and Property (17) concerning Assumption 2.). Then, using Lemma 1, Fact2 follows from Fact1.

Fact3. The Fourier operators of Families (a)-(b) satisfy (ii)-(iii). Indeed notice that the family {F (j) (α, x, y); α ∈ A, j = 1, . . . , 4} satises (D 0 ), and consequently, the assumptions of Proposition 2 are fullled (see Lemma 9 below and apply it to the case where (ξ p ) p∈P is any of the above Families (a)-(b)). Then, using Proposition 2, Fact3 follows from Fact2.

Fact4. Finally the Fourier operators of Families (a)-(b) satisfy Hypothesis (N-A'), and so (N-A) (see Lemma 2 and see also the end of Section 2).

The proof of Proposition 4 is now complete, provided that we give the proof of the next lemma.

2

Lemma 9. Assume that (ξ p ) p∈P and its derivative with respect to p fulll (D m 0 ) with some m 0 ∈ IN . Let First, Condition (C1) of Proposition 2 is exactly (D-F). Then, concerning Condition (C2) of Proposition 2, let us prove that the following properties are valid:

1. the map t → Q θ,p (t) is continuous from IR into L(B β , B 1 ) uniformly in (θ, p) ∈ Θ × P;

2. for all t ∈ IR, the map (θ, p) → Q θ,p (t) is continuous from Θ × P into L(B β , B 1 ).

Then one obviously has Q θ,p (t) -Q θ 0 ,p 0 (t 0 ) B β ,B 1 → 0 when (t, θ, p) → (t 0 , θ 0 , p 0 ), which completes the proof of Lemma 9.

Let us rst introduce the real number E β := c β κ β + b 1 , where b 1 := sup θ∈Θ π θ (V ) < +∞ from (VG1) and κ β and c β are dened in (D-F). Then, using (D-F), V ≥ 1 and β > 0, we obtain

∀θ ∈ Θ, Q θ V β ≤ E β V β .
On the other hand, let us state the following obvious inequality:

∀a ∈ IR, |e ia -1| ≤ min(2, |a|) ≤ 2|a| α .
Now recall that 0 < β < 1 and let us dene 0 < α ≤ 1 such that β + α/(m 0 + ε) ≤ 1 where ε > 0 is dened in (D m 0 ).

1) Let us dene

∆ := Q θ,p (t) -Q θ,p (t 0 ).
One has for all f ∈ B β and x ∈ E:

|∆f (x)| ≤ E e itξp(x,y) -e it 0 ξp(x,y) |f (y)| Q θ (x, dy) ≤ 2 |t -t 0 | α f B β E |ξ p (x, y)| α V (y) β Q θ (x, dy) ≤ 2 C α m 0 +ε ξ |t -t 0 | α f B β E (V (x) + V (y)) α m 0 +ε V (y) β Q θ (x, dy) ≤ 2 1+ α m 0 +ε C α m 0 +ε ξ |t -t 0 | α f B β V (x) α m 0 +ε Q θ V β (x) + Q θ V β+ α m 0 +ε (x) ≤ 2 1+ α m 0 +ε C α m 0 +ε ξ |t -t 0 | α f B β E β + E β+ α m 0 +ε V (x) β+ α m 0 +ε from which we deduce ∆f B 1 ≤ D ξ |t -t 0 | α f B β
where D ξ does not depend on (θ, p).

2) In the same way, let us dene ∆ := Q θ,p (t) -Q θ 0 ,p 0 (t). Let x denote some d-dimensional column vector (x d , . . . , x 1 ) and x d-1 denote the associated (d -1)-dimensional column vector (x d , . . . , x 2 ) . We have for all f ∈ B β and x ∈ E:

∆f (x) = IR exp itξ p x, x, g(θ) + z x d-1 f x, g(θ) + z x d-1 f Z (z) dz - IR exp itξ p 0 x, x, g(θ 0 ) + z x d-1 f x, g(θ 0 ) + z x d-1 f Z (z) dz = IR exp itξ p x, y x d-1 f y x d-1 f Z (y -x, g(θ) ) dy - IR exp itξ p 0 x, y x d-1 f y x d-1 f Z (y -x, g(θ 0 ) ) dy |∆f (x)| ≤ IR f y x d-1 (f Z (y -x, g(θ) ) -f Z (y -x, g(θ 0 ) )) dy + 2|t| α IR (ξ p -ξ p 0 ) x, y x d-1 α f y x d-1 f Z (y -x, g(θ 0 ) ) dy. Thus one has |∆f (x)| ≤ f B β (|θ -θ 0 | I 1 + |t| α |p -p 0 | α I 2 )
where, thanks to dierentiation under the integral sign and Assumptions (E)-(F), I 1 satises for some θ ∈ IR such that

| θ -θ| ≤ | θ -θ 0 | I 1 ≤ sup θ∈Θ | x, g (1) ( θ) | sup z∈IR f (1) Z (z) f Z (z) IR V y x d-1 β f Z (y -x, g( θ) ) dy = sup θ∈Θ | x, g (1) ( θ) | sup z∈IR f (1) Z (z) f Z (z) Q θ V β (x) ≤ E β V (x) 1 γ +β sup θ∈Θ g (1) ( θ) sup z∈IR f (1) Z (z) f Z (z)
and where I 2 satises on the other hand

I 2 ≤ 2 IR V (x) + V y x d-1 α m 0 +ε V y x d-1 β f Z (y -x, g(θ 0 ) ) dy = 2 1+ α m 0 +ε V (x) α m 0 +ε Q θ 0 V β (x) + Q θ 0 V β+ α m 0 +ε (x) ≤ 2 1+ α m 0 +ε E β + E β+ α m 0 +ε V β+ α m 0 +ε (x). Since 0 < β ≤ 1 -1/γ, one has ∆f B 1 ≤ D ξ (|θ -θ 0 | + |t| α |p -p 0 | α ) f B β . 2
A Proof of Theorem 3 of Subsection 6.1

The investigation of the case |u| > 2 √ ln n is similar to the one of [HLP], so that the details are omitted. By contrast, the case |u| ≤ 2 √ ln n is quite dierent. First let us introduce for all θ ∈ Θ and u

∈ IR, |u| ≤ 2 √ ln n τ = τ n (u, θ) := α 0 + σ(θ) √ n u and ς n (u, θ) := σ(θ) √ n u = τ -α 0 .
For the sake of simplicity, let us dene for all θ ∈ Θ and u

∈ IR, |u| ≤ 2 √ ln n P n,θ (u) := P θ √ n σ(θ) (α n -α 0 ) ≤ u = P θ {α n ≤ τ } , Q n,θ (u) := P θ M (1) n (τ ) ≥ 0 .
At a rst stage we prove that

sup θ∈Θ sup |u|≤2 √ ln n |P n,θ (u) -Q n,θ (u)| = o(n -1 2 ) (A)
and then we determine A θ such that

sup θ∈Θ sup |u|≤2 √ ln n Q n,θ (u) -N (u) + η(u)n -1 2 A θ (u) = o(n -1 2 ) (B)
to complete the proof of Theorem 3.

Let us prove that (A) holds true. It follows from (HYP.1) that there exists some real r.v.

α n such that | α n -τ | < |α n -τ | and 0 = M (1) n (τ ) + ( αn -τ ) M (2)
n ( α n ). Next, introducing the event {M

(2) n ( α n ) > 0} and its complement, one has

P n,θ (u) = P θ M (1) n (τ ) ≥ 0, M (2) n ( α n ) > 0 + P θ αn ≤ τ, M (2) n ( α n ) ≤ 0 , so that |P n,θ (u) -Q n,θ (u)| ≤ 2 P θ M (2) n ( α n ) ≤ 0 .
Introducing the events {M

(2)

n ( α n ) < M (2) n (α 0 ) -| α n -α 0 |l(θ)}, {M (2)
n (α 0 ) ≤ m 2 (θ)/2} and their complements, where the function l(•) is dened in (A.1), one has

P θ M (2) n ( α n ) ≤ 0 ≤ P 1 + P 2 + P 3 ,
where (P i ) i=1,2,3 denote

P 1 := sup θ∈Θ sup |u|≤2 √ ln n P θ M (2) n ( α n ) < M (2) n (α 0 ) -| α n -α 0 |l(θ) P 2 := sup θ∈Θ P θ M (2) n (α 0 ) ≤ m 2 (θ) 2
P 3 := sup θ∈Θ sup |u|≤2 √ ln n m 2 (θ) 2 -| α n -α 0 |l(θ) < M (2) n ( α n ) ≤ 0 .
• Introducing the event {W n ≥ l(θ)} and its complement, it is easy to check from (A.1) that P 1 = o(n -1/2 ).

• One has

P 2 ≤ P θ ( √ n/σ 2 (θ)) M (2) n (α 0 ) -m 2 (θ) ≤ -b √ n where b := inf θ∈Θ m 2 (θ)/2σ 2 (θ), b > 0 from (A.2), which implies P 2 = o(n -1/2 ).
Then it only remains to determine A θ such that

sup θ∈Θ sup |u|≤2 √ ln n |P θ {B 1± n,u,θ } -N (u) + η(u)n -1 2 A θ (u) | = o(n -1 2 ).
Let us introduce

∆ ± n (u, θ) := √ n σ n,u (θ) m 2 (θ)ς n (u, θ) + ς n (u, θ) 2 2 m 3 (θ) ± c -u = u σ n,u (θ) -1 σ 1 (θ) + σ(θ) m 3 (θ) 2 ς n (u, θ) ± σ(θ) l(θ) ς n (u, θ) 2 2 -1 so that P θ {B 1± n,u,θ } = 1 -P θ √ n σ n,u (θ) M n (u, θ) < -u -∆ ± n (u, θ) .
From the last property of (A.3) applied to v = -u -∆ ± n (u, θ), we obtain

sup θ∈Θ sup |u|≤2 √ ln n P θ {B 1± n,u,θ } ≤ -u -∆ ± n (u, θ) -N (u + ∆ ± n (u, θ)) + η(u + ∆ ± n (u, θ))n -1 2 E θ (-u -∆ ± n (u, θ)) = o(n -1 2 ).
From the rst property of (A.3), both ∆ + n (u, θ) and ∆ - n (u, θ) admit the following expansion:

sup θ∈Θ sup |u|≤2 √ ln n ∆ ± n (u, θ) -σ 1 (θ)D θ (u) + σ(θ) 2 2σ 1 (θ) m 3 (θ) u u n -1 2 = o(n -1 2 ),
and hence sup

θ∈Θ sup |u|≤2 √ ln n |P θ {B 1+ n,u,θ } -P θ {B 1- n,u,θ }| = o(n -1 2 ).
Finally we dene the polynomial function A θ as follows to obtain (36):

sup θ∈Θ sup |u|≤2 √ ln n N (u) + η(u)n -1 2 A θ (u) - N (u + ∆ + n (u, θ)) -η(u + ∆ + n (u, θ))n -1 2 E θ (-u -∆ + n (u, θ)) = o(n -1 2 ). 2
B The regeneration method versus Fourier techniques (combined with operator perturbation theorems)

This appendix is to convince that the regeneration method cannot easily be applied to our context.

Recall that in this paper we have studied the behavior of P θ,µ θ {S n (p)/(σ θ,p √ n) ≤ u} uniformly in (θ, p) ∈ Θ × P and u ∈ IR thanks to Fourier techniques, where S n (p) is dened by (1), i.e. S n (p) := n k=1 ξ p (X k-1 , X k ). In this appendix, we want to highlight the drawbacks of using the regeneration method to investigate the same issue.

Let us mention that there are mainly two constraints we must deal with: limit theorems must be obtained uniformly in both the model parameter θ and the technical parameter p, and they concern bivariate functions ξ p (with a view to making statistical inference).

To the best of our knowledge, on the one hand, limit theorems with an eective control of the constants which are obtained thanks to the regeneration method only deal with Berry-Esseen theorem and only concern univariate additive functionals n k=1 ξ p (X k ) (cf. [START_REF] Bertail | A renewal approach to markovian U-statistics[END_REF][START_REF] Douc | Bounds on regeneration times and limit theorems for subgeometric Markov chains[END_REF]). On the other hand, limit theorems concerning bivariate additive functionals which are obtained thanks to the regeneration method do not control the constants (cf. [START_REF] Jensen | Asymptotic expansions for strongly mixing Harris recurrent Markov chains[END_REF]).

There is one way to bypass the bivariate constraint: we may consider directly the double sequence (Y n ) n≥1 where Y n := (X n-1 , X n ). However, as explained in B.1, this choice induces too strong restrictions on the model. In the same way, considering the simple sequence (X n ) n≥0 as in B.2 induces very restrictive conditions on the initial probability, excepted for models with an atom. Moreover note that, in atomic models, the uniform control for bivariate functionals has not been investigated by regenerative methods.

In the sequel, we drop the parameters (θ, p) for the sake of notational simplicity. Recall that (X n ) n≥0 is a Markov chain with state space (E, E) and transition kernel Q.

B.1 Application of the results of [BC11] to the sequence (Y n ) n≥1

Before explaining why applying the usual regeneration method to the sequence (Y n ) n≥1 = (X n-1 , X n ) n≥1 induces restrictions, let us briey explain how it works. The sequence (Y n ) n≥1 is a Markov chain with state space (E × E, E ⊗ E) and transition kernel P dened by:

∀F ∈ E ⊗ E, P (x, y); F := E 1 F (y, z)Q(y, dz).
Let us recall that (Y n ) n≥1 is said to be ψ-irreducible for some positive measure ψ on (E × E, E ⊗ E) if, for all F ∈ E ⊗ E, we have:

ψ(F ) > 0 ⇒ ∀(x, y) ∈ E × E, +∞ n=2 P (x,y) Y n ∈ F > 0,
and (Y n ) n≥1 is said to be Harris-recurrent under the ψ-irreducibility assumption if, for all F ∈ E ⊗ E:

ψ(F ) > 0 ⇒ ∀(x, y) ∈ E × E, P (x,y) +∞ n=1 1 F (Y n ) = +∞ = 1.
where Z 1,a , Z 2,b and Z 3,c are independent and distributed according to the following measures Assume that we can state the asymptotic negligibility of Z 1,a and Z 3,c . Then it would only remain to control the behavior of Z 2,b . Let us notice that ( ξ(B j ), τ A (j + 1)) j≥1 is a twodimensional i.i.d. sequence which is lattice in one component. Although this sequence is lattice in one component, we can establish a limit theorem associated to Z 2,b when b → +∞ by applying the results of [START_REF] Dubinskaite | Limit theorems in IR k , i[END_REF][START_REF] Dubinskaite | Limit theorems in IR k , ii[END_REF][START_REF] Dubinskaite | Limit theorems in IR k , iii[END_REF] as in [BC04, lem. 6.5]. Then, at a rst glance, considering the sequence (Y n ) n≥1 instead of the sequence (X n ) n≥0 seems helpful.

However, assuming the existence of an atom for the double sequence (Y n ) n≥1 induces very restrictive conditions on the model: Lemma 10. If (Y n ) n≥1 possesses an accessible atom A, then there exists a state b ∈ E such that A := A 1 × {b} for some A 1 ∈ E. Furthermore, whatever the initial state y ∈ E, the chain (X n ) n≥0 visits {b} with strictly positive probability, that is:

∀y ∈ E, +∞ n=1
Q n y, {b} > 0.

Proof of Lemma 10. To avoid the assumption of the existence of an accessible atom (and hence the strong conditions of Lemma 10), we may use the regeneration method constructed via the splitting technique under a minorization condition. However in the same way, supposing that P satises a minorization condition induces also very restrictive conditions on the model.

Under the ψ-irreducibility assumption, we suppose that (Y n ) n≥1 is Harris-recurrent and satises a minorization condition, i.e. there exist a measurable function 0 ≤ h < 1 on E × E and a positive measure ν on (E × E, E ⊗ E) such that ν(h) > 0 and: ∀(x, y) ∈ E × E, ∀F ∈ E ⊗ E, P (x, y); F ≥ h(x, y) ν(F ). In conclusion, since P ((x, y); .) does not depend on x ∈ E, (Y n ) n≥1 can possess an accessible atom or satisfy a minorization condition only under very restrictive conditions on (X n ) n≥0 (cf. Lemmas 10 and 11). For instance, whenever (X n ) n≥0 is dominated by the Lebesgue measure on IR d (as the AR(1) processes dened in (28)), (Y n ) n≥1 cannot possess an accessible atom and cannot satisfy a minorization condition: in these quite usual models, the results of [START_REF] Bertail | A renewal approach to markovian U-statistics[END_REF] cannot be applied to (Y n ) n≥1 .

limit theorem when b → +∞ in spite of this one-dependence, we can apply the usual Nagaev-Guivarc'h method, as in [Jen89, th. 2] where Jensen manages to obtain non-parametric results on bivariate functions thanks to a judicious combination between regeneration and Fourier methods.

However, this method induces an unusual Cramér assumption (this condition concerns equally the lattice v.a. τ A (j)) as well as an intricate covariance matrix (in particular, this matrix depends on the variance of τ A (j)).

Furthermore, as explained in Introduction, Jensen must assume some block moment conditions which are far from being easy to be veried, except when the initial probability µ is either concentrated at one point in the atom A (i.e. there exists x 0 ∈ A such that µ = δ x 0 ), or is dominated by some multiple of the stationary probability π.

B.2.2

Discussion under a minorization condition for (X n ) n≥0

From now on, assume that (X n ) n≥0 is ψ-irreducible, Harris-recurrent, has no accessible atom but satises a minorization condition, i.e there exist a measurable function 0 ≤ h < 1 on E and a positive measure ν on (E, E) such that ν(h) > 0 and:

∀x ∈ E, ∀F ∈ E, Q x; F ≥ h(x) ν(F ). (44)
When assuming that the chain (X n ) n≥0 only satises the minorization condition (44), we must construct a new Markov chain ( Xn ) n≥0 whose state space is E ×{0, 1} and which possesses the atom A := E × {1} using the splitting method of Nummelin. Then we make all the preceding job with this new chain ( Xn ) n≥0 . In particular, ( Xn ) n≥0 must satisfy some complex block moment conditions which can be easily veried only under the following strong restrictions on the initial probability μ of this split chain: the initial probability μ is either concentrated at one point in E × {1}, or dominated by some multiple of the stationary probability of the split chain ( Xn ) n≥0 .

Let us recall that μ is dened as follows:

∀F ∈ E, μ((F, i)) :=

   F (1 -h(x)) µ(dx) if i = 0 F h(x) µ(dx) if i = 1.
Then, it is easy to see5 that, whatever x 0 ∈ E, there exist no probability measure µ for the chain (X n ) n≥0 which can match with μ := δ {x 0 }×{1} for the split chain ( Xn ) n≥0 . In other words, applying regeneration method with explicit moment condition whenever the chain only satises a minorization condition imposes the domination of the initial probability µ by some multiple of the stationary probability π.

In conclusion, even in non-parametric cases, the regeneration method is not as ecient as our method. Unlike regeneration results, our results are obtained under weak conditions on the

Figure 1 :

 1 Figure 1: The Gaussian linear example of 5.2: for u := -1 and u := 1, graphs of • the empirical distribution of √ n( θn -1/2)/σ(1/2), i.e. the estimation by Monte-Carlo methods of n → P 1/2,δ 0 √ n( θn -1/2)/σ(1/2) ≤ u ,

  Q θ,p (t) be the Fourier operator dened by (8) where the transition kernel Q θ is the particular one given in (40). Then, Conditions (C1) and (C2) of Proposition 2 hold true.Proof of Lemma 9. Let us prove that the family (ξ p ) p∈P veries the assumptions of Proposition 2 with B := B β → B := B 1 where β is dened in Assumption (F) (see page 15 for the denition of the spaces).

τ

  i ) ∈ B, T A (1) = a P µ Z 2,b ∈ B :A (j) = n -a -c P µ Z 3,c ∈ B :i ) ∈ B, τ A (2) > c .

  We obviously have∀(x, y) ∈ E × E, ∀ b ∈ E, P (x, y); {b} × E = δ y,b .Consequently, if (x, y) and (x , y ) are two elements of the atom A, then we have y = y . In other words we have A = A 1 × {b} for some A 1 ∈ E and b ∈ E. Besides we obtain for all (x, y)∈ E × E +∞ n=1 Q n y, {b} = +∞ n=2 P (x,y) (X n-1 , X n ) ∈ E × {b} ≥ +∞ n=2 P (x,y) Y n ∈ A > 0, since ψ(A) > 0 and (Y n ) n≥1 is ψ-irreducible. 2 B.1.2Discussion under a minorization condition for (Y n ) n≥1

  The Markov kernel P of (Y n ) n≥1 satises a minorization condition if and only if there exists b ∈ E such that Q(b, {b}) > 0.Proof of Lemma 11. Assume that there existsb ∈ E such that Q(b, {b}) > 0. Dene h(x, y) := 1 {(b,b)} (x, y) and ν := Q(b, {b}) δ (b,b) . Let F ∈ E ⊗ E. If (b, b) / ∈ F , then ν(F ) = 0 and P (b, b); F ≥ 0. If (b, b) ∈ F , then P (b, b); F ≥ P (b, b); {(b, b)} = Q(b, {b}) = ν(F ).Hence the desired minorization condition holds. Conversely, assume that P satises a minorization condition. At a rst stage, assume that∃ b ∈ E such that [ h(b, y) > 0 ⇒ y = b ] .(43)Then from h(b, •)1 E\{b} (•) ≡ 0 and (42), we obtainh(b, b)Q(b, {b}) = E h(b, z)Q(b, dz) = P h(b, b) ≥ h(b, b)ν(h),thus Q(b, {b}) ≥ ν(h) > 0, which ends the proof of Q(b, {b}) > 0. Then it only remains to prove (43): let (a, b) ∈ E 2 be such that h(a, b) > 0, dene the following set T := {y ∈ E; h(b, y) > 0}, and let us prove that T = {b}. Let g(x, y) := h(x, y)1 E\{b} (x), then we obtain from g(b, •) ≡ 0 and (42):0 = E g(b, z)Q(b, dz) = P g(a, b) ≥ h(a, b)ν(g),thus ν(g) = 0. Let us dene f (x, y) := h(x, y)1 {b} (x). Since ν(h) > 0, we obtain ν(f ) = ν(h) -ν(g) > 0, which implies in particular that T is not empty. Let y 0 ∈ T , then we obtain from f (x, •)1 E\{b} (x) ≡ 0 and (42):δ b (y 0 )P f (b, b) = Ef (y 0 , z)Q(y 0 , dz) = P f (b, y 0 ) ≥ h(b, y 0 )ν(f ) > 0, which implies y 0 = b, and thus (43). 2

  (X n ) n∈IN * a sequence of i.i.d. E-valued r.v. whose common distribution depends on θ ∈ Θ, and {ξ p (•); p ∈ P} a family of measurable functions from E into IR. The following assertions are obviously equivalent: (a) The family { n k=1 ξ p (X k ); n ∈ IN * , p ∈ P} fullls Hypothesis R(m) if and only if for all (θ, p) ∈ Θ × P, E θ [ξ p (X 1 )] = 0, and sup (θ,p)∈Θ×P E θ [|ξ p (X 1 )| m ] < +∞. (b) The family { n k=1 ξ p (X k ); n ∈ IN * , p ∈ P} fullls Hypothesis (N-A) if and only if, for any compact subset K 0 of IR * , one has

  (9) links the characteristic function of S n (p) to the iterated Fourier operator Q θ,p (t) n . Thus, according to Equality (9), Hypothesis R(m) requires to study the behavior of the application t → Q θ,p (t) n near 0. A natural assumption to do it is to assume that there exists a Banach space B which contains the function 1 E and on which (Q θ ) θ∈Θ acts continuously (i.e.

	∀θ ∈ Θ, Q θ ∈ L(B)) and (Q θ ) θ∈Θ satises the following uniform strong ergodicity properties
	(ERG.1)-(ERG.2):
	ERG. 1. : {π θ ; θ ∈ Θ} is bounded in B .
	ERG. 2. : The transition kernel (Q θ ) θ∈Θ has a spectral gap on B (uniformly in θ), that is

and in the paper intitled "Regularity of the characteristic function of additive functionals for iterated function systems. Statistical applications"', which is to be submitted very soon (authors: D.Guibourg and D.Ferré).

One dierence is that the asymptotic bias b θ,p = 0 in the i.i.d. case.

A set A ∈ E is said to be π θ 0 -full if π θ 0 (A) = 1 and Q θ 0 -absorbing if Q θ 0 (a, A) = 1 for all a ∈ A.

A small part of this work has been announced in [Fer10, note without proof].

Indeed, if μ(E × {0}) = 0, then h = 1 µ-a.s, which is in contradiction with the fact that h < 1.

Note that the transition kernel Q θ has the following representation:

where x denotes the column vector (x d , . . . , x 1 ) and y denotes the column vector (y d , . . . , y 1 ) . Then, as already mentioned, the dominated model Condition (S ) is not fullled in the multidimensional case (d = 1).

Next, let us assume that the probability density f Z of Z 1 fullls the following conditions:

(A) ∀z ∈ IR, f Z (z) > 0;

(B) for all θ ∈ Θ, E θ,µ θ [Z 1 ] = 0;

(C) for all θ ∈ Θ, E θ,µ θ [|Z 1 | 10 ] < +∞;

(D) f Z is 4-time-dierentiable on IR;

(E) for j = 1, . . . , 4, f

Z /f Z is a bounded function; (F) for all 9 < γ ≤ 10, there exists 0 < β ≤ 1 -1/γ such that, for all x 0 ∈ IR, there exists a neighborhood V x 0 of x 0 and a positive measurable function q x 0 satisfying

Actually, under (E), it is sucient to assume in (C) that there exists some ε > 0 such that for all θ ∈ Θ, E θ,µ θ [Z 9+ε 1 ] < +∞. Furthermore, note that Assumption (E) can be relaxed provided that the order of the moment of Z 1 is increased. However, Assumption (E) as above is satised in several interesting models and thus it does not need relaxing. Notice that Götze and Hipp [GH94, th 1.5] assume that, under (E), Z 1 admits a moment of order 15.

Finally, the vector g := (g 1 , . . . , g d ) is supposed to have the following properties:

Let us dene for all 9 < γ ≤ 10

Lemma 8. Under the previous conditions, (X n ) n≥d-1 belongs the class of Models (M) with the function V dened in (41).

Proof of Lemma 8. Under (C), one has for all θ ∈ Θ and x ∈ IR d ,

d} for n suciently large, and where the real number d is dened by d := inf θ∈Θ m 2 (θ)/(4l(θ)) > 0, so that

Therefore the estimate (A) holds true.

In a second and last step, let us determine A θ such that (B) holds true. There exists some real r.v.

Let us introduce the r.v.

where the r.v. W n and the function l(•) are dened in (A.1).

Consider the following events

and notice that Q n,θ (u) = P θ { B 1 n,u,θ } and the following facts

• one obviously has M

(1)

n (α 0 )), and hence from (A.1), one has

Discussion under existence of an atom for (Y n ) n≥1

Under the ψ-irreducibility assumption, we suppose that (Y n ) n≥1 is Harris-recurrent and regenerative, i.e. there exists A ∈ E ⊗ E such that ψ(A) > 0 and ∀(x, y) ∈ A, ∀(x , y ) ∈ A, ∀F ∈ E ⊗ E, P (x, y); F = P (x , y ); F .

Such a set A is called an accessible atom for the Markov chain (Y n ) n≥1 . Notice that the assumption of Harris-recurrence is equivalent to assuming that

). The hitting time to A and the successive return times to A are dened as follows

and we also dene the number of visits of (Y n ) n≥1 before time n ≥ 1 to the set A by l n :=

) for all n ≥ 1, we have:

where the blocks of observations between consecutive visits to the atom A are denoted by B j := (Y 1+T A (j) , . . . , Y T A (j+1) ), and where ξ(B j ) :=

Introduce τ A (j) := T A (j) -T A (j -1) for all j ≥ 2. Let us remark that (B j , τ A (j + 1)) j≥1 is an i.i.d. sequence (this fact follows from the strong Markov property), and thus ( ξ(B j ), τ A (j + 1)) j≥1 is also an i.i.d. sequence.

We obtain for all Borel set B

Application of the results of [Jen89]

As explained just before, the application of [START_REF] Bertail | A renewal approach to markovian U-statistics[END_REF] to the double sequence (Y n ) n≥1 fails to provide Berry-Esseen theorem for many general models. That is why in this subsection, we directly deal with the simple sequence (X n ) n≥0 .

B.2.1

Discussion under existence of an atom for (X n ) n≥0

By now, assume that (X n ) n≥0 is ψ-irreducible, Harris-recurrent and that (X n ) n≥0 has an accessible atom denoted by A, i.e. ψ(A) > 0 and

and for all n ≥ 0, l n := n i=0 1 A (X i ). Then, for all n ≥ 1, we have:

where the blocks of observations between consecutive visits to the atom A are denoted by B j := (X 1+T A (j) , . . . , X T A (j+1) ), and where ξ(B j-1 , B j ) :=

Let τ A (j) denote τ A (j) := T A (j) -T A (j -1) for all j ≥ 2. Let us remark that although (B j , τ A (j + 1)) j≥1 is an i.i.d. sequence, the sequence ( ξ(B j-1 , B j ), τ A (j + 1)) j≥1 is no longer i.i.d. but one-dependent.

We obtain for all Borel set B

where Z 1,a , Z 2,b and Z 3,c are distributed according to the following measures

Notice that Z 1,a , Z 2,b and Z 3,c are no longer independent and that ( ξ(B j-1 , B j ), τ A (j + 1)) j≥1 is a two-dimensional one-dependent sequence which is lattice in one component. To establish a initial probability µ of the chain (X n ) n≥0 . In particular, they hold true in the simple case where µ is a Dirac distribution at x 0 ∈ E. Furthermore, the eective control of constants in the regenerative method probably still needs much work to be done (even for univariate theorem, cf. [START_REF] Bertail | A renewal approach to markovian U-statistics[END_REF]).