D Guibourg
email: denis.guibourg@ens.insa-rennes.fr

L Hervé

Multidimensional renewal theory in the non-centered case. Application to strongly ergodic Markov chains

Keywords:

Let (Sn) n≥0 be a R d -valued random walk (d ≥ 2). Using Babillot's method [2], we give general conditions on the characteristic function of Sn under which (Sn) n≥0 satisfies the same renewal theorem as in the independent case (i.e. the same conclusion as in the case when the increments of (Sn) n≥0 are assumed to be independent and identically distributed). This statement is applied to additive functionals of strongly ergodic Markov chains under the non-lattice condition and (almost) optimal moment conditions.

Introduction

Let (S n ) n≥0 be a R d -valued random walk. Renewal theory gives the behavior, as a → +∞, of the positive measures U a (•) defined on the Borel σ-algebra B(R d ) of R d as follows :

∀a ∈ R d , ∀A ∈ B(R d ), U a (A) = +∞ n=1 E 1 A (S n -a) . (1)
To define the renewal measure U a (•), the sequence (S n ) n≥0 has to be transient: for independent or Markov random walks, this leads to consider the following cases: The behavior of U a (•) also depends on the usual lattice or non-lattice conditions.

This work is the continuation of [START_REF] Guibourg | Théorème de renouvellement pour chaînes de Markov fortement ergodiques. Applications aux modèles itératifs Lipschitziens[END_REF] (case d = 1) and [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF] (centered case in dimension d ≥ 3). More specifically, in this paper, we consider the non-centered case in dimension d ≥ 2, and we present some general assumptions involving the characteristic function of S n , under which we have the same conclusion as in the classical renewal theorem for random walks with independent and identically distributed (i.i.d.) increments. By using the weak spectral method [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF], this result is then applied to additive functionals of strongly ergodic Markov chains. This work is greatly inspired by Babillot's paper [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF]. Before presenting our results, we give a brief review of well-known multidimensional renewal theorems in both independent and Markov settings, as well as some general comments on Fourier's method. To that effect we introduce some notations which will be repeatedly used afterwards:

-•, • is the canonical scalar product on R d , -• is the associated euclidean norm on R d , -L d (•) is the Lebesgue-measure on R d , -C c (R d , C) is the set of complex-valued continuous compactly supported functions on R d , -the Fourier transform of any Lebesgue-integrable function f : R d → C is defined as follows: ∀t ∈ R d , f (t) := L d (e -i t,• f ), -H is the set of complex-valued continuous Lebesgue-integrable functions on R d , whose Fourier transform is compactly supported and infinitely differentiable on R d , -for any R > 0, we denote by B R := B(0, R) the open ball: B(0, R) := {t ∈ R d : t < R}, -for any 0 < r < b, we denote by K r,b the annulus K r,b := {t ∈ R d : r < t < b}.

Renewal theory for random walks with i.i.d. non-centered increments.

Let (X n ) n≥1 be a sequence of i.i.d. non-centered random variables (r.v) taking values in R d , and let S n = X 1 + . . . + X n . In dimension d ≥ 2, the renewal theorem was first established by Ney and Spitzer [START_REF] Ney | The Martin boundary for random walk[END_REF] in the lattice case. Extension to the non-lattice case was obtained by Doney [START_REF] Doney | An analogue of the renewal theorem in higher dimensions[END_REF] under Cramer's condition, and by Stam [START_REF] Stam | Renewal theory in r dimensions[END_REF] under the weaker non-lattice condition. Setting m d := max( d-1 2 , 2), Stam's statement writes as follows:

E X 1 m d < ∞, m := E[X 1 ] = 0 ⇒ ∀g ∈ C c (R d , R), lim τ → +∞ τ d-1 2 U τ m (g) = C L d (g) (2)
where C is a positive constant depending on the first and second moments of X 1 . In the lattice case, Property (2) still holds, but L d (•) must be replaced with the product of counting and Lebesgue measures both defined on some sublattices of R d . Stam's proof is based on the local limit theorem (LLT) due to Spitzer [22, Th. P7.10] 1 . More precisely, this LLT is applied to study the difference

+∞ n=1 n (d-1)/2 E[g(S n -a)] -E[g(T n -a)] ,
where the r.v. T n are defined as the partial sums of a i.i.d. sequence of Gaussian r.v. having the same first and second moments as X 1 . Then (2) is deduced from the Gaussian case.

Fourier techniques in renewal theory (Breiman's method).

The weak convergence in (2) can be established by investigating the behavior of U a (h) for h ∈ H. In fact the inverse Fourier formula gives (without any assumption on the model):

∀h ∈ H, E h(S n -a) = (2π) -d R d ĥ(t) E[e i t,Sn ] e -i t,a dt.

(3) This is the starting point of Fourier's method in probability theory. In the i.i.d. case, using [START_REF] Benda | A central limit theorem for contractive stochastic dynamical systems[END_REF] and the formula E[e i t,Sn ] = E[e i t,X 1 ] n , the potential U a (h) given by ( 1) is equal to the following integral:

U a (h) = I(a) := (2π) -d K ĥ(t) φ(t) 1-φ(t) e -i t,a dt with φ(t) = E[e i t,X 1 ],

where K is the support of ĥ. More precisely, since E[X 1 ] = 0 and d ≥ 2, the integrand in I(a) is integrable at 0. Thus I(a) is well-defined provided that |φ(t)| < 1 for all t = 0: this is the non-lattice condition. Fourier's method also applies to the lattice case by considering a periodic summation in [START_REF] Breiman | Probability Classic in Applied Mathematics[END_REF]. The renewal theorem then follows from the study of the integrals I(τ m) when τ → +∞. This method, introduced by Breiman [START_REF] Breiman | Probability Classic in Applied Mathematics[END_REF] in dimension d = 1, was extended to d ≥ 2 by Babillot [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] in the general setting of Markov random walks (see below).

Renewal theory for Markov random walks.

Let (E, E) denote a measurable space, and let (X n , S n ) n∈N be an E × R d -valued Markov random walk (MRW), namely: (X n , S n ) n∈N is a Markov chain and its transition kernel P satisfies the following additive property (in the second component):

∀(x, s) ∈ E × R d , ∀A ∈ E, ∀B ∈ B(R d ), P (x, s), A × B = P (x, 0), A × (B -s) . (5)
As usual we set S 0 = 0. When (X n ) n≥0 is strongly ergodic and S 1 is non-centered, Babillot gives in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] some (operator-type) moment and non-lattice conditions for the additive component (S n ) n to satisfy the renewal conclusion in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF]. Recall that the strong ergodicity condition states that the transition kernel Q of (X n ) n≥0 admits an invariant probability measure π, and that there exists a Banach space (B, • B ), composed of π-integrable functions on E and containing the function 1 E , such that π defines a continuous linear form on B and

lim n → +∞ sup f ∈B, f ≤1 Q n f -π(f )1 E B = 0. (6)
The proof in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] is based on Fourier techniques and the usual Nagaev-Guivarc'h spectral method involving the semi-group of Fourier operators associated with (X n , S n ) n∈N , namely:

∀n ∈ N, ∀t ∈ R d , ∀x ∈ E, Q n (t)f (x) := E (x,0) e i t,Sn f (X n ) ,
where E (x,0) denotes the expectation under the initial distribution (X 0 , S 0 ) ∼ δ (x,0) . The operators Q n (t) act (for instance) on the space of bounded measurable functions f : E → C. The semi-group property writes as follows:

∀(m, n) ∈ N 2 , Q m+n (t) = Q m (t) • Q n (t). In particular we have Q n (t) = Q 1 (t) n .
This property is the substitute for MRWs of the formula

E[e i t,Sn ] = E[e i t,X 1 ] n of the i.i.d. case.
The content of the paper.

Section 2 focuses on Fourier's method. More specifically we consider a general sequence (X n , S n ) n∈N (not necessarily a MRW) of random variables taking values in E × R d . In substance our non-centered condition writes as follows: Sn , Theorem 1 states that there exists some positive constant C (specified later) such that we have

m := lim n E[S n ]/n exists in R d and is nonzero. Let f : E →[0, +∞) be such that E[f (X n )] < ∞ for every n ≥ 1. Under a general hypothesis, called R(m), on the functions t → E f (X n ) e i t,
∀g ∈ C c (R d , R), τ d-1 2 +∞ n=1 E f (X n ) g(S n -a) -→ C L d (g)
when a := a(τ ) ∈ R d goes to infinity "around the direction m" in the sense (specified later) defined in [START_REF] Thirion | Propriétés de mélange du flot des chambres de Weyl des groupes de Ping-Pong[END_REF] (for instance, take a := τ m with τ → +∞). Actually Hypothesis R(m), introduced in [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF] (centered case), contains the tailor-made conditions to prove renewal theorems via Fourier's method. The proof of Theorem 1 borrows the lines of [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] with the following improvements. First, the distribution-type arguments and the modified Bessel functions used in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] are replaced with elementary computations. Second, the (asymmetric) dyadic decomposition, partially developed in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF][START_REF] Babillot | Le noyau potentiel des chaînes semi-markoviennes[END_REF] to study integrals of type (4), is detailed in this work.

Section 3 is devoted to the Markov context. Specifically, we assume that (X n ) n∈N is a Markov chain satisfying one of the three following classical strong ergodicity assumptions: [START_REF] Meyn | Markov chains and stochastic stability[END_REF]), -(X n ) n≥0 is a strictly contractive Lipschitz iterative model (see [START_REF] Duflo | Random Iterative Models[END_REF]).

-(X n ) n≥0 is ρ-mixing (see [20]), -(X n ) n≥0 is V -geometrically ergodic (see
Let ξ be a R d -valued measurable function, and let S n = ξ(X 1 )+. . .+ξ(X n ). Then the sequence (X n , S n ) n∈N is a special instance of MRW. As already used in [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF], the weak spectral method [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF] allows us to reduce Hypothesis R(m) to a non-lattice condition and to some (almost) optimal moment conditions on ξ, which are much weaker than those in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF].

Theorem 1 should supply further interesting applications, not only in Markov models but also in dynamical systems associated with quasi-compact Perron-Frobenius operators. On that subject, recall that the renewal theorems yield the asymptotic behavior of counting functions arising in the geometry of groups, as already developed for instance in [START_REF] Lalley | Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits[END_REF][START_REF] Dal'bo | Comportement asymptotique du nombre de géodésiques fermées sur la surface modulaire en courbure non constante. Études spectrales d'opérateurs de transfert et applications[END_REF][START_REF] Thirion | Propriétés de mélange du flot des chambres de Weyl des groupes de Ping-Pong[END_REF].

Renewal theory in the non-centered case (Fourier method)

For any A ⊂ R d , g : A → C, and τ ∈ (0, 1], we define the following quantities in [0, +∞]:

g 0,A = sup x∈A |g(x)| and g τ,A := sup |g(x) -g(y)| x -y τ , (x, y) ∈ A 2 , x = y .
We say that g is τ -Hölder on A if f is ⌊m⌋-times continuously differentiable on O, Each partial derivative of order j = 0, . . . , ⌊m⌋ of f is bounded on O, Each partial derivative of order ⌊m⌋ of f is τ -hölder on O.

Define ∇f := ( ∂f ∂x i ) 1≤i≤d if m ≥ 1, and
Hess f := ( ∂ 2 f ∂x i ∂x j ) 1≤i,j≤d (Hessian matrix) if m ≥ 2.
Let (Ω, F , P) be a probability space. We denote by (E, E) a measurable space, and we consider a sequence (X n , S n ) n≥0 of E×R d -valued random variables defined on Ω. Throughout, we assume d ≥ 2.

Hypothesis R(m). Given m ∈ [2, +∞) and f : E →[0, +∞) a measurable function satisfying

∀n ≥ 1, E[f (X n )] < ∞, (7)
we say that Hypothesis R(m) holds if the following conditions are fulfilled:

(i) There exists R > 0 such that, for all t ∈ B R and all n ≥ 1, we have:

E f (X n ) e i t,Sn = λ(t) n L(t) + R n (t), (8)
where λ(0) = 1, the functions λ( Below we assume that m = 0: this is our non-centered condition. In fact, under Hypothesis R(m) and additional mild conditions (see [13, Prop. 1]), we have m = lim n E[S n ]/n, so that m may be viewed as a nonzero mean vector in R d . Below we also assume that the symmetric matrix Σ is positive-definite. In the Markov setting of Section 3, Σ is linked to some covariance matrix, see (36).

Hypothesis (H). Setting m d := max( d-1 2 , 2), there exists a real number m > m d such that Hypothesis R(m) holds. We have L(0) = 0, m = 0, and Σ := -Hess λ(0) is positive-definite.

Theorem 1 Assume that Hypothesis (H) holds. Then, for each function a : [0, +∞) → R d such that

A := lim τ →+∞ a(τ ) -τ m √ τ exists in R d , (9)
the family {V τ (•), τ ∈ (0, +∞)} of positive measures on R d defined by

∀A ∈ B(R d ), V τ (A) := (2πτ ) d-1 2 +∞ n=1 E f (X n ) 1 A (S n -a(τ )) (10)
weakly converges to C L d (•) as τ → +∞, where C := C(L, m, Σ, A) ∈ (0, +∞) is given by:

C(L, m, Σ, A) := L(0) det S S m exp S m, SA 2 -S m 2 SA 2 2 S m 2 with S := Σ -1 2 .
Condition [START_REF] Ferré | Limit theorems for stationary Markov processes with L 2 -spectral gap[END_REF], introduced in [START_REF] Thirion | Propriétés de mélange du flot des chambres de Weyl des groupes de Ping-Pong[END_REF], specifies what we called "around the direction m" in Introduction. Obviously a(τ ) = τ m satisfies [START_REF] Ferré | Limit theorems for stationary Markov processes with L 2 -spectral gap[END_REF].

Remark 1 Theorem 1 may be extended to the lattice case: Hypothesis R(m)(ii) must be adapted, and L d (•) is replaced with the product of the counting measure and the Lebesgue measure both associated with some sublattices of R d , see [START_REF] Guibourg | Théorèmes de renouvellement pour des fonctionnelles additives associées à des chaînes de Markov fortement ergodiques[END_REF]Sect. 2.5].

The next subsections are devoted to the proof of Theorem 1.

Some reductions and Fourier techniques

• Change of coordinates. Let e 1 denote the first vector of the canonical basis of R d , and let T be any isometric linear map in R d such that T ( m) = m e 1 . Up to replace S n with T S n , one may assume without loss of generality that m = m e 1 . This leads to replace

λ(•), L(•), R n (•), h(•), Σ with λ • T -1 , L • T -1 , R n • T -1 , h • T -1 , T • Σ • T -1 .
• The function w. For all x = (x 1 , x 2 , . . . , x d ) ∈ R d , we set x ′ := (x 2 , . . . , x d ) ∈ R d-1 . The following function w(•) will play an important role:

∀x ∈ R d , w(x) = -ix 1 + x ′ 2 . ( 11
)
Remark 2 We have:

∀x ∈ R d , |w(x)| ≥ |x 1 | 3/4 x ′ 1/2
. Thus 1/w is integrable at 0.

Remark 3 Some simple facts on the function λ(•) of ( 8) can be deduced from Hypothesis (H).

First, since m d ≥ 2, we have

λ(t) = 1 + i m t 1 -1 2 Σt, t + o( t 2 ). (12)
Second, since L(•) is continuous on B R and L(0) = 0, one may suppose (up to reduce R) that we have: ∀t ∈ B R , L(t) = 0. By Hypothesis R(m)(ii), the last property then implies that, for all t ∈ B R \ {0}, the series n≥1 λ(t) n converges. Hence:

∀t ∈ B R \ {0}, |λ(t)| < 1. Third the function v 0 (•) := 1 -λ(•) is in C m b B R , C
), and thanks to -i∇λ(0) = m = m e 1 , we have ∀j ∈ {2, . . . , d}, ∂v 0 ∂x j (0) = 0.

Finally, up to reduce R, we deduce from ( 12) that there exist positive constants α, β such that:

∀t ∈ B R , α|w(t)| ≤ |v 0 (t)| ≤ β|w(t)|.
• Use of the space H. Thanks to the well-known results on weak convergence of positive measures (see [START_REF] Breiman | Probability Classic in Applied Mathematics[END_REF]), Theorem 1 will hold provided that we prove the following property:

∀h ∈ H, lim τ →+∞ (2πτ ) d-1 2 +∞ n=1 E f (X n ) h(S n -a(τ )) = C(L, m, Σ, A) R d h(x)dx. ( 13
)
• Integral decomposition. Let h ∈ H be fixed and let b > 0 such that ĥ(t) = 0 when t > b.

Next consider any real number ρ such that 0 < ρ < min(R, b) and any function χ

∈ C ∞ b (R d , R) compactly supported in B R such that χ(t) = 1 when t ≤ ρ. For each t ∈ R d we set E n (t) := E[f (X n ) e i t,Sn ]. The inverse Fourier formula gives (2π) d E f (X n ) h(S n -a) = R d ĥ(t) E n (t) e -i t,a dt.
Under Hypothesis (H), we prove below that, for every a ∈ R d , the following series

I(a) := (2π) d +∞ n=1 E f (X n ) h(S n -a)
converges, and that I(a) decomposes as the sum of three integrals called E(a), E 1 (a) and I 1 (a). The integrals E(a) and E 1 (a) are error terms, while I 1 (a) is the main part of I(a). In fact we have

I(a) = E(a) + B R χ(t) ĥ(t) λ(t) 1 -λ(t) L(t)) e -i t,a dt (14)
with

E(a) := R d ĥ(t) 1 B R (t) χ(t) n≥1 R n (t) + 1 K ρ,b (t) 1 -χ(t) n≥1 E n (t) e -i t,a dt.
Next we obtain

I(a) = E(a) + E 1 (a) + I 1 (a) (15)
with

E 1 (a) := t ≤R χ(t) ĥ(t)λ(t) L(t) -ĥ(0) L(0) 1 -λ(t) e -i t,a dt + ĥ(0) L(0) t ≤R χ(t) λ(t) -1 -i m t 1 + 1 2 Σt, t (1 -λ(t))(-i m t 1 + 1 2 Σt, t ) e -i t,a dt (16)
and

I 1 (a) := ĥ(0) L(0) t ≤R χ(t) -i m t 1 + 1 2 Σt, t e -i t,a dt.
Such equalities are established in [13, p. 389] (centered case). By using Hypothesis (H), the proof of ( 14) borrows the same lines. To obtain [START_REF] Hervé | A Berry-Esseen theorem on M for geometrically ergodic Markov chains[END_REF], use the fact that for t ≤ R, t = 0, we have

| N n=1 λ(t) n | ≤ 2/(b|w(t)|)
, and the fact that 1/w is integrable at 0 (cf. Remarks 2-3).

Property [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF] then follows from [START_REF] Hervé | A Berry-Esseen theorem on M for geometrically ergodic Markov chains[END_REF] and the next properties (17) ( 18) and ( 23).

Study of the first error term E(a)

Here we prove that we have when a → +∞:

E(a) = o( a -d-1 2 ). ( 17
) For u ∈ C m b (R d , C) and α = (α 1 , . . . , α d ) ∈ N d such that |α| := d i=1 α i ≤ ⌊m⌋,
we denote by ∂ α the derivative operator defined by :

∂ α := ∂ |α| ∂x α 1 1 . . . ∂x α d d = ∂ α 1 1 . . . ∂ α d d where ∂ j := ∂ ∂x j . The following proposition is classical. Let O be a bounded open subset of R d . Proposition 1 Let m ∈ (0, +∞) and τ = m -⌊m⌋. Assume that u is a function in C ⌊m⌋ b (R d , C) compactly supported in O and that its restriction to O is in C m b (O, C).
Then the following properties hold:

(i) u ∈ C m b (R d , C
), and for α ∈ N d , |α| = ⌊m⌋, we have:

[∂ α u] τ,R d = [∂ α u] τ,O = [∂ α u] τ,O (ii) ∃C ∈ (0, +∞), ∀a ∈ R d , a m |û(a)| ≤ C u 0,R d + |α|=⌊m⌋ [∂ α u] τ,R d .
Proof of [START_REF] Lalley | Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits[END_REF]. Define:

∀t ∈ B R , R(t) := +∞ n=1 R n (t) and ∀t ∈ R d \ {0}, E(t) := +∞ n=1 E n (t), ∀t ∈ R d , F (t) := 1 B R (t) ĥ(t) χ(t) R(t) and G(t) := 1 K ρ,b (t) ĥ(t) (1 -χ(t)) E(t) if t = 0 0 if t = 0.
We have

∀a ∈ R d , E(a) = F (a) + Ĝ(a). Note that χ ĥ ∈ C ∞ c (R d , C
) is compactly supported in the closed ball B R , and that, by

Hypothesis (H), we have R ∈ C m b (B R , C) with m > m d . By applying Proposition 1 to u := F , we obtain a m | F (a)| = O(1), thus F (a) = o( a -(d-1)/2 ) when a → +∞ since m > (d -1)/2. The same result holds for Ĝ(a) (replace B R with K ρ,b ).

Study of the second error term E 1 (a)

In this subsection we prove that we have when a → +∞:

E 1 (a) = o( a -(d-1)/2 ). (18)
Let E 11 (a) and E 12 (a) denote the two integrals in the right hand side of ( 16), so that we have:

E 1 (a) := E 11 (a) + E 12 (a). Define: ∀t ∈ B R , θ 1 (t) = χ(t) ĥ(t)λ(t) L(t) -ĥ(0) L(0) . Then E 11 (a) = q 1 (a) with q 1 := 1 B R θ 1 /v 0 , (19)
where

v 0 (t) = 1 -λ(t). Next define: ∀t ∈ B R , θ 2 (t) := χ(t) λ(t) -1 -i m t 1 + 1 2 Σt, t and ṽ0 (t) := -i m t 1 + 1 2 Σt, t . Then E 12 (a) = q 2 (a) with q 2 := 1 B R θ 2 /(v 0 ṽ0 ). (20)
Unfortunately, since q 1 and q 2 are not defined at 0, q 1 (a) and q 2 (a) cannot be studied by the elementary arguments of Subsection 2.2. This fact constitutes the main difficulty of the proof in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF]. Below, we present the two key results (Propositions 2 and 3) to obtain [START_REF] Meyn | Markov chains and stochastic stability[END_REF]. In the next subsection, these two propositions are also used to obtain the desired result for the main part I 1 (a) (by difference with the Gaussian case, see Lemma 1).

Recall that w(•) is defined in [START_REF] Guibourg | Théorème de renouvellement pour chaînes de Markov fortement ergodiques. Applications aux modèles itératifs Lipschitziens[END_REF]. In the two next propositions, we consider any real numbers m > m d and r > 0.

Proposition 2 Let θ and v be complex-valued functions on B r such that:

• θ ∈ C m b B r , C with compact support in B r and θ(0) = 0, • v ∈ C m b B r , C and ∀j ∈ {2, . . . , d}, (∂ j v)(0) = 0 (21) ∃(a, b) ∈ (R * + ) 2 , ∀x ∈ B r , a|w(x)| ≤ |v(x)| ≤ b|w(x)|. ( 22
)
Then q := 1 Br θ/v is integrable on R d and lim a →+∞ a (d-1)/2 q(a) = 0.

Proposition 3 In addition to the hypotheses of Proposition 2, we consider another complexvalued function ṽ on B r satisfying the same hypotheses as v(•). Moreover we assume that all the first and second partial derivatives of θ vanish at 0. Then q := 1 Br θ/(v ṽ) is integrable on R d and lim a →+∞ a (d-1)/2 q(a) = 0.

The proofs of Propositions 2-3, based on dyadic decompositions, are partially presented in [START_REF] Babillot | Le noyau potentiel des chaînes semi-markoviennes[END_REF][START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF]. Since dyadic decomposition is not familiar to probabilistic readers, these proofs are detailed in Appendix A for the sake of completeness.

Proof of [START_REF] Meyn | Markov chains and stochastic stability[END_REF]. Proposition 2 applied with θ := θ 1 and v := v 0 (see Rk. 3) gives q 1 (a) = o( a -(d-1)/2 ) when a → +∞. Similarly Proposition 3 applied with θ := θ 2 , v := v 0 and ṽ := ṽ0 gives q 2 (a) = o( a -(d-1)/2 ). Then [START_REF] Meyn | Markov chains and stochastic stability[END_REF] follows from (19) (20).

Study of the main part I 1 (a) of I(a)

Let a : R + → R d be a measurable function satisfying [START_REF] Ferré | Limit theorems for stationary Markov processes with L 2 -spectral gap[END_REF]. Here we prove that

lim τ →+∞ τ (d-1)/2 I 1 (a(τ )) = (2π) (d+1)/2 C(L, m, Σ, A) ĥ(0). ( 23
)
The proof of ( 23) in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] involves the modified Bessel functions and some related computations partially made in the book [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF]. Here we present a direct and simpler proof of (23) based on the next proposition. We denote by S(R d ) the so-called Schwartz space.

Proposition 4 Let w ∈ R d \ {0}, and let p : R + → R d such that

P := lim τ →+∞ p(τ )/ √ τ exists in R d . Then we have for all function k ∈ S(R d ) lim τ →+∞ τ d-1 2 R d k(u) e -i u,τ w+p(τ ) -i w, u + u 2 du = 2 π d+1 2 w k(0) exp - P 2 w 2 -P, w 2 4 w 2 .
The proof of Proposition 4 (again based on Propositions 2-3) is presented below. Let us first apply Proposition 4 to establish [START_REF] Stam | Renewal theory in r dimensions[END_REF].

Proof of [START_REF] Stam | Renewal theory in r dimensions[END_REF]. Since m = m e 1 , one can rewrite (9) as follows:

a(τ ) = τ m e 1 + √ τ b(τ ) with b : [0, +∞) → R d such that lim τ →+∞ b(τ ) = A. Denote by λ 1 , . . . , λ d the (positive) eigenvalues of Σ. Let ∆ := diag( √ λ 1 , . . . , √ λ d
) and let P be any orthogonal d × d-matrix such that

P -1 ΣP = ∆ 2 = diag(λ 1 , . . . , λ d ).
Observe that Σt, t = ∆ 2 P -1 t, P -1 t = ∆P -1 t 2 . Set ℓ := ∆ -1 P -1 e 1 . By using the variable t = P ∆ -1 u, one obtains (use t 1 = P ∆ -1 u, e 1 = u, ℓ )

I 1 (a) = 2 ĥ(0) L(0) (det ∆) -1 χ(P ∆ -1 u)e -i ∆ -1 u,P -1 a -i 2 m ℓ, u + u 2 du. Set ζ(x) := χ(P ∆ -1 x) (x ∈ R d ) and p(τ ) := √ 2τ ∆ -1 P -1 b(2τ ) (τ > 0)
. From the equality ∆ -1 u, P -1 e 1 = u, ℓ , it follows that

I 1 (a(τ )) = 2 ĥ(0) L(0) (det ∆) -1 χ(P ∆ -1 u)e -i ∆ -1 u , P -1 τ m e 1 + √ τ b(τ ) -i 2 m ℓ, u + u 2 du = 2 ĥ(0) L(0) (det ∆) -1 ζ(u)e -i u , 2( τ 2 ) m ℓ+p( τ 2 ) -i 2 m ℓ, u + u 2 du. Now ∆ -1 = P -1 Σ -1 2 P gives ℓ = P -1 Σ -1 2 e 1 , so ℓ = Σ -1 2 e 1 and Σ -1 2 m = m ℓ . Moreover, we have ζ(0) = χ(0) = 1 and lim τ →+∞ p(τ )/ √ τ = P with P := √ 2 ∆ -1 P -1 A = √ 2 P -1 Σ -1 2 A.
From Proposition 4, applied with P previously defined, w := 2 m ℓ = 2P -1 Σ -1 2 m, and finally with the function k := ζ, one obtains:

lim τ →+∞ ( τ 2 ) d-1 2 I 1 (a(τ )) = 2 ĥ(0) L(0) (det Σ) -1 2 2π d+1 2 2 Σ -1 2 m × exp - Σ -1 2 m 2 Σ -1 2 A 2 -Σ -1 2 m, Σ -1 2 A 2 2 Σ -1 2 m 2
, from which we easily deduce [START_REF] Stam | Renewal theory in r dimensions[END_REF].

Proof of Proposition 4. Let U be an isometric linear map on

R d such that U ( w) = w e 1 . Let τ > 0. The change of variable v = (v 1 , v ′ ) = U (u) in the integral of Proposition 4 gives R d k(u) e -i u,τ w+p(τ ) -i w, u + u 2 du = R d k(U -1 (v)) e -iτ w v 1 e -i v,U (p(τ )) -i w v 1 + v 2 dv,
and by hypothesis we know that lim τ →+∞ U (p(τ ))/ √ τ = U (P). Set U (P) := (ℓ 1 , ℓ ′ ) with

ℓ 1 ∈ R et ℓ ′ ∈ R d-1 .
As U is isometric, we have U (P) = P and ℓ 1 = U (P), e 1 = P, U -1 ( e 1 ) = P, w / w . Thus

ℓ ′ 2 = P 2 -P, w 2 / w 2 = w 2 P 2 -P, w 2 / w 2 . Next, let us set U (p(τ )) := u(τ ) = (u 1 (τ ), • • • , u d (τ )), and u ′ (τ ) = (u 2 (τ ), • • • , u d (τ )). Then R d k(u)e -i u,τ w+p(τ ) -i w, u + u 2 du = R d k(U -1 (v)) e -i(τ w +u 1 (τ ))v 1 e -i u ′ (τ ),v ′ -i w v 1 + v 2 dv.
Observe that the function k • U -1 is in S(R d ) and that

lim τ →+∞ u 1 (τ )/τ = 0 and lim τ →+∞ u ′ (τ )/ √ τ = ℓ ′ . (24)
Consequently Proposition 4 will be established if we prove that, for any µ ∈ R, µ = 0, and any h ∈ S(R d ), we have:

lim τ →+∞ τ (d-1)/2 R d h(v) e -i(τ µ+u 1 (τ ))v 1 e -i u ′ (τ ),v ′ -iµv 1 + v 2 dv = 2π (d+1)/2 e -ℓ ′ 2 /4 |µ| h(0). ( 25
)
Lemma 1 Property (25) holds with

h(v) = H(v) := e -(µ 2 v 2 1 + v ′ 4 )/2
, namely:

J µ (τ ) := R d e -(µ 2 v 2 1 + v ′ 4 )/2 e -i(τ µ+u 1 (τ ))v 1 e -i u ′ (τ ),v ′ -iµv 1 + v ′ 2 dv ∼ τ →+∞ 2π (d+1)/2 e -ℓ ′ 2 /4 |µ| τ (d-1)/2 . ( 26
)
Let us first assume that Lemma 1 is valid, and let us deduce ( 25) from [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF]. To that effect, we shall proceed by difference and use Propositions 2 and 3. For any τ > 0, we define d(τ

) := (τ µ + u 1 (τ ), u ′ (τ )). Moreover we set ∀v ∈ R d \ {0}, ∆(v) := h(v) -h(0)H(v) -iµv 1 + v 2 -h(0) v 2 1 H(v) (-iµv 1 + v 2 )(-iµv 1 + v ′ 2 ) . (27)
We have:

R d h(v) e -i(τ µ+u 1 (τ ))v 1 e -i u ′ (τ ),v ′ -iµv 1 + v 2 dv -h(0)J µ (τ ) = R d e -i(τ µ+u 1 (τ ))v 1 e -i u ′ (τ ),v ′ ∆(v) dv = R d e -i d(τ ), v ∆(v) dv.
Thanks to [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF], Property (25) will be established provided that we prove the following:

lim τ →+∞ τ d-1 2 R d e -i d(τ ), v ∆(v) dv = 0. (28)
Let us consider the functions

G 1 and G 2 in S(R d ) defined by G 1 (v) = h(v) -h(0)H(v) and G 2 (v) = v 2 1 H(v). One can easily check that G 1 (0) = G 2 (0) = 0, and ∀j ∈ {1, . . . , d}, (∂ j G 2 )(0) = 0 and (∂ 2 1 G 2 )(0) = 2, ∀(j, ℓ) ∈ {1, . . . , d} 2 \ {(1, 1)}, (∂ 2 jℓ G 2 )(0) = 0. with ∂ 2 jℓ := ∂ 2 ∂x j ∂x ℓ . Next, let g ∈ S(R d ) such that g(0) = -1/µ 2 , and define ∀v = (v 1 , . . . , v d ) ∈ R d , s(v) := -iµv 1 + v 2 -iµv 1 + v ′ 2 g(v),
where v ′ = (v 2 , . . . , v d ). One can easily check that s(0) = 0, ∀j ∈ {1, . . . , d}, (∂ j s)(0) = 0, (∂ 2 1 s)(0) = -2µ 2 g(0) = 2 and ∀(j, l) ∈ {1, . . . , d} 2 \ {(1, 1)}, (∂ 2 j l s)(0) = 0. Therefore the first and second derivatives of the difference G 3 := G 2 -s vanish at 0. Rewrite ∆(v) as:

∀v ∈ R d \ {0}, ∆(v) = G 1 (v) -iµv 1 + v 2 -h(0) G 3 (v) (-iµv 1 + v 2 )(-iµv 1 + v ′ 2 ) -h(0)g(v). Let γ ∈ C ∞ b (R d , [0, 1]
) be compactly supported and such that γ |B = 1 for some closed ball B of R d centered at 0. Since γg ∈ S(R d ) and lim τ →+∞ d(τ ) = +∞, we have

lim τ →+∞ d(τ ) d-1 2 R d e -i d(τ ), v γ(v) g(v) dv = 0.
Moreover Propositions 2 and 3 yield the following properties:

lim τ →+∞ d(τ ) (d-1)/2 R d e -i d(τ ), v γ(v)G 1 (v) -iµv 1 + v 2 dv = 0, lim τ →+∞ d(τ ) (d-1)/2 R d e -i d(τ ), v γ(v)G 3 (v) (-iµv 1 + v 2 )(-iµv 1 + v ′ 2 ) dv = 0.
Note that we have d(τ

) ∼ τ →+∞ |µ|τ from d(τ ) = (τ µ + u 1 (τ ), u ′ (τ )
) and ( 24). Hence:

lim τ →+∞ τ (d-1)/2 R d e -i v,d(τ ) γ(v) ∆(v) dv = 0. ( 31
)
Now observe that all the derivatives of γ(•) are bounded on R d and that ∆ is defined on R d \ {0} as the quotient of a function of S(R d ) by a polynomial function (cf. ( 27)). Since 1 -γ(•) vanishes on the closed ball B (centered at 0), we deduce that

(1 -γ)∆ ∈ S(R d ). So the Fourier transform of (1 -γ)∆ is in S(R d ) and we have lim τ →+∞ τ (d-1)/2 R d e -i d(τ ), v (1 -γ(v))∆(v)dv = 0. (32)
Hence (28) follows from (31) and (32). As already mentioned, we have (28) ⇒ (25), so that Proposition 4 is proved.

Proof of Lemma 1. It suffices to prove Lemma 1 in case µ = 1 (if not, set

w 1 = µv 1 , w ′ = v ′ ). Since +∞ 0 v -3/4 1 e -v 2 1 /2 dv 1 < ∞ and R d-1 v ′ -1/2 e -v ′ 4 /2 dv ′ < ∞ (use 1 2 < d -1 for the second integral), it follows from the Fubini-Tonelli theorem that R d e -(v 2 1 + v ′ 4 )/2 |-iv 1 + v ′ 2 | dv < ∞ (cf. Rk. 2). Next we have: ∀n ∈ N * , ∀b ∈ R n , (2π) -n/2 R n e -x 2 /2 e -i x,b dx = e -b 2 /2 (33a) ∀v ′ = 0, +∞ 0 e (iv 1 -v ′ 2 )u du = 1 -iv 1 + v ′ 2 . (33b)
From (33b), Fubini's theorem and (33a), it follows that

J 1 (τ ) = R d-1 e -i u ′ (τ ),v ′ e -v ′ 4 /2 R e -i(τ +u 1 (τ ))v 1 e -v 2 1 /2 -iv 1 + v ′ 2 dv 1 dv ′ = R d-1 e -i u ′ (τ ),v ′ e -v ′ 4 /2 +∞ 0 e -v ′ 2 u R e -i(τ +u 1 (τ )-u)v 1 e -v 2 1 /2 dv 1 du dv ′ = √ 2π R d-1 e -i u ′ (τ ),v ′ e -v ′ 4 /2 +∞ 0 e -v ′ 2 u e -(τ +u 1 (τ )-u) 2 /2 du dv ′ .
By using the following obvious equality

v ′ 2 u + τ + u 1 (τ ) -u 2 /2 = 1 2 u + v ′ 2 -(τ + u 1 (τ )) 2 -v ′ 4 + 2 τ + u 1 (τ ) v ′ 2 ,
and by setting y ′ = 2(τ + u 1 (τ )) v ′ (for τ large enough), we obtain:

J 1 (τ )/ √ 2π = R d-1 e -i u ′ (τ ),v ′ e -(τ +u 1 (τ )) v ′ 2 +∞ 0 e -1 2 u+ v ′ 2 -(τ +u 1 (τ )) 2 du dv ′ = R d-1 e -i u ′ (τ ),v ′ e -(τ +u 1 (τ )) v ′ 2 +∞ v ′ 2 -(τ +u 1 (τ )) e -x 2 2 dx dv ′ = 2(τ + u 1 (τ ) -d-1 2 × R d-1 e -i u ′ (τ ) √ 2(τ +u 1 (τ )) ,y ′ - y ′ 2 2 +∞ |y ′ | 2 2(τ +u 1 (τ )) -(τ +u 1 (τ ))
e -x 2 2 dx dy ′ .

Finally, since τ + u 1 (τ ) ∼ τ →+∞ τ by ( 24), Lebesgue's theorem and (33a) give

lim τ →+∞ τ (d-1)/2 J 1 (τ ) = √ 2π 2 -(d-1)/2 R d-1 e -i ℓ ′ √ 2 ,y ′ e -y ′ 2 2 +∞ -∞ e -x 2 2 dx dy ′ = √ 2π 2 -(d-1)/2 √ 2π ( √ 2π) d-1 e -ℓ ′ 2 /4 = 2π (d+1)/2 e -ℓ ′ 2 /4 .
Hence the desired property for J 1 (τ ).

Applications to additive functionals of Markov chains

In this section, (X n ) n≥0 denotes a Markov chain with state space (E, E), transition kernel Q(x, dy) and initial distribution µ. We assume that Q admits an invariant probability measure, denoted by π. We denote by P µ the probability distribution of (X n ) n≥0 with respect to the initial distribution µ. The associated expectation is denoted by E µ . Finally we consider a measurable function ξ : E → R d and we define the associated additive functional:

∀n ≥ 1, S n = ξ(X 1 ) + • • • + ξ(X n ). (34)
The next moment conditions on ξ will ensure that ξ is π-integrable. We can then define the first moment vector m of ξ w.r.t. to π. We assume that m is nonzero, that is

m = E π [ξ(X 0 )] = E ξ dπ = 0. (35)
Set ξ c := ξ -m, and for each n ≥ 1 define the following random variable, taking values in the set of nonnegative symmetric d × d-matrices:

S ⊗2 n,c = n k,ℓ=1 ξ c (X k ) ξ c (X ℓ ) * ,
where * stands for the transposition. The next conditions on ξ will also enable us to define the following nonnegative symmetric d × d-matrix:

Σ = m • m * + lim n 1 n E µ [S ⊗2 n,c ]. (36)
Finally we use the following standard (Markov) nonlattice condition:

Nonlattice Condition.

We say that ξ is nonlattice if there do not exist any (b,

H, A, θ) with b ∈ R d , H = R d a closed subgroup in R d , A ∈ E a π-full Q-absorbing 2 set
, and finally θ : E → R d a bounded measurable function, such that we have ∀x ∈ A, ξ(y) + θ(y) -θ(x) ∈ b + H Q(x, dy) -p.s.

Our first application concerns ρ-mixing Markov chains, namely: the strong ergodicity property (6) holds on the usual Lebesgue space L 2 (π) (see [START_REF] Rosenblatt | Markov processes. Structure and asymptotic behavior[END_REF]). For instance, this condition is fulfilled if (X n ) n≥0 is ergodic, aperiodic and satisfies the so-called Doeblin condition.

Hypothesis H ρ . The Markov chain (X n ) n≥0 is ρ-mixing and µ = π. 3 Moreover ξ is nonlattice and satisfies the following moment condition

∃ε 0 > 0, E π [ ξ(X 0 ) m d +ε 0 ] < +∞. ( 37
)
Our second application concerns V -geometrically ergodic Markov chains, namely: given some unbounded function V : E →[1, +∞[, Property [START_REF] Doney | An analogue of the renewal theorem in higher dimensions[END_REF] holds on the weighted supremumnormed space (B V , • V ) composed of all the measurable functions f : E → C satisfying : [START_REF] Meyn | Markov chains and stochastic stability[END_REF]. In particular we have π(V ) < ∞.

f V := sup x∈E |f (x)|/V (x) < ∞, see
Hypothesis H V . The Markov chain (X n ) n≥0 is V -geometrically ergodic and µ(V ) < ∞. 4 Moreover ξ is nonlattice and satisfies the following domination condition:

∃ ε 0 > 0, sup x∈R d ξ(x) m d +ε 0 V (x) < ∞. (38)
Our last application concerns Lipschitz iterative models. Here we suppose that (E, d) is a complete metric space in which every closed ball is compact. The space E is equipped with its borel σ-algebra E. Consider a measurable space (G, G) and a sequence {ϑ n } n≥1 of G-valued i.i.d. random variables. Let F : (E × G, E ⊗ G) →(E, E ) be jointly measurable and Lipschitz continuous in the first variable. Then, given X 0 an E-valued r.v. independent of the sequence {ϑ n } n≥1 , the associated Lipschitz iterative model (LIM) is the sequence (X n ) n∈N of r.v. recursively defined as follows (starting from X 0 ):

∀n ≥ 1, X n := F (X n-1 , ϑ n ). ( 39
)
The LIM (X n ) n∈N is said to be strictly contractive when

C := sup (x,y)∈E 2 , x =y d F (x, ϑ 1 ), F (y, ϑ 1 ) d(x, y) < 1 almost surely. ( 40
)
2 Recall that A ∈ E is said to be π-full if π(A) = 1, and Q-absorbing if Q(a, A) = 1 for all a ∈ A. 3 The stationarity condition µ = π may be replaced with dµ = φdπ provided that the density function φ is in L p (π) for some p > p ε 0 , with p ε 0 ∈ (1, +∞) depending on ε0. See [13, cor. 4] for details. 4 For instance this condition holds when µ = π or µ is the Dirac distribution δx at some x ∈ E.

Hypothesis H

Lim . The Markov chain (X n ) n∈N is a strictly contractive LIM satisfying ∃s ≥ 0, ∃ε 0 > 0, E d F (x 0 , ϑ 1 ), x 0 (s+1)m d +ε 0 < ∞ (41)
where x 0 is some point in E. Moreover we assume that E (d(X 0 , x 0 ) (s+1)m d +ε 0 < ∞5 and that ξ is nonlattice and satisfies

∃S ≥ 0, ∀(x, y) ∈ E × E, ξ(x) -ξ(y) ≤ S d(x, y) 1 + d(x, x 0 ) + d(y, x 0 ) s . ( 42
)
Corollary 1 Assume that one of the set of Hypotheses H ρ , H V or H Lim holds. Then m in ( 35) is well-defined, the limit in (36) exists, and Σ is invertible. Moreover, for any set A ∈ B(R d ) whose boundary has zero Lebesgue-measure, we have:

lim τ →+∞ (2πτ ) d-1 2 +∞ n=1 P µ S n -τ m ∈ A = L d (A) (det Σ) 1 2 Σ -1 2 m . ( 43
)
Proof. We apply Theorem 1 with f = 1 E . The fact that m in ( 35) is well-defined is obvious under Hypotheses H ρ or H V . Under Hypothesis H Lim , we have ξ(•) ≤ ξ(x 0 ) + S(1 + d(•, x 0 )) s+1 , so that ξ(•) is π-integrable under Conditions (40) (41), see [START_REF] Duflo | Random Iterative Models[END_REF][START_REF] Benda | A central limit theorem for contractive stochastic dynamical systems[END_REF]. The others conditions of Hypothesis (H) follow from the results proved in [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF] (centered case). Indeed, note that

ξ c = ξ -m is π-centered. Then the existence of Σ c := lim n 1 n E µ [S ⊗2 n,c
] is proved in [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF]Sect. 4] under the three above hypotheses. Moreover, since ξ is nonlattice, so is ξ c . The matrix Σ c is then definite from [14, p. 437], thus so is Σ in (36). Now define

S n,c = ξ c (X 1 ) + • • • + ξ c (X n ). We have ∀n ≥ 1, ∀t ∈ R d , E µ [e i t,Sn ] = e in t, m E µ [e i t,Sn,c ]. ( 44
)
From [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF]Sect. 4], under anyone of Hypotheses H ρ , H V or H Lim , the sequence (X n , S n,c ) n≥0 satisfies Hypothesis R(m) for some m > m d . Denote by λ c (•), L c (•), and R n,c (•) the associated complex-valued functions 6 . Then the sequence (X n , S n ) n≥0 satisfies Hypothesis R(m) with λ(•), L(•), and R n (•) given by:

λ(t) := e i t, m λ c (t), L(t) = L c (t), R n (t) := e in t, m R n,c (t). (45)
The fact that n≥1 R n (•) (resp.

n≥1 E[e i t,Sn ]) converges on the ball B R (resp. on the annulus K r,b ) and defines a function in

C m b (B R , C) (resp. in C m b (K r,b , C)
) can be easily derived from (45) (resp. ( 44)) and the spectral formulas given in [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF]. Finally we know that λ c (0

) = 1, ∇λ c (0) = 0 (since ξ c is π-centered), and that L c (0) = 1 in case f = 1 E . Thus -i∇λ(0) = m, Hess λ(0) = -(Σ c + m • m * ), and L(0) = 1. Consequently Hypothesis (H) is fulfilled.
Actually, under each of hypotheses H ρ , H V or H Lim , Theorem 1 applies for a large class of nonnegative functions f (use the refinements stated in [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF]). Moreover Corollary 1 can be extended to the lattice case, see [START_REF] Guibourg | Théorèmes de renouvellement pour des fonctionnelles additives associées à des chaînes de Markov fortement ergodiques[END_REF].

The domination or moment condition on ξ in Hypotheses H ρ , H V or H Lim involves the optimal order m d of the i.i.d. case [START_REF] Stam | Renewal theory in r dimensions[END_REF] (up to ε 0 > 0). Corollary 1 can be derived from the results of [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] but under much stronger moment conditions (the comparisons with [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF] in terms of moment conditions are the same as in [START_REF] Guibourg | A renewal theorem for strongly ergodic Markov chains in dimension d ≥ 3 and in the centered case[END_REF]).

For example, consider in R 3 the autoregressive model X n = AX n-1 +ϑ n , where X 0 , ϑ 1 , ϑ 2 , . . . are R 3 -valued i.i.d. random variables, and A is a contractive matrix of order 3. Clearly, taking d(x, y) = x -y , the sequence (X n ) n≥0 is a strictly contractive LIM. Now let us consider S n = X 1 + . . . + X n (i. e. ξ(x) = x). Condition (42) is fulfilled with S = 1 and s = 0. Consequently, if we have E X 0 2+ε 0 + ϑ 1 2+ε 0 < +∞ for some ε 0 > 0, then (41) holds and X 0 satisfies the moment condition stated in Hypothesis H Lim . Then, if ξ(x) = x is non-lattice, we have (43) with m = E π [X 0 ] and Σ defined in (36).

Additional remarks in Markov setting.

Theorem 1 may be applied to general Markov random walks (X n , S n ) n≥0 , see [START_REF] Dal'bo | Comportement asymptotique du nombre de géodésiques fermées sur la surface modulaire en courbure non constante. Études spectrales d'opérateurs de transfert et applications[END_REF]. Above we have only considered the special instance of additive functionals

S n = ξ(X 1 ) + • • • + ξ(X n ).
For general MRWs, Hypothesis R(m) involves the increments Y n := S n -S n-1 in place of the function ξ, e.g. see [START_REF] Ferré | Limit theorems for stationary Markov processes with L 2 -spectral gap[END_REF] for MRWs associated with ρ-mixing driving Markov chains. In particular, for the three above Markov models, Hypothesis R(m) is investigated in [START_REF] Ferré | Limit theorems for stationary Markov processes with L 2 -spectral gap[END_REF][START_REF] Hervé | A Berry-Esseen theorem on M for geometrically ergodic Markov chains[END_REF] for bivariate additive functionals of type S n = n k=1 ψ(X k-1 , X k ). Under some strong non-lattice conditions, asymptotic refinements of (43) have been obtained under the V -geometrical ergodicity assumption in [START_REF] Fuh | Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks[END_REF], and under the uniform ergodicity assumption in [START_REF] Uchiyama | Asymptotic estimates of the Green functions and transition probabilities for Markov additive processes[END_REF]. By using the weak spectral method [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF], some results of [START_REF] Fuh | Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks[END_REF][START_REF] Uchiyama | Asymptotic estimates of the Green functions and transition probabilities for Markov additive processes[END_REF] could be improved in terms of moment conditions.

The notion of convergence cone (see [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF]) is not investigated here. In fact, this study requires to define the Laplace kernels (in place of the Fourier kernels). The definition of such kernels needs some exponential (operator-type) moment conditions. Consequently the usual spectral method applies as stated in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF].

A Proof of Propositions 2-3

This appendix completes and details some arguments summarized in [START_REF] Babillot | Théorie du renouvellement pour des chaînes semi-markoviennes transientes[END_REF]. Roughly speaking, the dyadic decomposition consists in writing an integral on R d as the sum of integrals on the dyadic annuli

D n := {2 n ≤ x < 2 n+1 }, n ∈ Z.
Here some asymmetric annuli (in place of D n ) must be considered to take into account the mean direction m = m e 1 . Let us first specify some notations.

For any real number m > 0, we set τ := m -⌊m⌋, and for any open subset

O of R d , any K ⊂ O and f ∈ C m b (O, C), we define f m,K = |β|≤⌊m⌋ ∂ β f 0,K + |β|=⌊m⌋ ∂ β f τ,K .
Let us recall that, for

x = (x 1 , x 2 , . . . , x d ) ∈ R d , we have set x ′ = (x 2 , . . . , x d ) ∈ R d-1 and w(x) = -ix 1 + x ′ 2 .
Next define the following:

for 0 < ω 0 < ω ′ 0 : Γ 0,ω 0 ,ω ′ 0 := x ∈ R d : ω 0 ≤ |w(x)| ≤ ω ′ 0 (46a) ∀k ∈ Z, ∀x = (x 1 , . . . , x d ) ∈ R d , D k (x) := ( x 1 4 k , x 2 2 k , . . . , x d 2 k ) (46b) ∀k ∈ Z, Γ k,ω 0 ,ω ′ 0 := D k (Γ 0,ω 0 ,ω ′ 0 ) = x ∈ R d : ω 0 4 k ≤ |w(x)| ≤ ω ′ 0 4 k , ( 46c
) ∀k ∈ Z, Γ k := Γ k, 1 4 ,1 = x ∈ R d : 1 4 k+1 ≤ |w(x)| ≤ 1 4 k (46d) Γ 0 := Γ 0, 1 8 ,2 ⊂ Γ -1 ∪ Γ 0 ∪ Γ 1 . (46e)
We will repeatedly use the following obvious inclusions:

∀k ∈ N, Γ k,ω 0 ,ω ′ 0 ⊂ B 0, (ω ′2 0 + ω ′ 0 ) 1/2 /2 k (47a) ∀k ∈ Z, D k ( Γ 0 ) ⊂ Γ k-1 ∪ Γ k ∪ Γ k+1 . (47b)
Now let us fix a real number r > 0 and k 0 ∈ N * such that

√ 2 2 k 0 -1 < r. For each k ≥ k 0 -1 we clearly have Γ k ⊂ B(0, √ 2 2 k ) ⊂ B(0, √ 2
2 k 0 -1 ) ⊂ B r . Moreover we have ∀k ≥ k 0 , D k ( Γ 0 ) ⊂ B 0, √ 2/2 k-1 ⊂ B r . (48)
For any function u : B r → C, we define the following functions:

∀k ∈ N, ∀x ∈ B r , u k (x) = u(D k x).
Observe that, if k ≥ k 0 , then u is defined on D k ( Γ 0 ), so that u k is well-defined on Γ 0 .

A.1 Construction of a partition of the unity on R d \ {0}

Starting with

Γ 0 = x ∈ R d : 1 8 ≤ |w(x)| ≤ 2 , let us define for all k ∈ Z Γ k := D k ( Γ 0 ) = x ∈ R d : 1 8 1 4 k ≤ |w(x)| ≤ 2 1 4 k . (49)
Γ k contains Γ k (cf. (46d)). Now let γ ∈ C ∞ b (R d , R + ) be compactly supported in Γ 0 , such that ∀x ∈ Γ 0 , γ(x) = 1. Note that γ • D -k ∈ C ∞ b (R d , R + ) is compactly supported in Γ k . Next we set ∀x ∈ R d , φ(x) = k∈Z γ(D -k x).
We have φ(0) = 0, and for x = 0:

x ∈ Γ k ⇔ k ∈ [ -ln 8-ln |w(x)| ln 4 , ln 2-ln |w(x)| ln 4
].

Since the length of the previous interval is 2, a point x = 0 belongs at most to three sets among the Γ k 's, and since

R d \ {0} = ∪ p∈Z Γ p = ∪ p∈Z Γ p , we have: 1 ≤ φ(x) < +∞.
Notice also that we have by definition of φ

∀ℓ ∈ Z, φ • D ℓ = φ. (50)
Proposition A.1 The positive function ρ := γ/φ is infinitely differentiable on R d \ {0}. Moreover ρ vanishes on R d \ Γ 0 ∪ {0} , and we have:

∀x ∈ R d \ {0}, k∈Z ρ(D -k x) = 1.
Proof. For p ∈ Z, we denote by

Int (Γ p ∪ Γ p+1 ) the interior of Γ p ∪ Γ p+1 . Since R d \ {0} = ∪ p∈Z Int (Γ p ∪ Γ p+1 ), it suffices to check that φ is C ∞ on the open subset Int(Γ p ∪ Γ p+1 ). Let x ∈ Int(Γ p ∪ Γ p+1 ), namely: 1 4 p+2 < |w(x)| < 1 4 p , and let k ≤ p -2. Since 1 4 p ≤ 1 16 1 4 k , we have |w(x)| < 1 8 1 4 k . Thus x ∈ Γ k , and so γ(D -k (x)) = 0. Similarly, if k ≥ p + 3, then 4 1 4 k ≤ 1 4 p+2 , thus |w(x)| > 2 1
4 k , and so x ∈ Γ k and γ(D -k (x)) = 0. It follows that the restriction of φ to Int(Γ p ∪Γ p+1 ) is the finite sum p+2 k=p-1 γ •D -k . This proves the first point of Proposition A.1. The two last assertions are obvious.

A.2 Proof of Proposition 2

Recall that r > 0, that k 0 ∈ N * is such that

√ 2 2 k 0 -1 < r, and that Γ k ⊂ B r for all k ≥ k 0 -1. Define C := ∪ j≥k 0 +1 Γ j = x ∈ R d , 0 < |w(x)| ≤ 4 -(k 0 +1)
and set r ′ = 2 -1/2 4 -(k 0 +1) . Then it can be easily seen that B(0, r

′ ) \ {0} ⊂ C ⊂ C ⊂ B r . Let us consider η ∈ C ∞ b (R d , R
), compactly supported in C, such that we have:

∀x ∈ B(0, r ′ ), η(x) = 1.
Recall that m d := max(2, (d -1)/2). Let m > m d and let θ and v be complex-valued functions on B r satisfying the hypotheses of Proposition 2. Recall that q := 1 Br θ/v. The function

q 1 := (1 -η)q is in C ⌊m⌋ b (R d , C
) and is compactly supported in K r ′ ,r . Moreover its restriction to K r ′ ,r is in C m b (K r ′ ,r , C). It follows from Proposition 1 applied with O = K r ′ ,r and u = q 1 that lim a →+∞ a (d-1)/2 q1 (a) = 0 (use m > (d -1)/2).

From the previous remark and the fact that ηθ satisfies the same hypotheses as θ, it suffices to prove Proposition 2 in the case when q is compactly supported in C. From now on, in addition to the assumptions of Proposition 2, we assume that q is compactly supported in C.

To prove that q is integrable and to estimate q, we use Proposition A.1. Observe that

D -k (C) = ∪ j≥k 0 +1 D -k (Γ j ) = ∪ ℓ≥k 0 +1-k Γ ℓ . Hence, if k ≤ k 0 -1, then D -k (C) ⊂ ∪ ℓ≥2 Γ ℓ . Since Γ 0 and ∪ ℓ≥2 Γ ℓ are disjoint, it follows from Proposition A.1 that, for each k ≤ k 0 -1, the function ρ • D -k vanishes on C . Moreover we have q(x) = 0 if x ∈ C ∪ {0}. Thus ∀x ∈ R d \ {0}, q(x) = +∞ k=k 0 ρ(D -k x) q(x).
For each k ≥ k 0 , we set:

∀x ∈ R d , ψ k (x) := ρ(x) q(D k x) if x ∈ Γ 0 0 if x / ∈ Γ 0 . = ρ(x) θ k (x) v k (x) if x ∈ Γ 0 0 if x / ∈ Γ 0 .
The following proposition is the key statement to prove Proposition 2.

Proposition A.2 For each k ≥ k 0 , we have ψ k ∈ C m b (R d , C), and there exists K > 0 such that:

∀k ≥ k 0 , ψ k m, Γ 0 ≤ K 2 k .
The proof of Proposition A.2 is postponed in Subsection A.4.

Proof of Proposition 2. Let k ≥ k 0 . Setting t = D -k x and using Proposition A.2, we obtain

R d ρ(D -k x) |q(x)|dx = ( 1 2 k ) d+1 R d ρ(t) |q(D k t)|dt ≤ KL d ( Γ 0 ) 2 kd . ( 51
)
Thus k≥k 0 R d ρ(D -k x) |q(x)|dx < ∞.
So q is integrable, and we have for all a ∈ R d :

q(a) := R d q(x) e -i x,a dx = +∞ k=k 0 R d ρ(D -k x) q(x)e -i x, a dx = +∞ k=k 0 ( 1 2 k ) d+1 R d ρ(t) q(D k t)e -i t, D k a dt = +∞ k=k 0 ( 1 2 k ) d+1 ψ k (D k a). (52)
From Proposition A.2 and Proposition 1 applied with u := ψ k and O := Int( Γ 0 ), one can deduce the following property.

∀k ≥ k 0 , ∀b ∈ R d , b m | ψ k (b)| ≤ CK 2 k . (53)
Moreover Proposition A.2 gives with

K ′ := KL d ( Γ 0 ): ∀k ≥ k 0 , ∀b ∈ R d , | ψ k (b)| ≤ K ′ 2 k .
Then, from (53) and m > d-1 2 , we obtain

∀k ≥ k 0 , ∀b ∈ R d , b d-1 2 | ψ k (b)| ≤ b m + 1 | ψ k (b)| ≤ (CK + K ′ ) 2 k .
Let a ∈ R d . By using the fact that a ≤ 4 k D k a , we obtain for all k ≥ k 0

a d-1 2 ( 1 2 k ) d+1 | ψ k (D k a)| ≤ (4 k ) d-1 2 ( 1 2 k ) d+1 (CK + K ′ ) 2 k = CK + K ′ 2 k , (54)
from which we deduce that

lim k 1 → +∞ a d-1 2 +∞ k=k 1 ( 1 2 k ) d+1 ψ k (D k a) = 0 uniformly in a ∈ R d . By (53), we have ∀a ∈ R d \ {0}, | ψ k (D k a)| ≤ 4 km C2 k a m
, and since m > d-1 2 , we obtain

∀k 1 > k 0 , lim a →+∞ a d-1 2 k 1 -1 k=k 0 ( 1 2 k ) d+1 ψ k (D k a) = 0.
From (52) and from the two previous properties, it follows that lim a →+∞ a d-1

2 q(a) = 0, as claimed in Proposition 2.

A.3 Proof of Proposition 3

Here the function q is defined by q := 1 Br θ/(v ṽ), with θ, v and ṽ satisfying the hypotheses of Proposition 3. As above one may suppose that q is compactly supported in C. The proof of Proposition 3 is then similar to the previous one, up to the following changes. First the function ψ k (•) is replaced with:

∀x ∈ R d , ψ k (x) := ρ(x) θ k (x)/(v k (x) ṽk (x)) if x ∈ Γ 0 0 if x / ∈ Γ 0 .
Second, Proposition A.2 is replaced with the following one (proved in Subsection A.4):

Proposition A.3 For each k ≥ k 0 , we have ψ k ∈ C m b (R d , C
), and there exists L > 0 such that, for all k ≥ k 0 , we have ψ k m, Γ 0 ≤ L (2 k ) 2-ν with ν := min(m -2, 1).

Note that ν > 0 since m > m d ≥ 2 by hypothesis. Next the term O(2 -kd ) in ( 51) is replaced with O(2 -k(d-1+ν) ): this yields the integrability of q and a formula analogous to (52). The term O(2 k ) of ( 53) is replaced with O(2 k(2-ν) ): this gives the property analogous to (54) with O(2 -kν ) (in place of O(2 -k )). We can then conclude as above.

A.4 Proof of Propositions A.2 and A.3

For any α

= (α 1 , . . . , α d ) ∈ N d , we set α! := α 1 ! . . . α d !. If γ = (γ 1 , . . . , γ d ) ∈ N d , the notation α ≤ γ means: ∀i = 1, . . . , d, α i ≤ γ i . We shall use Leibniz's formula, namely: if Ω is an open subset of R d and if γ ∈ N d is such that |γ| := γ 1 + . . . + γ d ≤ k (k ∈ N * ), then we have for all f, g ∈ C k b (Ω, C) ∂ γ (f • g) = β≤γ γ β ∂ β f ∂ γ-β g, (55)
where γ β = γ! β!(γ-β)! We shall also use repeatedly the following lemma. Below we denote by θ, v, ṽ three functions from B r into C, we consider a real number m > 2, and we set τ := m -⌊m⌋.

Lemma A.2 If θ ∈ C m b (B r , C) satisfies θ(0) = 0, then there exist a constant C > 0 such that, for all k ≥ k 0 , we have |β| ≤ ⌊m⌋ ⇒ ∂ β θ k 0, Γ 0 ≤ C 2 -k (56a) |β| = ⌊m⌋ ⇒ ∂ β θ k τ, Γ 0 ≤ C 2 -k (56b)
Proof. Since the first partial derivatives of θ are bounded on B r , there exists M > 0 such that:

∀x ∈ B r , |θ(x)| ≤ M x . From (48), it follows that θ k 0, Γ 0 ≤ M √ 2/2 k-1 . Let β ∈ N d such that 1 ≤ |β| ≤ ⌊m⌋, and let x ∈ B r . We have |∂ β θ k (x)| ≤ 1 2 k|β| |∂ β θ(D k x)|.
So (56a) follows from the fact that the partial derivatives of θ of order j = 1, . . . , ⌊m⌋ are bounded on B r . Now assume that |β| = ⌊m⌋ ≥ 2. Then we have for all (x, y) ∈ Γ 0 × Γ 0 :

(∂ β θ k )(x) -(∂ β θ k )(y) ≤ 4 -k (∂ β θ)(D k x) -(∂ β θ)(D k y) ≤ 4 -k ∂ β θ τ,Br x -y τ , hence we have (56b). Lemma A.3 If θ ∈ C m b (B r , C
) satisfies θ(0) = 0 and if all the first and second partial derivatives of θ vanish at 0, then there exist a constant D > 0 such that, for all k ≥ k 0 , we have with ν := min(m -2, 1):

|β| ≤ ⌊m⌋ ⇒ ∂ β θ k 0, Γ 0 ≤ D 2 -k(2+ν) (57a) |β| = ⌊m⌋ ⇒ ∂ β θ k τ, Γ 0 ≤ D 2 -k(2+ν) . (57b)
Proof. By hypothesis we have θ(0) = 0, (∂ j θ)(0) = 0, and (∂ 2 jℓ θ)(0) = 0 for every (j, ℓ) ∈ {1, . . . , d} 2 . If |β| ≤ 2, then (57a) holds since there exists a constant K β > 0 such that :

∀x ∈ B r , |∂ β θ(x)| ≤ K β x 2+ν-|β| . For 2 < |β| ≤ ⌊m⌋, (57a) is obvious. Now assume that |β| = ⌊m⌋. Let (x, y) ∈ Γ 0 2 . We have |(∂ β θ k )(x) -(∂ β θ k )(y)| ≤ 2 -k⌊m⌋ |(∂ β θ)(D k x) -(∂ β θ)(D k y)| ≤ 2 -k⌊m⌋ ∂ β θ τ,Br ( 1 2 k ) τ x -y τ .
Since ⌊m⌋ + τ = m, this gives (57b). 21) and ( 22), then there exist c, c ′ , c ′′ > 0 such that, for all k ≥ k 0 , we have: 21) and the mean value inequality (notice that the second order partial derivatives of v are bounded on B r ), there exists M > 0 such that:

Sublemma A.1 If v ∈ C m (B r , C) satisfies (
|β| ≤ ⌊m⌋ ⇒ ∂ β v k 0, Γ 0 ≤ c 4 -k (58a) |β| ≤ ⌊m⌋ -1 ⇒ ∂ β v k 1, Γ 0 ≤ c ′ 4 -k (58b) |β| = ⌊m⌋ ⇒ ∂ β v k τ, Γ 0 ≤ c ′′ 4 -k (58c) Proof. From (22), we obtain |v k (x)| ≤ b |w(D k x)| ≤ b 4 -k |w(x)|, hence v k 0, Γ 0 ≤ 2b 4 -k . Moreover we have ∀x ∈ B r , (∂ 1 v k )(x) = 4 -k (∂ 1 v)(D k x), thus ∂ 1 v k 0, Γ 0 ≤ 4 -k ∂ 1 v 0,Br . Next we have: ∀j ∈ {2, . . . , d}, ∀x ∈ B r , (∂ j v k )(x) = 2 -k (∂ j v)(D k x). From (
∀x ∈ B r , |(∂ j v)(x)| ≤ M x . From (48), it follows that ∂ j v k 0, Γ 0 ≤ M 2 √ 2 4 -k . This proves (58a) for |β| = 1. Now, if 2 ≤ |β| ≤ ⌊m⌋, then ∀x ∈ B r , (∂ β v k )(x) = ( 1 2 k ) β 1 ( 1 4 k ) β 2 +...+β d (∂ β v)(D k x), 21
and therefore we have [START_REF] Smith | A frequency function form of the central limit theorem[END_REF]. So there exists M > 0 such that: ∀j ∈ {1, . . . , d}, ∀x ∈ B r , |(∂ j V )(x)| ≤ M x . This fact and the mean value inequality applied on B(0,

∂ β v k 0, Γ 0 ≤ 1 4 k ∂ β v 0,Br . The proof of (58a) is then complete. Let us first prove (58b) in case β = 0. Set V (x) = v(x) -(∂ 1 v)(0) x 1 for x ∈ B r . We have (∂ 1 V )(0) = 0, and (∂ 2 V )(0) = • • • = (∂ d V )(0) = 0 thanks to
|(∂ β v k )(x) -(∂ β v k )(y)| ≤ 1 2 |β|k |(∂ β v)(D k x) -(∂ β v)(D k y)| ≤ M 2 -|β|k D k x -D k y ≤ M 4 -k x -y .
This yields (58b). The proof of (58c) is similar to that of (56b).

Lemma A.4 If v ∈ C m b (B r , C) satisfies ( 21) and ( 22), then there exist a constant E > 0 such that, for all k ≥ k 0 , we have

|β| ≤ ⌊m⌋ ⇒ ∂ β (1/v k ) 0, Γ 0 ≤ E4 k (59a) |β| = ⌊m⌋ ⇒ ∂ β (1/v k ) τ, Γ 0 ≤ E4 k . (59b)
Proof. From ( 22) and (49), we have 1/v k 0, Γ 0 ≤ 2.4 k+1 a . Let j ∈ {1, . . . , d}. From (58a) and the previous inequality, we obtain k . v k ) = 0, Leibniz's formula gives:

∂ j ( 1 v k ) 0, Γ 0 ≤ ∂ j v k 0, Γ 0 1 v k 2 0, Γ 0 ≤ c
∂ γ ( 1 v k ) = -1 v k β≤γ, β =γ γ β ∂ β ( 1 v k ). ∂ γ-β (v k ). (60)
Thus we have (use the induction hypothesis and (58a)): The proof of (59a) is complete. To prove (59b), we need the following. Sublemma A.2 Under the hypotheses of Lemma A.4, there exist a constant c > 0 such that, for all k ≥ k 0 , we have

∂ γ ( 1 v k ) 0, Γ 0 ≤ 1 v k 0, Γ 0 β≤γ, β =γ γ β ∂ β ( 1 v k ) 0, Γ 0 ∂ γ-β (v k ) 0, Γ 0 ≤ 2.
|β| ≤ ⌊m⌋ -1 ⇒ ∂ β (1/v k ) 1, Γ 0 ≤ c4 k . ( 61
)
Proof. If β = 0, then we have for all (x, y) ∈ Γ 0 2 (use (58b) and proceed as for (59a))

| 1 v k (x) - 1 v k (y) | = |v k (x) -v k (y)| |v k (x)||v k (y)| ≤ 1 a 2 4 2k+3 v k 1, Γ 0 x -y ≤ 64c ′ a 2 4 k x -y .
Hence [ 1 v k ] 1, Γ 0 ≤ 64c ′ 4 k /a 2 . Now, let us consider the case |β| = 1. Let j ∈ {1, . . . , d}. By using Lemma A.1, (58a) (59a) and the previous inequality, we obtain

∂ j 1 v k 1, Γ 0 = (∂ j v k ) • 1 v 2 k 1, Γ 0 ≤ ∂ j v k 0, Γ 0 1 v 2 k 1, Γ 0 + ∂ j v k 1, Γ 0 1 v 2 k 0, Γ 0 ≤ ∂ j v k 0, Γ 0 2 1 v k 0, Γ 0 1 v k 1, Γ 0 + ∂ j v k 1, Γ 0 1 v k 0, Γ 0 2 ≤ 2 c 4 k d4 k 64c ′ a 2 4 k + c ′ 4 k (d4 k ) 2 := c ′′′ 4 k .
This gives (61) for |β| = 1. To complete the proof of (61), let us again proceed by induction. Assume that, for some ℓ ∈ {1, . . . , ⌊m⌋ -2} and for all β ∈ N d such that |β| ≤ ℓ, there exists

D β > 0 such that ∂ β ( 1 v k ) 1, Γ 0 ≤ D β 4 k . ( 62
)
Let γ ∈ N d such that |γ| = ℓ + 1, and let β ≤ γ such that β = γ. By applying Lemma A.1, we get (below • 0 and [• ] 1 stand for • 0, Γ 0 and [• ] 1, Γ 0 respectively) :

1 v k • ∂ β ( 1 v k ) • ∂ γ-β (v k ) 1 ≤ 1 v k 1 ∂ β ( 1 v k ) • ∂ γ-β (v k ) 0 + 1 v k 0 ∂ β ( 1 v k ) • ∂ γ-β (v k ) 1 ≤ 1 v k 1 ∂ β ( 1 v k ) 0 ∂ γ-β (v k ) 0 + 1 v k 0 ∂ β ( 1 v k ) 1 ∂ γ-β (v k ) 0 + ∂ β ( 1 v k ) 0 ∂ γ-β (v k ) 1 .
From (58a) (58b) (59a) and (62) (observe that |β| ≤ ℓ), we get

1 v k ∂ β ( 1 v k ) ∂ γ-β (v k ) 1 ≤ L β 4 k for some L β > 0. From (60), it follows that [∂ γ ( 1 v k )]
1, Γ 0 ≤ D γ 4 k for some D γ > 0.

Finally we prove (59b). Let γ ∈ N d be such that |γ| = ⌊m⌋. If β ≤ γ, β = 0, β = γ (thus |β| ≤ ⌊m⌋ -1 and |γ -β| ≤ ⌊m⌋ -1), we have

1 v k ∂ β ( 1 v k ) ∂ γ-β (v k ) 1, Γ 0 ≤ E 4 k
for some E > 0 (use Lemma A.1 as above and (58a) (58b) (59a) (61)). Set

s k = - 1 v k β≤γ, β =0,β =γ γ β ∂ β ( 1 v k ) ∂ γ-β (v k ).
The previous remark shows that [s k ] 1, Γ 0 ≤ E ′ 4 k for some E ′ > 0. The same inequality holds for [s k ] τ, Γ 0 because Γ 0 is bounded. Finally, from (60), we get

∂ γ ( 1 v k ) = -1 v 2 k ∂ γ v k + s k

1 .

 1 d ≥ 3 and E[S 1 ] = 0 (centered case), 2. d ≥ 1 and E[S 1 ] = 0 (non-centered case): in this case, the behavior of U a (•) is specified when a → +∞ in the direction of E[S 1 ].

  [g] τ,A < ∞. Moreover, for any open subset O of R d and every m ∈ N * , we denote by C m b (O, C) the vector space composed of m-times continuously differentiable functions f : O → C with bounded partial derivatives on O. If m ∈ (0, +∞) \ N, we set τ := m -⌊m⌋ where ⌊m⌋ is the integer part of m, and we denote by C m b (O, C) the vector space composed of functions f : O → C satisfying the three following conditions:

  •) and L(•) are in C m b B R , C , and the series n≥1 R n (•) uniformly converges on the open ball B R and defines a function in C m b B R , C . (ii) For all 0 < r < b, the series n≥1 E f (X n ) e i •,Sn uniformly converges on the annulus K r,b and defines a function in C m b K r,b , C . Under Hypothesis R(m), we set m := -i∇λ(0) and Σ := -Hess λ(0).

Lemma A. 1

 1 Let σ ∈]0, 1] and let O be an open subset of R d . For any f, g ∈ C σ b (O, C), we have f g ∈ C σ b (O, C) and f g σ,O ≤ f 0,O g σ,O + g 0,O f σ,O . Recall that Γ 0 = x ∈ R d : 1/8 ≤ |w(x)| ≤ 2and that, for any function u : B r → C, we have set: ∀k ∈ N, ∀x ∈ B r , u k (x) = u(D k x). Observe that all the partial derivatives of the function ρ(•) are bounded on Γ 0 . From this fact and from (55), Propositions A.2 and A.3 easily follow from the next Lemmas A.2, A.3 and A.4.

√ 2 2 k- 1 )

 21 imply that there exists C ′ > 0 such that we have:∀(x, y) ∈ Γ 0 2 , |V (D k x) -V (D k y)| ≤ C ′ 2 k D k x -D k y . Since v k (x) = V (D k x) + (∂ 1 v)(0) x 1 4 k , we obtain |v k (x) -v k (y)| ≤ C ′′ 4 k x -y for some C ′′ > 0. Thus v k 1, Γ 0 ≤ C ′′ 4 k .Now we establish (58b) in case |β| ∈ {1, . . . , ⌊m⌋-1}. Since all the partial derivatives of order |β| + 1 of v are bounded on B r , there exists M > 0 such that we have for all (x, y) ∈ Γ 0 2 :

4 k 4 2k+3 a 2

 42 = 64c a 2 4 k . Now, let us proceed by induction. Let ℓ ∈ {1, . . . , ⌊m⌋ -1}, and assume that, for all β ∈ N d such that |β| ≤ ℓ, there existsC β > 0 such that ∂ β ( 1 v k ) 0, Γ 0 ≤ C β 4 k . Let γ ∈ N d such that |γ| = ℓ + 1. Since ∂ γ (v -1

  , and Lemma A.1 and (59a) (58c) give[∂ γ v k /v 2 k ] τ, Γ 0 ≤ E ′′ 4 k for some E ′′ > 0.Remark A.1 Lemmas A.2 and A.4, and so Proposition A.2, hold when m = 1. Consequently the conclusion of Proposition 2 is fulfilled as soon as m > max(1, (d -1)/2). However the condition m > max(2, (d -1)/2) (i.e. m > m d ) seems to be necessary to prove Proposition 3.

This LLT, established by Fourier techniques, extends to d ≥

the one-dimensional result of[START_REF] Smith | A frequency function form of the central limit theorem[END_REF].

This condition holds when X0 ∼ δx. Under Assumptions (40) (41), it holds for X0 ∼ π, see[START_REF] Duflo | Random Iterative Models[END_REF][START_REF] Benda | A central limit theorem for contractive stochastic dynamical systems[END_REF].

They are derived from the perturbation theorem due to Keller and Liverani[START_REF] Keller | Stability of the Spectrum for Transfer Operators[END_REF]: λc(t) is the perturbed eigenvalue of the Fourier operators associated with (Xn, Sn,c) n≥0 , while Lc(•) and Rn,c(•) are linked to some spectral projections.


F R Éd

Éric Chapoton

Éric Patras

ENVELOPING ALGEBRAS OF PRELIE ALGEBRAS, SOLOMON IDEMPOTENTS AND THE MAGNUS FORMULA

We study the internal structure of enveloping algebras of preLie algebras. We show in particular that the canonical projections arising from the Poincaré-Birkhoff-Witt theorem can be computed explicitely. They happen to be closely related to the Magnus formula for matrix differential equations. Indeed, we show that the Magnus formula provides a way to compute the canonical projection on the preLie algebra. Conversely, our results provide new insights on classical problems in the theory of differential equations and on recent advances in their combinatorial understanding.

Introduction

PreLie algebras play an increasingly important role in various fields of pure and applied mathematics, as well as in mathematical physics. They appear naturally in the study of differential operators: the preLie product underlying the theory can be traced back to the work of Cayley in relation to tree expansions of compositions of differential operators. PreLie structures were defined formally much later in the work of Vinberg (they are sometimes refered to in the litterature as Vinberg algebras). They were rediscovered in deformation theory by Gerstenhaber [START_REF] Gerstenhaber | The cohomology structure of an associative ring[END_REF], who introduced their current name. They appear currently in differential geometry, control theory, operads, perturbative quantum field theory... see e.g. [START_REF] Agrachev | Chronological algebras and nonstationary vector fields[END_REF][START_REF] Chapoton | Pre-Lie algebras and the rooted trees operad Internat[END_REF][START_REF] Ebrahimi-Fard | A Magnus-and Fer-type formula in dendriform algebras[END_REF] for some applications and further references on the subject.

In most applications, preLie algebras structures appear through their representations, and their action can therefore be canonically lifted to the enveloping algebra of the underlying Lie algebra. The structure of enveloping algebras of preLie algebras, which are the subject of the present article, is richer than the structure of arbitrary enveloping algebras of Lie algebras. In the particular case of preLie algebras L, the usual Poincaré-Birkhoff-Witt (PBW) theorem can indeed be refined: there exists on S(L), the symmetric algebra over L another associative, simply defined, product, making S(L) the enveloping algebra of L. Recall that, for Lie algebras, the PBW theorems asserts only that the canonical map from S(L) to U (L), the enveloping algebra constructed as the quotient of the tensor algebra by the ideal generated by Lie brackets, is a linear isomorphism; see e.g. [START_REF] Reutenauer | Free Lie algebras[END_REF] for details on the subject.

In the context of differential operators and trees, the definition of the product on S(L) first appeared in the work of Grossman-Larson [START_REF] Grossman | Hopf algebraic structure of families of trees[END_REF]. The dual structure was discovered by Connes-Kreimer in the framework of perturbative quantum field theory [START_REF] Kreimer | On the Hopf algebra structure of perturbative quantum field theories[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem I: The Hopf algebra structure of graphs and the main theorem[END_REF]. The work of Chapoton-Livernet [START_REF] Chapoton | Pre-Lie algebras and the rooted trees operad Internat[END_REF] showed that free preLie algebras can be realized as Connes-Kreimer preLie algebras of trees, from which it follows that the enveloping algebra of a free preLie algebra L can be realized as a Grossman-Larson algebra, that is, as the symmetric algebra S(L) provided with a new product associated to the process of insertion of trees. The general construction of the enveloping algebra of a preLie algebra L as S(L) is recent and due to Guin-Oudom, to which we refer for details [START_REF] Guin | On the Lie enveloping algebra of a pre-Lie algebra[END_REF].

In the present paper, we study the internal structure of S(L) in relation to the PBW theorem. The classical combinatorial study of enveloping algebras, due originally to Solomon [START_REF] Solomon | On the Poincaré Birkhoff Witt theorem[END_REF] and developped further in the work of Reutenauer and coauthors [START_REF] Reutenauer | Theorem of Poincaré-Birkhoff-Witt, logarithm and representations of the symmetric group whose orders are the Stirling numbers[END_REF][START_REF] Reutenauer | Free Lie algebras[END_REF], shows that the canonical projections on the components of the direct sum decomposition of U (L) induced by the PBW isomorphisms and the natural decomposition of the symmetric algebra into homogeneous components:

S(L) = n∈N S n (L),
where L is the free Lie algebra over an arbitrary set of generators, give rise to a family of orthogonal projections (i.e. idempotents) in the symmetric group group algebras Q[S n ]. These idempotents, the Solomon idempotents, play a key role in many fields of mathematics, partially due to their close connexion to enveloping algebra structures: free Lie algebras, control theory, group theory (both in the study of finite Coxeter groups and the general theory of transformation groups), iterated integrals, Baker-Campbell-Hausdorff-type formulas (they were discovered independently of Solomon by Mielnik-Plebanski in this setting [START_REF] Mielnik | Combinatorial approach to Baker Campbell Hausdorff exponents[END_REF]), noncommutative symmetric functions, homological algebra, and so on... See e.g. [START_REF] Reutenauer | Free Lie algebras[END_REF][START_REF] Gelfand | Noncommutative symmetric functions[END_REF] for further details on the subject. These idempotents have been given many names in the litterature: canonical idempotents (as canonical projections in the enveloping algebras), Barr idempotent (for the first idempotent of the series), eulerian idempotents (because, according to the seminal work of Solomon, their coefficients in the symmetric group algebras are related to the eulerian numbers). The one of "Solomon idempotents" fits the best with the practice for other Lie idempotents (e.g. the Dynkin, Garsia, Klyachko or Garsia-Reutenauer idempotents).

In the particular case of enveloping algebras of free preLie algebras, new structures and formulas arise in relation to the Solomon idempotents, due to the possibility of realizing these enveloping algebras as S(L) equipped with a new product. This is the subject of the present article, which relates in particular the first Solomon idempotent to the Magnus series familiar in the theory of differential equations.

PreLie and enveloping algebras

In this article, all vector spaces are over an arbitrary ground field k of characteristic zero. Definition 1. A preLie algebra is a vector space L equipped with a bilinear map such that, for all x, y, z in L:

(x y) z -x (y z) = (x z) y -x (z y).
The vector space L is then equipped with a Lie bracket [x, y] := x yy x. We write L Lie for the associated Lie algebra, when we want to emphasize that we view L as a Lie algebra. Most of the time this will be clear from the context, and we will then simply write L although making use of the Lie algebra structure. For example, we will write U (L) and not U (L Lie ) for the enveloping algebra of L Lie . We will also denote by the right action of the universal enveloping algebra of L Lie on L that extends the pre-Lie product: ∀a, b ∈ L, (b)a := b a. Notice that this action is well defined since the product makes L a module over the Lie algebra L Lie :

∀a, b, c ∈ L, ((c)b)a -((c)a)b = (c b) a -(c a) b = c (b a -a b) = (c)[b, a].
Recall that one can consider S(L), the symmetric algebra over L, equipped with a product * induced by , as the enveloping algebra of L Lie . Here, we realize S(L) = n∈N S n (L) as the subspace of symmetric tensors in T (L) := n∈N L ⊗n , so that S n (L) identifies with the S n -invariant tensors in L ⊗n : S n (L) = (L ⊗n ) Sn , where S n stands for the symmetric group of order n -the permutation group of [n] := {1, ..., n}. For a ∈ L, we write a n for its n-fold symmetric tensor power a ⊗n in S(L). More generally, for l 1 , ..., l n ∈ L, we write l 1 ...l n for the n-fold symmetric tensor product:

1 n! σ l σ(1) ⊗ ... ⊗ l σ(n)
, where σ runs over S n . An enveloping algebra carries the structure of a cocommutative Hopf algebra for which the elements of the Lie algebra identify with the primitive elements. The corresponding coproduct on S(L) is given by: for arbitrary a 1 , ..., a n ∈ L,

∆(a 1 ...a n ) = I a I ⊗ a J
where, for a subset I of [n], a I := i∈I a i , where I runs over the (possibly empty) subsets of [n] and J := [n] -I. The product * is associative but not commutative and is defined as follows:

(a 1 ...a l ) * (b 1 ...b m ) = f B 0 (a 1 B 1 )...(a l B l ),
where the sum is over all functions f from {1, ..., m} to {0, ..., l} and B i := f -1 (i). The increasing filtration of S(L) by the degree is respected by the product * , but the direct sum decomposition into graded components is not.

A symbol means that we consider the completion of a graded vector space: for example, L stands for the product n L n , where L n stands for the linear span of preLie products of length n of elements of L. The series exp(a) := n a n n! belongs for example to S(L), and so on. A graded vector space

V = n∈N V n is connected if V 0 = k.
Since we are interested in universal properties of preLie algebras, we assume from now on that L is a free preLie algebra. It follows that both L and L Lie are naturally graded by the lengths of preLie products. For clarity, we will distinguish notationally between:

• U (L), the enveloping algebra of L defined as the quotient of the tensor algebra T (L) over L by the ideal generated by the sums [x, y]x ⊗ y + y ⊗ x, x, y ∈ L. The product on U (L) is written simply • (so that e.g. for x, y ∈ L, x • y = x ⊗ y in U (L)). • S * (L), the symmetric algebra over L equipped with the structure of an enveloping algebra of L Lie by the product * . • S(L), the symmetric algebra over L equipped with the structure of a polynomial algebra.

Recall that the product of l 1 , ..., l n ∈ L in S(L) is written l 1 l 2 ...l n .

We write S * ,n (L) and S n (L) for the symmetric tensors of length n in S * (L) and S(L); we write S ≥n (L) for sums of symmetric tensors of length at least n. The coproduct on S * (L) and S(L) for which L is the primitive part is written ∆ in both cases. It provides S * (L) with the Hopf algebra structure arising from its enveloping algebra structure, whereas S(L) inherits a bicommutative Hopf algebra structure. Both S * (L) and S(L) are free cocommutative coalgebras over L. We refer to [START_REF] Milnor | On the structure of Hopf algebras Ann[END_REF][START_REF] Guin | On the Lie enveloping algebra of a pre-Lie algebra[END_REF] for details. The natural linear isomorphism (the PBW isomorphism) from S(L) to U (L) obtained by identifying S(L) with the symmetric tensors in T (X) is written π nat :

π nat (l 1 ...l n ) = 1 n! σ∈Sn l σ(1) • l σ(2) • ... • l σ(n) .
It is a coalgebra isomorphism since it maps L to L and since both S(L) and U (L) are cofree cocommutative coalgebras over L. Similarly, the identity map Id from S * (L) to S(L) is (obviously) a coalgebra isomorphism.

The canonical isomorphism, written is, between U (L) and S * (L) maps a product of elements of L x 1 ...x n in U (L) to their product x 1 * ... * x n in S * (L). Take care that is is not the inverse of the PBW map π nat .

All these maps are summarized by the following diagram:

S * (L) Id -→ S(L) πnat -→ U (L) is -→ S * (L).
Recall at last from [START_REF] Patras | Homothéties simpliciales[END_REF][START_REF] Patras | L'algèbre des descentes d'une bigèbre graduée[END_REF] the following two results:

Lemma 1. The Solomon first idempotent sol 1 , that is the map from U (L) (resp. S * (L)) to L orthogonally to the image of S ≥2 (L) by π nat (resp. is • π nat ) can be expressed as:

sol 1 = log ⋆ (Id),
where Id is the identity map of U (L), resp. S * (L). More generally, the map to the image of S n (L) orthogonally to the image of the other graded components S i (L), i = n, reads

sol n = sol ⋆n 1 n! .
The Lemma generalizes to arbitrary enveloping algebras of graded connected Lie algebras Reutenauer's computation of the Solomon idempotent in [START_REF] Reutenauer | Theorem of Poincaré-Birkhoff-Witt, logarithm and representations of the symmetric group whose orders are the Stirling numbers[END_REF]. Here the logarithm is computed in the convolution algebras of linear endomorphisms of U (L) and S * (L). That is, for f, g ∈ End(U (L)) (resp. End(S * (L))):

f ⋆ g := m • (f ⊗ g) • ∆,
where m stands for the product in U (L), resp. S * (L).

Proposition 2. The inverse maps inv, resp inv S to the PBW isomorphisms π nat and is • π nat are given by:

inv(x) := n sol ⊗n 1 n! • ∆ [n] ,
where ∆ [n] is the iterated coproduct map from U (L) to U (L) ⊗n . The same formula holds for inv S .

The preLie PBW theorem

Let us consider now the combinatorial PBW problem for preLie algebras, that is the particularization of Solomon's PBW problem for Lie algebras [START_REF] Solomon | On the Poincaré Birkhoff Witt theorem[END_REF] to the preLie case: compute explicit formulas for the canonical decomposition of the enveloping algebra S * (L) induced by the isomorphism is

• π nat with S(L) = n S n (L).
We are interested in the general form of the decomposition, that is, the way an arbitrary element l = l 1 ...l n of S * (L) decomposes into l = sol 1 (l) + ... + sol n (l). We can therefore assume that the l i are algebraically independent in L. Moreover, by the classical polarization argument, setting a := l 1 + ... + l n , we get that l is the l 1 , ..., l n -multilinear component of a n n! ; similarly, sol i (l) identifies to the l 1 , ..., l n -multilinear component of sol i ( a n n! ) and of sol i (exp(a)). In the end, the computations can therefore be handled in the sub free preLie algebra of L over a and amount to computing the sol i (exp(a)).

We get:

exp(a) = Id(exp(a)) = (exp ⋆ • log ⋆ (Id))(exp(a)),
where we recall that log

⋆ (Id) = sol 1 . Since ∆(a n ) = i≤n n i a i ⊗ a n-i , a := exp(a)
is a group-like element in S * (L) (i.e. ∆(a) = a ⊗ a). However, the logarithm of a group-like element x is a primitive element since

∆(log * (x)) = log * (x ⊗ x) = log * (x ⊗ 1) • (1 ⊗ x) = log * (x) ⊗ 1 + 1 ⊗ log * (x)
and therefore log ⋆ (Id)(a) = log * (a) is a primitive element in S * (L) (here log ⋆ , resp. log * means that we compute the logarithm for the ⋆ product, resp. the * product, and so on). We get finally the first, polarized, form of the preLie PBW decomposition:

a = n π [n] • sol ⊗n 1 • ∆ [n] n! (a) = n (log * (a)) * n n! ,
where π [n] stands for the iterated * product from (S * (L)) ⊗n to S * (L), resp. ∆ [n] for the iterated coproduct from S * (L) to (S * (L)) ⊗n . Now, the l 1 , ..., l n -multilinear part sol 1 (l 1 ...l n ) of log * (a) reads

sol 1 (l 1 ...l n ) = n i=1 (-1) i-1 i I 1 ,...,I i l I 1 * ... * l I i .
More generally, for higher components of the PBW decomposition, we have:

sol i (l 1 ...l n ) = n j=i s(j, i) j! I 1 ,...,I j l I 1 * ... * l I j ,
where the s(j, i) are Stirling numbers of the first kind.

The proof of the general case follows from a Ψ-ring argument as for the geometric computation of the sol i in [START_REF] Patras | Construction géométrique des idempotents eulériens. Filtration des groupes de polytopes et des groupes d'homologie de Hochschild[END_REF]. Notice first that

Ψ k := Id ⋆k = (exp ⋆ (log ⋆ (Id))) ⋆k = exp ⋆ (k log ⋆ Id) = n k n sol n .
Besides, with the same notation as above for ordered partitions, we get from the definition of the ⋆ product and the group-like behaviour of a:

Ψ k (l 1 ...l n ) = n j=1 I 1 ,...,I j k j l I 1 * ... * l I j .
From the definition of the Stirling numbers s(n, k) of the first kind,

x(x -1)...(xj + 1) = 1≤i≤j s(j, i)x j , we get:

Ψ k (l 1 ...l n ) = n i=1   n j=i s(j, i) j! I 1 ,...,I j l I 1 * ... * l I j   k i
and finally, by identification of the coefficient of k i :

sol i (l 1 ...l n ) = n j=i s(j, i) j! I 1 ,...,I j l I 1 * ... * l I j ,
which concludes the proof.

The preLie PBW decomposition and the Magnus element

Definition 2. The Magnus element in the free preLie algebra L over a single generator a is the (necessarily unique) solution Ω to the equation:

a Ω exp(Ω) -1 = Ω.
The terminology is motivated by the so-called Magnus solution to an arbitrary matrix (or operator) differential equation X ′ (t) = A(t)X(t), X(0) = 1: X(t) = exp(Ω(t)), where

Ω ′ (t) = ad Ω(t) exp ad Ω(t) -1 A(t) = A(t) + n>0 B n n! ad n Ω(t) (A(t)),
where ad stands for the adjoint representation and the B n for the Bernoulli numbers. The link with preLie algebras follows from the observation that (under the hypothesis that the integrals and derivatives are well-defined), for arbitrary time-dependent operators, the preLie product

M (t) N (

sol 1 (I 1 Proposition 3 .

 113 l 1 ...l n ) = ,...,I i l I 1 * ... * l I i where the I 1 , ..., I i run over all ordered partitions of {l 1 , ..., l n } (I 1 ... I i = {l 1 , ..., l n }) where ∀j ∈ [n], I j = ∅ and, for any subset I of [1, n], l I := i∈I l i . We get the first part of the following Proposition: The PBW decomposition of a generic element l 1 ...l n ∈ S * (L) reads:

t) := t 0 [

 0 N (u), M ′ (u)]du satisfies (M (t) N (t)) ′ = ad N (t) M ′ (t). The Magnus formula rewrites therefore:Ω ′ (t) = A(t) Ω exp(Ω) -1 ′where Ω exp(Ω)-1 is computed in the enveloping algebra of the preLie algebra of time-dependent operators. See e.g. the recent works by K. Ebrahimi-Fard and D. Manchon for further insights and an up-to-date point of view on the Magnus formula, in particular[START_REF] Ebrahimi-Fard | A Magnus-and Fer-type formula in dendriform algebras[END_REF].

Theorem 4 .

 4 The Magnus element identifies with sol 1 (exp(a)):Ω = sol 1 (exp(a)) = log * (exp(a)).Indeed, we have:sol 1 (exp(a)) = log * (a) = n>0 (-1) n-1 (a -1) * n n .Let us write a = b + o(1) to mean that a and b in S * (L) are equal up to an element of S * ,≥2 (L). We then have, by definition of the * product:sol 1 (exp(a)) = (a + o(1)) * n (-1) n-1 (a -1) * n-1 n = a * n (-1) n-1 (a -1) * n-1 n + o(1).Moreover, for an arbitrary element b of S * ,n (L) the product a * b reads a b + o(1). Finally, since sol 1 (exp(a)) ∈ L, we getsol 1 (exp(a)) = a n (-1) n-1 (a -1) * n-1 n = a log * (a) a -1 ,from which the theorem follows.

Acknowledgements. This work originated with discussions at the IESC conference Institute of Cargèse with K. Ebrahimi-Fard, F. Hivert, F. Menous, J.-Y. Thibon and the other participants to the CNRS PEPS program "Mould calculus". We thank them warmly for the stimulating exchanges, as well as the IESC for its hospitality.