Jérôme Dedecker

Florence Merlevède

Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary α-dependent sequences

Keywords: Empirical process, Wasserstein distance, central limit theorem, moments inequalities, stationary sequences, intermittency Mathematics Subject Classification (2010). 60F17, 60E15, 60G10

Behavior of the Wasserstein distance between the empirical and the marginal distributions of stationary α-dependent sequences

Jérôme Dedecker, Florence Merlevède

Introduction

Let (Xi) i∈Z be a stationary sequence of integrable real-valued random variables, with common marginal distribution µ. Let µn be the empirical measure of {X1, . . . , Xn}, that is

µn = 1 n n k=1 δX k .
In this paper, we study the behavior of the quantity W1(µn, µ) for a large class of stationary sequences, where W1(µ1, µ2) is the Wasserstein distance of order 1 between two probability measures µ1, µ2 having finite first moments. The precise definition is as follows:

W1(µ1, µ2) = inf π∈M (µ 1 ,µ 2 )
|x -y|π(dx, dy) ,

where M (µ1, µ2) is the set of probability measures on R 2 with marginal distributions µ1 and µ2.

The distance W1 belongs to the general class of minimal distances, as the total variation distance.

Since the cost function c1(x, y) = |x -y| is regular, W1 can be used to compare two singular measures, which is not possible with the total variation distance, whose cost function is given by the discrete metric c0(x, y) = 1 x =y . The quantity W1(µn, µ) appears very frequently in statistics, and can be understood from many points of view:

• The well known dual representation of W1 implies that W1(µn, µ) = sup

f ∈Λ 1 1 n n k=1 (f (X k ) -µ(f )) , (1.2)
where Λ1 is the set of Lipschitz functions f from R to R such that |f (x) -f (y)| ≤ |x -y|. Hence, W1(µn, µ) is a measure of the concentration of µn around µ through the class Λ1.

• In the one dimensional setting the minimization problem (1.1) can be explicitely solved, and leads to the expression

W1(µn, µ) = 1 0 |F -1 n (t) -F -1 (t)|dt , (1.3)
where Fn and F are the distribution functions of µn and µ, and F -1 n and F -1 are their usual generalized inverses. Hence W1(µn, µ) is the L 1 -distance between the empirical quantile function F -1 n and the quantile function of µ.

• Starting from (1.3), it follows immediately that W1(µn, µ) = R |Fn(t) -F (t)|dt .

(1.4)

Hence W1(µn, µ) is the L 1 -distance between the empirical distribution function Fn and the distribution function of µ.

At this point, it should be clearly quoted that, if (1.3) and (1.4) have no analogue in higher dimension, the dual expression (1.2) is very general and holds if the Xi's take their values in a Polish space X , as soon as the cost function c is a lower semi-continuous metric (the class Λ1 being the class of 1-Lipschitz functions from X to R with respect to c).

Assume now that the sequence (Xi) i∈Z is ergodic. Since µ has a finite first moment, it is well known that W1(µn, µ) converges to zero almost surely, and that E(W1(µn, µ)) converges to zero (this is a uniform version of Birkhoff's ergodic theorem, which can be easily deduced from the Glivenko-Cantelli theorem for ergodic sequences). However, without additional asumptions on µ the rate of convergence can be arbitrarily slow.

The purpose of this paper is to give some conditions under which the central limit theorem (CLT) holds (meaning that √ nW1(µn, µ) converges in distribution to a certain law), and to prove some inequalities for W1(µn, µ) p when p ≥ 1 (von Bahr-Esseen type inequalities for p ∈ (1, 2) and Rosenthal type inequalities for p > 2). We will do this for the class of α-dependent sequences, which is quite natural in this context, since the related dependency coefficients are defined through indicator of half lines. Hence our results apply to mixing sequences in the sense of Rosenblatt [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF], but also to many other dependent sequences including a large class of one dimensional dynamical systems. We shall illustrate our results through the examples of Generalized Pomeau-Manneville maps, as defined in [START_REF] Dedecker | Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains[END_REF].

The central limit question for √ nW1(µn, µ) has been already investigated for dependent sequences in the papers by Dédé [START_REF] Dédé | An empirical central limit theorem in L 1 for stationary sequences[END_REF] and Cuny [START_REF] Cuny | Limit theorems under the Maxwell-Woodroofe condition in Banach spaces[END_REF] (see Sections 4 and 5 for more details). This is not the case of the upper bounds for W1(µn, µ) p, even for sequences of independent and identically distributed (i.i.d.) random variables (except for p = 1, see for instance [START_REF] Bobkov | One-dimensional empirical measures, order statistics and Kantorovich transport distances[END_REF]). Hence, for p > 1, our moment bounds seem to be new even in the i.i.d. context.

Thanks to the relation (1.4), the central limit question for √ nW1(µn, µ) is closely related to the empirical central limit theorem in L 1 (dt), as first quoted by del Barrio, Giné and Matrán [START_REF] Del Barrio | Central limit theorems for the Wasserstein distance between the empirical and the true distributions[END_REF]. We shall deal with the more general central limit question for L 1 (m)-valued random variables in the separate Section 4. In Section 5, we shall express some of our conditions in terms of the quantile function of X0, in the spirit of Doukhan, Massart and Rio [START_REF] Doukhan | The functional central limit theorem for strongly mixing processes[END_REF]. It will then be easier to compare our conditions for the CLT to previous ones in the literature.

For r > 1, the quantity W r r (µn, µ) may be defined as in (1.1), with the cost function cr(x, y) = |x -y| r instead of c1 (Wr is the Wasserstein distance of order r). In the i.i.d. case, some sharp upper bounds on E(W r r (µn, µ)) are given in the recent paper [START_REF] Bobkov | One-dimensional empirical measures, order statistics and Kantorovich transport distances[END_REF]. In particular, if µ has an absolutely component with respect to the Lebesgue measure which does not vanishes on the support of µ, then the optimal rate n -r/2 can be reached. But in general, the rate can be much slower. Note that for W r r (µn, µ) there is no such nice dual expression as (1.2). However the minimization problem can still be explicitely solved and implies that Wr(µn, µ) is the L r -distance between F -1 n and F -1 . There is no simple way to express W r r (µn, µ) in terms of Fn and F (as in (1.4)), but the following upper bound due to Èbralidze [START_REF] Èbralidze | Inequalities for the probabilities of large deviations in terms of pseudomoments, (Russian)[END_REF] holds:

W r r (µn, µ) ≤ κr R |x| r-1 |Fn(x) -F (x)|dx , (1.5)
where κr = 2 r-1 r. Starting from this inequality, we shall also give some upper bounds on W r r (µn, µ) p for p ≥ 1, but it is very likely that these bounds can be improved by assuming the existence of an absolutely regular component for µ, as in [START_REF] Bobkov | One-dimensional empirical measures, order statistics and Kantorovich transport distances[END_REF].

To be complete, let us mention the recent paper by Fournier and Guillin [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF], who give some upper bounds for E(W r r (µn, µ)) in any dimension, starting from an inequality which can be viewed as a d-dimensional analogue of (1.5). Note that the case of ρ-mixing sequences is also considered in this paper.

Definitions and notations

In this section, we give the notations and definitions which we will used all along the paper.

Let us start with the notation an bn, which means that there exists a numerical constant C not depending on n such that an ≤ Cbn, for all positive integers n.

Stationary sequences and dependency coefficients

Let (Ω, A, P) be a probability space, and T : Ω → Ω be a bijective bi-measurable transformation preserving the probability P. Let F0 be a sub-σ-algebra of A satisfying F0 ⊆ T -1 (F0). We say that the couple (T, P) is ergodic if any A ∈ A satisfying T (A) = A has probability 0 or 1.

Let X0 be an F0-measurable and integrable real-valued random variable with distribution µ. Define the stationary sequence X = (Xi) i∈Z by Xi = X0 • T i .

Let us first define the tail and quantile functions of the random variable X0.

Definition 2.1. The tail function H : R + → [0, 1] of X0 is defined by

H(t) = P(|X0| > t). The quantile function Q : [0, 1] → R + of X0 is the generalized inverse of H, that is Q(u) = inf {t ≥ 0 : f (t) ≤ u} .
Let us now define the dependency coefficients of the sequence (Xi) i∈Z . These coefficients are less restrictive than the usual mixing coefficients of Rosenblatt [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]. Definition 2.2. For any integrable random variable Z, let

Z (0) = Y -E(Z). For any random variable Y = (Y1, • • • , Y k ) with values in R k and any σ-algebra F, let α(F, Y ) = sup (x 1 ,...,x k )∈R k E k j=1 (1 Y j ≤x j ) (0) F -E k j=1 (1 Y j ≤x j ) (0) 1 .
For the stationary sequence X = (Xi) i∈Z , let

α k,X (n) = max 1≤l≤k sup n≤i 1 ≤...≤i l α(F0, (Xi 1 , . . . , Xi l )). (2.1)
Note that α 1,X (n) is then simply given by

α 1,X (n) = sup x∈R E (1 Xn≤x |F0) -F (x) 1 , (2.2)
where F is the distribution function of µ.

All the results of Section 3 below involve only the coefficients α 1,X (n), except for the Rosenthal bounds (Subsection 3.4) for which the coefficient α 2,X (n) is needed.

Intermittent maps

Let us first recall the definition of the generalized Pomeau-Manneville maps introduced in [START_REF] Dedecker | Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains[END_REF].

Definition 2.3. A map θ : [0, 1] → [0, 1] is a generalized Pomeau-Manneville map (or GPM map) of parameter γ ∈ (0, 1) if there exist 0 = y0 < y1 < • • • < y d = 1 such that, writing I k = (y k , y k+1 ), 1. The restriction of θ to I k admits a C 1 extension θ (k) to I k . 2. For k ≥ 1, θ (k) is C 2 on I k , and |θ (k) | > 1. 3. θ (0) is C 2 on (0, y1], with θ (0) (x) > 1 for x ∈ (0, y1], θ (0) (0) = 1 and θ (0) (x) ∼ cx γ-1 when
x → 0, for some c > 0.

4. θ is topologically transitive.

The third condition ensures that 0 is a neutral fixed point of θ, with θ(x) = x+c x 1+γ (1+o(1)) when x → 0. The fourth condition is necessary to avoid situations where there are several absolutely continuous invariant measures, or where the neutral fixed point does not belong to the support of the absolutely continuous invariant measure. The following well known example of GPM map with only two branches has been introduced by Liverani, Saussol and Vaienti [START_REF] Liverani | A probabilistic approach to intermittency[END_REF]:

y 0 = 0 y 1 y 2 y 3 y 4 = 1
θ(x) = x(1 + 2 γ x γ ) if x ∈ [0, 1/2[ 2x -1 if x ∈ [1/2, 1]. (2.3)
As quoted in [START_REF] Dedecker | Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains[END_REF], a GPM map θ admits a unique invariant absolutely continuous (with respect to the Lebesgue measure) probability ν with density h. Moreover, it is ergodic, has full support, and x γ h(x) is bounded from above and below.

We shall illustrate each result of Section 3 by controlling, on the probability space ([0, 1], ν), the quantity W1(μn, µ), where

μn = 1 n n k=1 δ g•θ k , (2.4)
θ is a GPM map, g is a monotonic function from (0, 1) to R (which can blow up near 0 or 1), and µ is the distribution of g.

To do this, we go back to the Markov chain associated to θ, as we describe now. Let first K be the Perron-Frobenius operator of θ with respect to ν, defined as follows: for any functions

u, v in L 2 ([0, 1], ν) ν(u • v • θ) = ν(K(u) • v) . (2.5)
The relation (2.5) states that K is the adjoint operator of the isometry U : u → u • θ acting on L 2 ([0, 1], ν). It is easy to see that the operator K is a transition kernel, and that ν is invariant by K. Let now Y = (Yi) i≥0 be a stationary Markov chain with invariant measure ν and transition kernel K. It is well known (see for instance Lemma XI.3 in [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF]) that on the probability space ([0, 1], ν), the random vector (θ, θ 2 , . . . , θ n ) is distributed as (Yn, Yn-1, . . . , Y1).

Let T be the shift operator from [0, 1] Z to [0, 1] Z defined by (T (x))i = xi+1, and let πi be the projection from [0, 1] Z to [0, 1] defined by πi(x) = xi. By Kolmogorov's extension theorem, there exists a shift-invariant probability P on ([0, 1] Z , (B([0, 1])) Z ), such that π = (πi) i≥0 is distributed as Y.

Let then X0 = g • π0 and Xi = X0 • T i = g • πi, and define F0 = σ(πi, i ≤ 0). From the above considerations, we infer that the two random variables W1(µn, µ) (defined on the probability space (R Z , P)) and W1(μn, µ) (defined on the probability space ([0, 1], ν)) have the same distribution. Hence, any information on the distribution of W1(μn, µ) can be derived from the distribution of W1(µn, µ).

From Proposition 1.17 (and the comments right after) in [START_REF] Dedecker | Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains[END_REF], we know that for any positive integer k, there exist two positive constants C and D such that, for any n > 0,

D n (1-γ)/γ ≤ α k,π (n) ≤ C n (1-γ)/γ .
Since Xi = g • πi, and since g is monotonic, it follows immediately that

α k,X (n) ≤ α k,π (n) ≤ C n (1-γ)/γ . (2.6)
This control of the coefficients α k,X (n) (for k = 1 or k = 2) and a control of the tail ν(|g| > t) are all we need to apply the results of Section 3 to the random variable W1(μn, µ).

CLT and moment bounds

In all this section, we use the notations of Sections 1 and 2.

Central limit theorem

Our first result is a central limit theorem for W1(µn, µ). It is a straightforward consequence of a CLT in L 1 (m) for the empirical distribution function given in Proposition 4.2 of Subsection 4.4 (it suffices to consider the case where m is the Lebesgue measure on R and to use the continuous mapping theorem).

Proposition 3.1. Assume that the couple (T, P) is ergodic, and that

∞ 0 ∞ k=0 min {α 1,X (k), H(t)} dt < ∞ . (3.1)
Then √ nW1(µn, µ) converges in distribution to the random variable |G(t)| dt, where G is a Gaussian random variable in L 1 (dt) whose covariance function may be described as follows: for any f, g in L∞(µ),

Cov f (t)G(t)dt, g(t)G(t)dt = k∈Z E f (t)g(s)(1 X 0 ≤t -F (t))(1 X k ≤s -F (s)) dtds . (3.2) Remark 3.1.
Let m be a nonnegative integer. As usual, the stationary sequence X is mdependent if σ(Xi, i ≤ 0) is independent of σ(Xi, i ≥ m + 1), and m = 0 corresponds to the i.i.d. case. In the m-dependent case, the condition (3.1) becomes simply

∞ 0 H(t) dt < ∞ , (3.3)
which is exactly the condition given by del Barrio, Giné and Matrán [START_REF] Del Barrio | Central limit theorems for the Wasserstein distance between the empirical and the true distributions[END_REF] in the i.i.d. case. Note that these authors also proved that, in the i.i.d. case, the condition (3.3) is necessary and sufficient for the stochastic boundedness of √ nW1(µn, µ). In the dependent context, other general criteria have been proposed by Dédé [START_REF] Dédé | An empirical central limit theorem in L 1 for stationary sequences[END_REF] and Cuny [START_REF] Cuny | Limit theorems under the Maxwell-Woodroofe condition in Banach spaces[END_REF]. We shall discuss these conditions in Sections 4 and 5, and show that, in the α-dependent case, the condition (3.1) is weaker than the corresponding condition obtained by applying the criteria by Dédé or Cuny.

Example. Let θ be a GPM map of parameter γ ∈ (0, 1/2), with absolutely continuous invariant probability ν. Let μn be defined as in (2.4), where g is a monotonic function from (0, 1) to R. Let then (Xi) i∈Z be the stationary sequence constructed in Subsection 2.2, whose dependency coefficients α k,X (n) satisfy (2.6). Note that H(t) = P(|X0| > t) = ν(|g| > t). From Subsection 2.2, Proposition 3.1 and Item 3 of Proposition 5.3, we infer that √ nW1(μn, µ) converges in distribution to the random variable |G(t)|dt, where G is a Gaussian random variable in L 1 (dt) as soon as

∞ 0 (H(t)) 1-2γ 2(1-γ) dt < ∞ . (3.4)
As a consequence:

1. If g is positive and non increasing on (0, 1), with

g(x) ≤ C x (1-2γ)/2 | ln(x)| b near 0, for some C > 0 and b > 1, then (3.4) holds.
2. If g is positive and non decreasing on (0, 1), with Recall from (1.2) that W1(μn, µ) = sup f ∈Λ 1 |μn(f ) -µ(f )|, so that the condition (3.4) allows to control the supremum of √ n(µn(f ) -µ(f )) over the class Λ1. Now if we only want a central limit theorem for √ n(µn(f ) -µ(f )) where f is an element of Λ1, then it follows from [START_REF] Dedecker | Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains[END_REF] that the condition

g(x) ≤ C (1 -x) (1-2γ)/(2-2γ) | ln(1 -x)| b
∞ 0 t(H(t)) 1-2γ 1-γ dt < ∞ . ( 3

Upper bounds for moments of order 1 and 2

In this section, we give some upper bounds for the quantities E(W1(µn, µ)) and W1(µn, µ) 2 in terms of the coeffcients α 1,X (k) and of the tail function H. For any t ≥ 0, let

Sα,n(t) = n k=0 min {α 1,X (k), H(t)} . (3.6) Proposition 3.2.
The following upper bounds hold:

E(W1(µn, µ)) ≤ 4 ∞ 0 min H(t) 2 , Sα,n(t) n dt , (3.7)
and

W1(µn, µ) 2 ≤ 2 √ 2 √ n ∞ 0 Sα,n(t) dt . (3.8)
Remark 3.2. As will be clear from the proof, one can also get some upper bounds involving the quantity

B(t) = F (t)(1 -F (t)) instead of H(t).
For instance, we can obtain an extension of the upper bound given in Theorem 3.5 of [START_REF] Bobkov | One-dimensional empirical measures, order statistics and Kantorovich transport distances[END_REF] to α-dependent sequences. We have chosen to express the upper bounds in terms of the function H, because they are easier to compute in the α-dependent case (see Remark 3.4 below).

The proof of Proposition 3.2 is based on the following elementary inequality applied to p = 1 and p = 2:

For any p ≥ 1, |Fn(t) -F (t)| dt p ≤ Fn(t) -F (t) p dt .
One could also start from this inequality in the case where p ∈ (1, 2) (resp. p > 2) by applying a von Bahr-Esseen bound (resp. a Rosenthal bound) to Fn(t) -F (t) p. However, this would give less satisfactory bounds than in Subsections 3.3 and 3.4, even in the i.i.d. case. For instance, in the i.i.d. case and p ∈ (1, 2), this would give

W1(µn, µ)) p p 1 n p-1 (H(t)) 1/p dt p . (3.9)
Note that the condition (H(t)) 1/p dt < ∞ is more restrictive than X0 p < ∞. Hence the upper bound (3.24) of Subsection 3.3 is always better than (3.9). Remark 3.3. Starting from Inequality (1.5) and following the proof of Proposition 3.2 we obtain the upper bounds

E(W r r (µn, µ)) ≤ 4 ∞ 0 t r-1 min H(t) 2 , Sα,n(t) n dt , (3.10)
and

W r r (µn, µ) 2 ≤ 2 √ 2 √ n ∞ 0 t r-1 Sα,n(t) dt . (3.11) Remark 3.4.
As a consequence of Proposition 3.2, the following upper bounds hold:

1. If (3.1) holds, then W1(µn, µ) 2 n -1/2 . 2. If α(k) = O(k -a ) for some a > 1, then E(W1(µn, µ))   n -a a+1 0 Q(u)du + 1 √ n 1 n -a a+1 Q(u) u a+1 2a du   , (3.12)
and

W1(µn, µ) 2 n -a 0 Q(u) √ u du + 1 √ n 1 n -a Q(u) u a+1 2a du . (3.13) 3. If α(k) = O(a k ) for some a < 1, then E(W1(µn, µ)) ln(n) n 0 Q(u)du + 1 √ n 1 ln(n) n Q(u)| ln(u)| √ u du , and
W1(µn, µ) 2 e -n 0 Q(u) √ u du + 1 √ n 1 e -n Q(u)| ln(u)| √ u du .
4. Assume that the α k 's converge to zero, but are not summable, and let

un = 1 n n k=1 α k . Then E(W1(µn, µ)) √ un 0 Q(u) du , (3.14)
and

W1(µn, µ) 2 un 0 Q(u) √ u du . (3.15)
Remark 3.5. In the m-dependent case, the inequality (3.12) holds with a = ∞, that is

E(W1(µn, µ)) n -1 0 Q(u)du + 1 √ n 1 n -1 Q(u) √ u du .
In particular, if

H(t) = O(t -1 (ln(t)) -a ) for some a > 1 (which implies that E(|X0|) < ∞), then Q(u) = O(u -1 | ln(u)| -a
), and consequently

E(W1(µn, µ)) 1 (ln(n)) a-1 .
Example (continued). We continue the example of Subsection 3.1.

1. If g is positive and non increasing on (0, 1), with

g(x) ≤ C x b near 0, for some C > 0 and b ∈ [0, 1 -γ), then Q(u) ≤ Du -b/(1-γ)
for some D > 0. Applying (3.12)-(3.13) and (3.14)-(3.15), the following upper bounds hold.

For γ ∈ (0, 1/2),

E(W1(μn, µ))      n -1/2 if b < (1 -2γ)/2 n -1/2 ln(n) if b = (1 -2γ)/2 n b+γ-1 if b > (1 -2γ)/2, and
W1(μn, µ) 2      n -1/2 if b < (1 -2γ)/2 n -1/2 ln(n) if b = (1 -2γ)/2 n (2b+γ-1)/2γ if (1 -2γ)/2 < b < (1 -γ)/2. For γ = 1/2, E(W1(μn, µ)) ln(n) n 1-2b 2
, and

W1(μn, µ) 2 ln(n) n 1-4b 2 if b < 1/4. For γ ∈ (1/2, 1), E(W1(μn, µ)) n b+γ-1 2γ
, and

W1(μn, µ) 2 ≤ Cn 2b+γ-1 2γ if b < (1 -γ)/2.
2. If g is positive and non decreasing on (0, 1), with

g(x) ≤ C (1 -x) b near 1, for some C > 0 and b ∈ [0, 1), then Q(u) ≤ Du -b
for some D > 0. Applying (3.12)-(3.13) and (3.14)-(3.15), the following upper bounds hold. For γ ∈ (0, 1/2),

E(W1(μn, µ))      n -1/2 if b < (1 -2γ)/2(1 -γ) n -1/2 ln(n) if b = (1 -2γ)/2(1 -γ) n (γ-1)(1-b) if b > (1 -2γ)/2(1 -γ), and
W1(μn, µ) 2      n -1/2 if b < (1 -2γ)/2(1 -γ) n -1/2 ln(n) if b = (1 -2γ)/2(1 -γ) n (γ-1)(1-2b)/2γ if (1 -2γ)/2(1 -γ) < b < 1/2. For γ = 1/2, E(W1(μn, µ)) ln(n) n 1-b 2
, and

W1(μn, µ) 2 ln(n) n 1-2b 2 if b < 1/2.
For γ ∈ (1/2, 1),

E(W1(μn, µ)) n (γ-1)(1-b) 2γ
, and W1(μn, µ) 2 n

(γ-1)(1-2b) 2γ if b < 1/2.
Proof of Proposition 3.2. Starting from (1.4), we immediately see that

E(W1(µn, µ)) ≤ Fn(t) -F (t) 1 dt and W1(µn, µ) 2 ≤ Fn(t) -F (t) 2 dt . (3.16) Let B(t) = F (t)(1 -F (t))
, and note first that

Fn(t) -F (t) 1 ≤ 1 X 0 ≤t -F (t) 1 = 2B(t) . (3.17)
On another hand

Fn(t) -F (t) 2 1 ≤ Fn(t) -F (t) 2 2 ≤ 1 n Var(1 X 0 ≤t ) + 2 n n k=1 |Cov(1 X 0 ≤t , 1 X k ≤t )| . (3.18)
Now, the two following upper bounds hold:

|Cov(1 X 0 ≤t , 1 X k ≤t )| ≤ E(1 X k ≤t |F0) -F (t) 1 ≤ α 1,X (k) , (3.19) |Cov(1 X 0 ≤t , 1 X k ≤t )| ≤ Var(1 X 0 ≤t ) = B(t) . (3.20)
From (3.17), (3.18), (3.19) and (3.20) it follows that

Fn(t) -F (t) 1 ≤ 2 min B(t) 2 , 1 n n k=0 min {α 1,X (k), B(t)}
and

Fn(t) -F (t) 2 ≤ 2 n n k=0 min {α 1,X (k), B(t)} .
These two upper bounds combined with (3.16) imply that

E(W1(µn, µ)) ≤ 2 min B(t) 2 , 1 n n k=0 min {α 1,X (k), B(t)} dt ≤ 4 ∞ 0 min H(t) 2 , Sα,n(t) n dt and W1(µn, µ) 2 ≤ 2 n n k=0 min {α 1,X (k), B(t)} dt ≤ 2 √ 2 √ n ∞ 0 Sα,n(t) dt ,
which are the desired inequalities.

A von Bahr-Esseen type inequality

In this section, we give some upper bounds for the quantity W1(µn, µ) p when p ∈ (1, 2) in terms of the coefficients α 1,X (k) and of the quantile function Q. For u ∈ (0, 1), let

α -1 1,X (u) = ∞ k=0 1 u≤α 1,X (k) . (3.21)
Proposition 3.3. For p ∈ (1, 2), the following inequality holds

W1(µn, µ) p p 1 n p-1 1 0 (α -1 1,X (u) ∧ n) p-1 Q p (u)du . (3.22)
Note that Inequality (3.22) writes also

W1(µn, µ) p p 1 n p-1 n k=0 1 (k + 1) 2-p α 1,X (k) 0 Q p (u)du .
Remark 3.6. Let r ≥ 1 and p ∈ (1, 2). Starting again from (1.5) and following the proof of Proposition 3.3, we obtain the upper bound This inequality seems to be new even in the i.i.d. case. It is noteworthy that the upper bound (3.24) is the same as the moment bound of order p for partial sums of i.i.d. random variables, which can be deduced from the classical inequality of von Bahr and Esseen [START_REF] Bahr | Inequalities for the rth absolute moment of a sum of random variables, 1 ≤ r ≤ 2[END_REF].

W r r (µn, µ) p p 1 n p-1 1 0 (α -1 1,X (u) ∧ n) p-1 Q pr (u)du . ( 3
Example (continued). We continue the example of Subsection 3.1.

1. Let p ∈ (0, 1), and let g be positive and non increasing on (0, 1), with

g(x) ≤ C x b near 0, for some C > 0 and b ∈ [0, (1 -γ)/p).
Applying Proposition 3.3, the following upper bounds hold. For γ ∈ (0, 1/p),

W1(μn, µ)) p      n (1-p)/p if b < (1 -pγ)/p (n (1-p) ln(n)) 1/p if b = (1 -pγ)/p n (pb+γ-1)/pγ if b > (1 -pγ)/p.
Moreover, if b = (1 -pγ)/p, Proposition 3.4 below gives the upper bound

P (W1(µn, µ) ≥ x) 1 n p-1 x p . (3.25)
For γ ∈ [1/p, 1), W1(μn, µ)) p n (pb+γ-1)/pγ .

2. Let p ∈ (0, 1), and let g be positive and non decreasing on (0, 1), with

g(x) ≤ C (1 -x) b near 1, for some C > 0 and b ∈ [0, 1/p).
Applying Proposition 3.3, the following upper bounds hold. For γ ∈ (0, 1/p),

W1(μn, µ)) p      n (1-p)/p if b < (1 -pγ)/(p(1 -γ)) (n (1-p) ln(n)) 1/p if b = (1 -pγ)/(p(1 -γ)) n (γ-1)(1-pb)/pγ if b > (1 -pγ)/(p(1 -γ)).
Moreover, if b = (1 -pγ)/(p(1 -γ)), Proposition 3.4 below gives the upper bound (3.25).

For γ ∈ [1/p, 1), W1(μn, µ)) p n (γ-1)(1-pb)/pγ .
Remark 3.8. The upper bound (3.25) is in accordance with a result by Gouëzel [START_REF] Gouëzel | Central limit theorem and stable laws for intermittent maps[END_REF]. He proved that, if g is exactly of the form g(x) = x -(1-pγ)/p and θ is the LSV map defined by (2.3), then for any positive real x,

lim n→∞ ν 1 n 1/p n k=1 g • θ k -ν(g) > x = P(|Zp| > x) ,
where Zp is a p-stable random variable such that limx→∞

x p P(|Zp| > x) = c > 0.
Proof of Proposition 3.3. For any n ∈ N, let us introduce the following notations:

Rn(u) = (min{q ∈ N * : α 1,X (q) ≤ u} ∧ n)Q(u) and R -1 n (x) = inf{u ∈ [0, 1] : Rn(u) ≤ x} .
The proof is based on the following proposition:

Proposition 3.4. For any positive integer n, any x > 0, and any η ∈ [1, 2[, the following inequality holds:

P (nW1(µn, µ) ≥ 6x) ≤ c1 n x R -1 n (x) 0 Q(u)du + c2 n x η 1 R -1 n (x) R η-1 n (u)Q(u)du , (3.26)
where c1 = 36 and c2 = 64(2 -η) -1 .

Before proving the proposition above, let us see how it entails Proposition 3.3. We have

nW1(µn, µ) p p = 6 p p ∞ 0 x p-1 P nW1(µn, µ) ≥ 6x dx .
Therefore applying Inequality (3.26) with η ∈ (p, 2) and using the fact that

u < R -1 n (x) ⇐⇒ x < Rn(u) , we get nW1(µn, µ) p p ≤ 6 p p nc1 1 0 Q(u) ∞ 0 x p-2 1 x<Rn(u) dxdu + 6 p p nc2 1 0 R η-1 n (u)Q(u) ∞ 0 x p-1-η 1 x≥Rn(u) dxdu ,
which gives the desired result since 1 < p < η < 2. Hence it remains to prove Proposition 3.4.

Proof of Proposition 3.4. Let v = R -1 n (x) , M = Q(v) (3.27)
and set gM (y) = (y ∧ M ) ∨ (-M ). For any integer i, let

X i = gM (Xi) and X i = Xi -X i . (3.28) Starting from (1.
2), we first notice that

nW1(µn, µ) = sup f ∈Λ 1 n i=1 (f (Xi) -E(f (Xi))) ≤ sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) + sup f ∈Λ 1 n i=1 f (Xi) -f (X i ) -E(f (Xi) -f (X i )) . Therefore nW1(µn, µ) ≤ sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) + n i=1 (|X i | + E(|X i |) . (3.29) Let now q = min{k ∈ N * : α 1,X (k) ≤ v} ∧ n . (3.30)
Since Rn is right continuous, we have Rn(R -1 n (w)) ≤ w for any w, hence

qM = Rn(v) = Rn(R -1 n (x)) ≤ x . (3.31) Assume first that q = n. Bounding f (X i ) -E(f (X i )) by 2M in (3.29), we obtain nW1(µn, µ) ≤ 2qM + n k=1 (|X k | + E(|X k |)) . (3.32)
Taking into account (3.31) this gives

P nW1(µn, µ) ≥ 6x ≤ 1 2x n k=1 E(|X k |) .
Writing ϕM (x) = (|x| -M )+, we have

n k=1 E(|X k |) ≤ n k=1 E(ϕM (X k )) . But Q ϕ M (X k ) ≤ Q |X k | 1 [0,v] ≤ Q1 [0,v] . Consequently n k=1 E(|X k |) ≤ n R -1 n (x) 0 Q(u)du . (3.33)
From (3.32) and (3.33), we infer that

P nW1(µn, µ) ≥ 6x ≤ n 2x R -1 n (x) 0 Q(u)du , (3.34)
which then proves the proposition in case where q = n.

From now on, we assume that q < n. Therefore q = min{k ∈ N * : α 1,X (k) ≤ v} and then α 1,X (q) ≤ v. Starting from (3.29), we first notice that

P (nW1(µn, µ) ≥ 6x) ≤ P sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) ≥ 5x + 2 x n k=1 E(|X k |) .
Therefore taking into account (3.33),

P (nW1(µn, µ) ≥ 6x) ≤ P sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) ≥ 5x + 2n x R -1 n (x) 0 Q(u)du . (3.35)
To control the first term on the right-hand side, we first notice that sup

f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) ≤ sup f ∈Λ 1 [n/q]q i=1 f (X i ) -E(f (X i )) + 2(n -[n/q]q)M ≤ R [n/q]q i=1 1 X i ≤t -E 1 X i ≤t dt + 2qM.
Using (3.31), it follows that

P sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) ≥ 5x ≤ P   R [n/q]q i=1 1 X i ≤t -E 1 X i ≤t dt ≥ 3x   .
For any integer i, define

Ui(t) = iq k=(i-1)q+1 1 X k ≤t -E 1 X k ≤t .
Consider now the σ-algebras Gi = Fiq and define the variables Ũi(t) as follows: Ũ2i-

1(t) = U2i-1(t) -E(U2i-1(t)|G 2(i-1)-1 ) and Ũ2i(t) = U2i(t) -E(U2i(t)|G 2(i-1)
). Substituting Ũi(t) to Ui(t), we obtain the inequality

[n/q]q i=1 1 X i ≤t -E 1 X i ≤t = [n/q] i=1 Ui(t) ≤ max 2≤2j≤[n/q] j i=1 Ũ2i(t) + max 1≤2j-1≤[n/q] j i=1 Ũ2i-1(t) + [n/q] i=1 |Ui(t) -Ũi(t)| . (3.36) Therefore P sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) ≥ 5x ≤ I1(n) + I2(n) + I3(n) , (3.37)
where

I1(n) = P   R [n/q] i=1 |Ui(t) -Ũi(t)| dt ≥ x   I2(n) = P R max 2≤2j≤[n/q] j i=1 Ũ2i(t) dt ≥ x I3(n) = P R max 1≤2j-1≤[n/q] j i=1 Ũ2i-1(t) dt ≥ x .
Using Markov's inequality and stationarity, we get

I1(n) ≤ n x R E E 1 X 1 ≤t |F-q -E 1 X 1 ≤t dt = n x M -M E E 1 X 1 ≤t |F-q -E 1 X 1 ≤t dt . But, sup t∈R E 1 X 1 ≤t |F-q -E 1 X 1 ≤t 1 = sup t∈R E 1 g M (X 1 )≤t |F-q -E 1 g(X 1 )≤t 1 ≤ α 1,X (q + 1) ,
where the inequality comes from the fact that gM is a nondecreasing function. Therefore,

I1(n) ≤ 2n x Q(v)α 1,X (q + 1) ≤ 2n x vQ(v) ≤ 2n x v 0 Q(u)du . (3.38)
We handle now the term I2(n) in the decomposition (3.37). Using again Markov's inequality, we get

I2(n) ≤ 1 x 2 Q(v) -Q(v) max 2≤2j≤[n/q] j i=1 Ũ2i(t) 2 dt 2 .
By Doob's maximal inequality, max

2≤2j≤[n/q] j i=1 Ũ2i(t) 2 2 ≤ 2 1 2 [n/q] i=1 Ũ2i(t) 2 2 ≤ 2 1 2 [n/q] i=1 U2i(t) 2 2 . Now 2 1 2 [n/q] i=1 U2i(t) 2 2 ≤ n q q k=1 1 X k ≤t -E 1 X k ≤t 2 2 ≤ 2n q-1 k=0 E 1 g M (X 0 )≤t -E 1 g M (X 0 )≤t 1 g M (X k )≤t -E 1 g M (X k )≤t .
Note that since gM is a nondecreasing function,

sup t∈R E 1 g M (X 0 )≤t -E 1 g M (X 0 )≤t 1 g M (X k )≤t -E 1 g M (X k )≤t ≤ sup t∈R |E ((1 X 0 ≤t -E (1 X 0 ≤t )) (1 X k ≤t -E (1 X k ≤t )))| . Moreover sup t∈R |E ((1X 0 >t -E (1X 0 >t)) (1X k >t -E (1X k >t)))| ≤ sup t∈R E (1X k >t|F0) -E (1X k >t) 1 = α 1,X (k) .
On an other hand, the following bound is also valid

E 1 g M (X 0 )≤t -E 1 g M (X 0 )≤t 1 g M (X k )≤t -E 1 g M (X k )≤t ≤ Var 1 g M (X 0 )≤t ≤ min E 1 g M (X 0 )≤t , E 1 g M (X 0 )>t .
So, overall, we get

I2(n) ≤ 2n x 2   Q(v) 0 q-1 k=0 α 1,X (k) ∧ P(gM (X0) > t) 1 2 + q-1 k=0 α 1,X (k) ∧ P(-gM (X0) ≥ t) 1 2 dt   2 ≤ 2n x 2   Q(v) 0 q-1 k=0 α 1,X (k) ∧ P(|X0| > t) 1 2 + q-1 k=0 α 1,X (k) ∧ P(|X0| ≥ t) 1 2 dt   2 .
We then derive that

I2(n) ≤ 8n x 2   Q(v) 0 q-1 k=0 α 1,X (k) ∧ H(t) 1/2 dt   2 = 8n x 2   Q(v) 0 q-1 k=0 H(t) 0 1 u≤α 1,X (k) du 1/2 dt   2 .
Using the fact that q-1 k=0 1 u≤α 1,X (k) = α -1 1,X (u) ∧ q, we then get

I2(n) ≤ 8n x 2   Q(v) 0 vq + H(t) v α -1 1,X (u) ∧ n du 1 2 dt   2 ≤ 16n x 2 vq(Q(v)) 2 + 16n x 2   Q(v) 0
H(t) v α -1 1,X (u) ∧ n du 1 2 dt   2 ≤ 16n x v 0 Q(u)du + 16n x 2   Q(v) 0
H(t) v α -1 1,X (u) ∧ n du 1/2 dt   2 . (3.39)
where for the last inequality we have used (3.31) and the fact that vQ(v) ≤ v 0 Q(u)du, since Q is non increasing. To handle the last term on the right-hand side, we proceed as follows. For any η in [1, 2), we first note that

H(t) v (α -1 1,X (u)∧n)du = H(t) v α -1 1,X (u) ∧ n Q η (u)Q -η (u)du ≤ 1 t η H(t) v α -1 1,X (u) ∧ n Q η (u)du ,
where the inequality comes from the fact that u < H(t) ⇐⇒ t < Q(u), and then u < H(t)

implies that Q -η (u) < t -η . Now, since u > v implies that α -1 1,X (u) ≤ α -1 1,X (v), we get H(t) v α -1 1,X (u) ∧ n du ≤ 1 t η α -1 1,X (v) ∧ n 2-η 1 v α -1 1,X (u) ∧ n η-1 Q η (u)du .
Therefore, since η/2 < 1,

  Q(v) 0
H(t) v α -1 1,X (u) ∧ n du 1/2 dt   2 ≤ α -1 1,X (v) ∧ n 2-η Q(v) 0 t -η/2 dt 2 1 v R -1 n (u) η-1 Q(u)du ≤ 2 2 -η α -1 1,X (v) ∧ n 2-η Q 2-η (v) 1 v R -1 n (u) η-1 Q(u)du . But, by (3.31), α -1 1,X (v) ∧ n 2-γ Q 2-γ (v) = R 2-γ n (v) ≤ x 2-γ . Therefore, 1 x 2   Q(v) 0
H(t) v α -1 1,X (u) ∧ n du 1/2 dt   2 ≤ 2 x η (2 -η) 1 v R -1 n (u) η-1 Q(u)du ,
which combined with (3.39) gives

I2(n) ≤ 16n x v 0 Q(u)du + 32n x η (2 -η) 1 v R -1 n (u) η-1 Q(u)du . (3.40)
With similar arguments, we get

I3(n) ≤ 16n x v 0 Q(u)du + 32n x η (2 -η) 1 v R -1 n (u) η-1 Q(u)du . (3.41)
Starting from (3.37) and using the upper bounds (3.38), (3.40) and (3.41), we derive that

P sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) ≥ 5x ≤ 34n x v 0 Q(u)du + 64n x η (2 -η) 1 v (R -1 n (u)) η-1 Q(u)du ,
which combined with (3.35) ends the proof of the proposition.

A Rosenthal type inequality

In this section, we give some upper bounds for the quantity W1(µn, µ) p when p > 2 in terms of the coefficients α 2,X (k) and of the quantile function Q. The function α -1 2,X is defined as in (3.21) by replacing the coefficient α 1,X (k) by α 2,X (k). Proposition 3.5. For p > 2, the following inequality holds:

W1(µn, µ) p p s p α,n n p/2 + 1 n p-1 1 0 α -1 2,X (u) ∧ n p-1 Q p (u)du , (3.42)
where sα,n = ∞ 0 Sα,n(t)dt
and Sα,n is the function defined in (3.6).

Note that Inequality (3.42) writes also

W1(µn, µ) p p s p α,n n p/2 + 1 n p-1 n k=0 (k + 1) p-2 α 1,X (k) 0 Q p (u)du .
Remark 3.9. Inequality (3.42) is similar to the Rosenthal inequality for partial sums given in Theorem 6.3 of Rio [START_REF] Rio | Théorie asymptotique des processus aléatoires faiblement dépendants[END_REF], with however two main differences:

• Firstly, the variance terms is not the same, but this is because we consider the quantity W1(µn, µ) and not only the partial sums, in accordance with the upper bounds for W1(µn, µ) 2 given in Subsection 3.2.

• Secondly, Rio's inequality is stated for α-mixing sequences in the sense of Rosenblatt [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF], and its proof relies on the coupling properties of these coefficients. Our result is valid for the larger class of α-dependent sequences as defined in 2.1 (with k = 2 for the index of the dependency), and the proof is based on a version of the Rosenthal inequality for martingales given in [START_REF] Merlevède | Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples[END_REF]. Note that Rio's inequality cannot be applied to GPM maps, because the associated Markov chain is not α-mixing in the sense of Rosenblatt.

Remark 3.10. Let r ≥ 1 and p > 2. Starting again from (1.5) and following the proof of Proposition 3.5, we obtain the upper bound

W r r (µn, µ) p p 1 n p/2 ∞ 0 t r-1 Sα,n(t)dt p + 1 n p-1 1 0 α -1 2,X (u) ∧ n p-1 Q rp (u)du . (3.43)
Remark 3.11. Inequality (3.42) implies in particular that if p > 2 and

1 0 α -1 2,X (u) p/2 Q p (u)du < ∞ , (3.44) then W1(µn, µ) p 1 √ n .
Remark 3.12. In the m-dependent case, Inequality (3.42) becomes

W1(µn, µ) p p 1 n p/2 ∞ 0 H(t)dt p + 1 n p-1 X0 p p .
This inequality seems to be new even in the i.i.d. case. Compared to the usual Rosenthal bound for sums of i.i.d. random variables, the variance term is replaced by the integral involving H, in accordance with the upper bound (3.8).

Example (continued). We continue the example of Subsection 3.1.

1. Let p > 2, and let g be positive and non increasing on (0, 1), with g(x) ≤ C x b near 0, for some C > 0 and b ∈ [0, (1 -γ)/p).

Applying Proposition 3.5, the following upper bounds hold.

For γ ∈ (0, 1/2)

W1(μn, µ)) p n -1/2 if b ≤ (2 -γ(p + 2))/2p n (pb+γ-1)/pγ if b > (2 -γ(p + 2))/2p. For γ ∈ [1/2, 1), W1(μn, µ)) p n (pb+γ-1)/pγ .
2. Let p ∈ (0, 1), and let g be positive and non decreasing on (0, 1), with

g(x) ≤ C (1 -x) b near 1, for some C > 0 and b ∈ [0, 1/p).
Applying Proposition 3.5, the following upper bounds hold. For γ ∈ (0, 1/2)

W1(μn, µ)) p n -1/2 if b ≤ (2 -γ(p + 2))/2p(1 -γ) n (γ-1)(1-pb)/pγ if b > (2 -γ(p + 2))/2p(1 -γ).
For γ ∈ [1/2, 1), W1(μn, µ)) p n (γ-1)(1-pb)/pγ . Remark 3.13. In the case where θ is the LSV map defined by (2.3) and g is the identity (which is a particular case of Item 2, b = 0, of the example above) all the rates for W1(μn, µ)) p given in Subsections 3.2, 3.3 and 3.4 have been obtained in Corollary 4.1 of [START_REF] Dedecker | Moment bounds for dependent sequences in smooth Banach spaces[END_REF] by using a different approach. Moreover, all the bounds are optimal in that case (see the discussion in Section 4.2 of [START_REF] Dedecker | Moment bounds for dependent sequences in smooth Banach spaces[END_REF]).

Proof of Proposition 3.5. Inequality (3.42) follows from Proposition 3.6 below.

Proposition 3.6. There exists a positive universal constant c such that, for any positive integer n, any x > 0, any η > 2 and any β ∈ (η -2, η), the following inequality holds:

P (nW1(µn, µ) ≥ x) ≤ c n η/2 x η s η α,n + n x 1+β/2 R -1 n (x) 0 R β/2 n (u)Q(u)du + c n x 1+η/2 1 R -1 n (x) R η/2 n (u)Q(u)du , (3.45)
where

Rn(u) = (min{q ∈ N * : α 2,X (q) ≤ u} ∧ n) Q(u) and R -1 n (x) = inf {u ∈ [0, 1] : Rn(u) ≤ x} . Indeed, nW1(µn, µ) p p = p ∞ 0 x p-1 P (nW1(µn, µ) ≥ x) dx n p/2 s p α,n + ∞ n 1/2 sα,n x p-1 P (nW1(µn, µ) ≥ x) dx . (3.46)
To handle the second term on the right-hand side, we apply (3.45) with η ∈ (2p -2, 2p) and β ∈ (η -2, 2p -2). This gives

∞ n 1/2 sα,n x p-1 P (nW1(µn, µ) ≥ x) dx n η/2 s η α,n ∞ n 1/2 sα,n x p-η-1 dx + n 1 0 R β/2 n (u)Q(u) ∞ 0 x p-β/2-2 1 u<R -1 n (x) du + n 1 0 R η/2 n (u)Q(u) ∞ 0 x p-η/2-2 1 u≥R -1 n (x) du .
Since u < R -1 n (x) ⇐⇒ x < Rn(u), the choice of η and β implies that, for any p > 2,

∞ n 1/2 sα,n x p-1 P (nW1(µn, µ) ≥ x) dx n p/2 s p α,n + n 1 0 R p-1 n (u)Q(u)du ,
which together with (3.46) give (3.42).

To complete the proof of Proposition 3.5, it remains to prove Proposition 3.6. With this aim, we proceed as for the proof of Proposition 3.4 with the following modification: in the definition of Rn (and then also of v defined in (3.27)), α 1,X is replaced by α 2,X , and in the definition of q given in (3.30), α 1,X is also replaced by α 2,X . Assuming first that q = n, we first notice, by following the proof of Proposition 3.4, that the bound (3.34) is still valid. In addition since

u < R -1 n (x) ⇐⇒ x < Rn(u), R -1 n (x) 0 Q(u)du ≤ x -β/2 R -1 n (x) 0 R β/2 n (u)Q(u)du , (3.47)
which combined with (3.34) proves the proposition in case where q = n.

From now on, we assume that q < n (therefore α 2,X (q) ≤ v). The bound (3.35) is still valid and combined with (3.47) gives

P (nW1(µn, µ) ≥ 6x) ≤ P sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) ≥ 5x + 2n x 1+β/2 R -1 n (x) 0 R β/2 n (u)Q(u)du . (3.48)
As in the proof of Proposition 3.4, the first term on the right-hand side can be handled with the help of the decomposition (3.37). Clearly since α 1,X (q) ≤ α 2,X (q) ≤ v, the term I1(n) in (3.37) satisfies the inequality (3.38). Therefore taking into account (3.47), it follows that

I1(n) ≤ 2n x 1+β/2 R -1 n (x) 0 R β/2 n (u)Q(u)du . ( 3

.49)

We handle now the term I2(n) in the decomposition (3.37). Using again Markov's inequality, we get that for any η > 2,

I2(n) ≤ 1 x η   Q(v) -Q(v) max 2≤2j≤[n/q] j i=1 Ũ2i(t) η dt   η .
Note that ( Ũ2i(t))i∈Z (resp. ( Ũ2i-1(t))i∈Z) is a stationary sequence of martingale differences with respect to the filtration (G2i) i∈Z (resp. (G2i-1) i∈Z ). By using the Rosenthal inequality of Merlevède and Peligrad [START_REF] Merlevède | Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples[END_REF] for martingales (see their Theorem 6), we get max

2≤2j≤[n/q] j i=1 Ũ2i(t) η (n/q) 1/η Ũ2(t) η + (n/q) 1/η    [n/q] k=1 1 k 1+2δ/η E0   k i=1 Ũ2i(t) 2   δ η/2    1/(2δ)
, where δ = min 1, (η -2) -1 . Since ( Ũ2i(t))i∈Z is a stationary sequence of martingale differences with respect to the filtration (G2i) i∈Z ,

E0   k i=1 Ũ2i(t) 2   = k i=1 E0 Ũ 2 2i (t) . Moreover E0 Ũ 2 2i (t) ≤ E0 U 2 2i (t) . Therefore E0   k i=1 Ũ2i(t) 2   r/2 ≤ k i=1 E0 U 2 2i (t) -E U 2 2i (t) r/2 + k i=1 E U 2 2i (t) .
By stationarity

k i=1 E U 2 2i (t) = k S q (t) 2 2 ,
where

S q (t) = q i=1 1 X i ≤t -E 1 X i ≤t .
It follows that max

2≤2j≤[n/q] j i=1 Ũ2i(t) r (n/q) 1/η S q (t) η + (n/q) 1/2 S q (t) 2 + (n/q) 1/η   [n/q] k=1 1 k 1+2δ/r D δ k,q (t)   1/(2δ)
, where

D k,q (t) = k i=1 E0 U 2 2i (t) -E U 2 2i (t) η/2 .
We have

D k,q (t) ≤q 2 k i=1 sup j≥ ≥(i-1)q+1 sup t∈R E0 (1 X ≤t -E(1 X ≤t ))(1 X j ≤t -E(1 X j ≤t )) -E (1 X ≤t -E(1 X ≤t ))(1 X j ≤t -E(1 X j ≤t )) η/2 ≤q 2 k i=1 sup j≥ ≥(i-1)q+1 sup t∈R E0 (1 X ≤t -E(1 X ≤t ))(1 X j ≤t -E(1 X j ≤t )) -E (1 X ≤t -E(1 X ≤t ))(1 X j ≤t -E(1 X j ≤t )) η/2 ≤q 2 k i=1 α 2/η 2,X (iq + 1) ,
where we have used the fact that gM is nondecreasing for the second inequality. Since β < η, Hölder's inequality gives

D k,q (t) q 2 k (η-β)/η k i=1 i β/2-1 α 2,X (iq + 1) 2/η . Therefore, since β > η -2,    Q(v) -Q(v) n 1/η q 1/η   [n/q] k=1 1 k 1+2δ/r D k,q (t) δ   1/(2δ) dt    η nq η-1 Q η (v)   [n/q] k=1 k δ(η-β)/η k 1+2δ/η   η/(2δ) [n/q] i=1 i β/2-1 α 2,X (iq + 1) nq η-1 Q η (v) [n/q] i=1 i β/2-1 α 2,X (iq + 1) . Note that since y < α -1 2,X (u) ⇐⇒ α 2,X (y) > u and α 2,X (q) ≤ v, [n/q] i=1 i β/2-1 α 2,X (iq + 1) = [n/q] i=1 i β/2-1 1 0 1 u<α 2,X (iq+1)
≤ v 0 [n/q] i=1 i β/2-1 1 i≤q -1 α -1 2,X (u) ≤ q -β/2 v 0 α -1 2,X (u) ∧ n β/2 du . Hence    Q(v) -Q(v) n 1/η q 1/η   [n/q] k=1 1 k 1+2δ/r D k,q (t) δ   1/(2δ) dt    η nq η-1-β/2 Q η (v) v 0 α -1 2,X (u) ∧ n β/2 du .
Using (3.31) and the fact that u < v ⇐⇒ Q(v) < Q(u), we infer that

   Q(v) -Q(v) n 1/η q 1/η   [n/q] k=1 1 k 1+2δ/r D k,q (t) δ   1/(2δ) dt    η nx η-β/2-1 v 0 R β/2 n (u)Q(u)du . (3.50)
On another hand, since

S q (t) 2 2 ≤ 2q q-1 k=0 E (1 g M (X 0 )≤t -E(1 g M (X 0 )≤t ))(1 g M (X k )≤t -E(1 g M (X k )≤t )) ,
proceeding as to bound I2(n) in the proof of Proposition 3.4, we infer that

Q(v) -Q(v) n 1/2 q 1/2 S q (t) 2 dt η n η/2 s η α,n . (3.51)
We prove now that

Q(v) -Q(v) n 1/η q 1/η S q (t) η dt η nx η-β/2-1 v 0 R β/2 n (u)Q(u)du + nx η/2-1 1 v R η/2 n (u)Q(u)du . (3.52) 20
With this aim, assume first that we can prove that

Q(v) -Q(v) S q (t) η dt q 1/2 Q(v) 0 H(t) 0 α -1 2,X (u) ∧ q η/2 du 1/η dt , (3.53)
then

Q(v) -Q(v) n 1/η q 1/η S q (t) η dt η A(n) + B(n) ,
where

A(n) = nvq η-1 Q η (v) and B(n) = nq η/2-1   Q(v) 0
H(t) v α -1 2,X (u) ∧ q η/2 du 1/η dt   η dx . Using (3.31), the fact that u < v ⇐⇒ Q(v) < Q(u) and that u < R -1 n (x) = v ⇐⇒ x < Rn(u), we successively derive A(n) nx η-1 vQ(v) nx η-1 v 0 Q(u)dx nx η-β/2-1 v 0 R β/2 n (u)Q(u)du . (3.54)
On the other hand, since u < H(t) ⇐⇒ t < Q(u), we have

B(n) ≤ nq η/2-1   Q(v) 0 1 t 1/2+1/η H(t) v α -1 2,X (u) ∧ q r/2 Q η/2+1 (u)du 1/η dt   η dx n(qQ(v)) η/2-1 1 v α -1 2,X (u) ∧ n η/2 Q η/2+1 (u)du .
Using (3.31), it follows that

B(n) nx η/2-1 1 v R η/2 n (u)Q(u)du .
This last upper bound together with (3.54) show that to prove (3.52) it suffices to prove (3.53). To prove this moment inequality, we use Corollary 2 in [START_REF] Dedecker | A new covariance inequality and applications[END_REF]. Since, for any t ∈ R,

|1 X 0 ≤t -E(1 X 0 ≤t )| ≤ 1, this gives S q (t) η ≤ 2qη Y (t) 1 0 γ -1 (u) ∧ q η/2 du 1/η , where Y (t) = 1 g M (X 0 )≤t -E(1 g M (X 0 )≤t ) and γ -1 (u) = ∞ k=0 1 u≤γ(k) with γ(k) = E0 1 g M (X k )≤t -E 1 g M (X k )≤t 1 . Since gM is nondecreasing γ(k) ≤ α 1,X (k) ≤ α 2,X (k) in such a way that γ -1 (u) ≤ α -1 2,X (u)
. Moreover, for any t ∈ R,

Y (t) 1 = 2P(gM (X0) ≤ t)P(gM (X0) > t) ≤ 2 min {P(|X0| ≥ -t), P(|X0| > t)} .
All these considerations end the proof of (3.53).

So, overall, we get

I2(n) x -η n η/2 s η α,n + nx -β/2-1 v 0 R β/2 n (u)Q(u)du + nx -η/2-1 1 v R η/2 n (u)Q(u)du .
With similar arguments, we can prove that

I3(n) x -η n η/2 s η α,n + nx -β/2-1 v 0 R β/2 n (u)Q(u)du + nx -η/2-1 1 v R η/2 n (u)Q(u)du .
Therefore starting from (3.37) and taking into account (3.49), (3.4) and (3.4), it follows that

P sup f ∈Λ 1 n i=1 f (X i ) -E(f (X i )) ≥ 5x x -η n η/2 s η α,n + nx -β/2-1 v 0 R β/2 n (u)Q(u)du + nx -η/2-1 1 v R η/2 n (u)Q(u)du ,
which combined with (3.48) ends the proof of Proposition 3.6.

4 Weak convergence of partial sums in L 1 (m)

Let (S, S, m) be a σ-finite measure space such that L 1 (S, S, m) is separable. In what follows, we shall denote by L 1 (m) the space L 1 (S, S, m).

We use the notations of Section 2. Let Y0 = {Y0(t), t ∈ S} be a random variable with values in L 1 (m), such that

Y0(t) 1 m(dt) < ∞ and Y0(t) m(dt) = 0 .
Define the stationary sequence Y = (Yi) i∈Z by Yi = Y0 • T i , and let

Sn = n k=1 Y k .

Previous results

If Y is a sequence of i.i.d. random variables, Jain [START_REF] Jain | Central limit theorem and related questions in Banach space[END_REF] proved that n -1/2 Sn satisfies the CLT (i.e. converges in distribution to an L 1 (m)-valued Gaussian random variable) if and only if

Y0(t) 2 m(dt) < ∞. (4.1)
Using a general result by de Acosta, Araujo and Giné [START_REF] De Acosta | On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces[END_REF], Dédé [START_REF] Dédé | An empirical central limit theorem in L 1 for stationary sequences[END_REF] proved that the CLT remains valid under (4.1) for stationary and ergodic martingale differences (meaning that E(Y1|F0) = 0 almost surely). Starting from a martingale approximation, she proved then that, if Y is ergodic, the CLT holds as soon as (4.1) holds and

k∈Z P0(Y k (t)) 2 m(dt) < ∞, (4.2)
where

P0(Y k (t)) = E(Y k (t)|F0) -E(Y k (t)|F-1).
In a recent paper, Cuny [START_REF] Cuny | Limit theorems under the Maxwell-Woodroofe condition in Banach spaces[END_REF] has given many new results concerning the behavior of partial sums of dependent sequences in Banach spaces of cotype 2. Among these results, he showed that, if Y is ergodic, Y0 is F0-measurable, (4.1) holds and n>0

E(Sn|F0) 2 n 3/2 m(dt) < ∞ , (4.3)
then the CLT and the weak invariance principle (WIP) hold. By WIP, we mean that the partial sum process {n -1/2 S [nt] , t ∈ [0, 1]} converges in distribution to an L 1 (m)-valued Wiener process in the space D L 1 (m) ([0, 1]) of L 1 (m)-valued càdlàg functions equipped with the uniform metric. As usual, an L 1 (m)-valued Wiener process with covariance Λ is a centered Gaussian process

W = {W (t), t ∈ [0, 1]} such that E( W (t) 2 L 1 (m)
< ∞ for all t ∈ [0, 1] and, for all f, g in L ∞ (m), Cov f (u)Wt(u) m(du), g(u)Ws(u) m(du) = min{s, t}Λ(f, g) (as usual, we identify a function f in L ∞ (m) with an element of the dual of L 1 (m)). Note that Cuny [START_REF] Cuny | Limit theorems under the Maxwell-Woodroofe condition in Banach spaces[END_REF] also proved that the WIP holds under (4.2), and that the almost sure invariance principle with rate o( √ n ln ln n) is true if either (4.2) of (4.3) holds. The condition (4.2) is the L 1 (m) version of Hannan's criterion [START_REF] Hannan | Central limit theorems for time series regression[END_REF], and the condition (4.3) is the L 1 (m) version of Maxwell-Woodroofe's criterion criterion [START_REF] Maxwell | Central limit theorems for additive functionals of Markov chains[END_REF]. If Y0 is F0-measurable, both criteria hold as soon as

∞ k=0 1 k + 1 E(Y k (t)|F0) 2 m(dt) < ∞ . (4.4)
As shown in [START_REF] Cuny | Limit theorems under the Maxwell-Woodroofe condition in Banach spaces[END_REF], if either (4.2) or (4.3) holds, there exists a stationary and ergodic sequence of martingale differences (Di) i∈Z with values in L 1 (m), such that, setting

Mn = n k=1 D k , max 1≤k≤n |S k (t) -M k (t)| m(dt) 2 = o( √ n) .
In the next subsections, we shall rather look for a martingale approximation in L 1 , in the spirit of Gordin [START_REF] Gordin | Abstracts of Communication, T.1:A-K, International Conference on Probability Theory[END_REF]. Our criterion will not be directly comparable to either (4.2) or (4.3), but its application to the empirical distribution function of α-dependent sequences will lead to weaker conditions (see Section 5 for a deeper discussion).

4.2 A central limit theorem in L 1 (m) for non-adapted sequences

In this section, we give an extension of Gordin's criterion [START_REF] Gordin | Abstracts of Communication, T.1:A-K, International Conference on Probability Theory[END_REF] for the central limit theorem to L 1 (m)-valued random variables. Theorem 4.1. Assume that, for m almost every t, the series

U (t) = ∞ k=1 E0(Y k (t)) and V (t) = - 0 k=-∞ Y k (t) -E0(Y k (t)) (4.5)
converge in probability, and let

D0(t) = k∈Z E0(Y k (t)) -E-1(Y k (t))
and Mn(t) For such a f and any (s1, . . . , s d ) in [0, 1] d , we get that

= n k=1 D0(t) • T k . If U (t) + V (t) 1 m(dt) < ∞ , (4.6
D0(t) = k∈Z E0(Y k (t)) -E-1(Y k (t)) and Z(t) • T = ∞ k=1 E0(Y k (t)) - 0 k=-∞ Y k (t) -E0(Y k (t)) . Now, if Z(t)
E f S [ns 1 ] √ n , . . . , S [ns d ] √ n -E f M [ns 1 ] √ n , . . . , M [ns d ] √ n ≤ d i=1 ci S [ns i ] (t) √ n - M [ns i ] (t) √ n 1 m(dt) ,
and it follows from (4.7) that

lim n→∞ E f S [ns 1 ] √ n , . . . , S [ns d ] √ n -E f M [ns 1 ] √ n , . . . , M [ns d ] √ n = 0 . (4.15)
Now, when (4.9) holds, Cuny [START_REF] Cuny | Limit theorems under the Maxwell-Woodroofe condition in Banach spaces[END_REF] proved that the process {n -

(m) ([0, 1]), • ∞) to R such that |ψ(x)| ≤ C(1 + x 2 ∞ ) for some positive constant C. Then lim n→∞ E ψ Sn √ n = E((ψ(W )) .
In particular Proof of Lemma 4.1. We first note that, for any positive random variable V ,

lim n→∞ E(Tn) = E max t∈[0,1] |Wt(s)| m(ds)
E max 1≤k≤n |S k (t)| m(dt) 2 V ≤ E √ V max 1≤k≤n |S k (t)| m(dt) 2 ≤ √ V max 1≤k≤n |S k (t)| 2 m(dt) 2 .
Taking V = 1T n >M , we obtain that

E (Tn1T n>M ) ≤ 1 n max 1≤k≤n |S k (t)| 1T n>M 2 m(dt) 2 (4.21)
Applying Inequality (3.12) in [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF] with λ = 0, we get that

max 1≤k≤n |S k (t)| 2 2 ≤ 16 n k=1 Y k (t) n i=k E k (Yi(t)) 1 ≤ 16nL(t) . ( 4

An invariance principle in L 1 (m) for the empirical distribution function

In this subsection, S = R, and m is a σ-finite measure on R equipped with the Borel σ-field.

As in Section 2, let X0 be an F0-measurable and integrable real-valued random variable with distribution function F . Define the stationary sequence X = (Xi) i∈Z by Xi = X0 • T i , and denote by

F X k |F 0 the conditional distribution function of X k given F0. The random variable Y k is then defined by Y k (t) = 1 X k ≤t -F (t), in such a way that Sn = n k=1 Y k = n(Fn -F ) ,
where Fn is the empirical distribution function of {X1, . . . , Xn}. Note that Y0 is a L 1 (m)-valued random variable as soon as E(|X0|) < ∞. When applied to α-dependent sequences as defined in Section 2, Theorem 4.4 yields the following result. (3.19). It remains to prove that the covariance operator Λ given in (4.10) can be expressed as in (4.27). As usual, we identify a function f in L ∞ (m) with an element of the dual of L 1 (m), and we write

f (Y k ) = f (t)Y k (t) m(dt).

Quantile conditions

As a consequence of the results by Dédé [START_REF] Dédé | An empirical central limit theorem in L 1 for stationary sequences[END_REF] or Cuny [START_REF] Cuny | Limit theorems under the Maxwell-Woodroofe condition in Banach spaces[END_REF] Hence, in that case, the condition (4.26) is weaker than the condition (5.2), and is in fact equivalent to the minimal condition to get the central limit theorem for partial sums of stationary α-dependent sequences of bounded random variables.

We shall now focus on the the case where m = λ is the Lebesgue measure on R. In that case, the condition (5.2) is equivalent to

∞ k=0 1 √ k + 1 ∞ 0 min{α 1,X (k), H(t)} dt . (5.5)
and the condition (4.26) is equivalent to (3.1). We shall see that the condition (3.1) is always weaker than the condition (5.5). The first step is to express (3.1) and (5.5) in terms of the quantile function of X0, as done in [START_REF] Doukhan | The functional central limit theorem for strongly mixing processes[END_REF] for the invariance principle of stationary α-mixing sequences. More precisely, we shall compare the three following conditions:

1 0 α -1 (u)Q 2 (u)du < ∞ , (5.6
)

1 0 α -1 (u)Q(u) u 0 α -1 (x) dx du < ∞ , (5.7
)

1 0 α -1 (u)Q(u) √ u du < ∞ , (5.8)
where for simplicity we denote by α -1 the function α -1 1,X defined in (3.21). The condition (5.6) has been introduced by Doukhan, Massart and Rio [START_REF] Doukhan | The functional central limit theorem for strongly mixing processes[END_REF], but in that paper the function α -1 is defined with the α-mixing coefficients of Rosenblatt [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]. These authors showed that (5.6) implies the functional central limit theorem for the Donsker line

   1 √ n [nt] k=1 X k -E(X k )), t ∈ [0, 1]    ,
and that it is optimal in a precise sense. The optimality of this condition has been further discussed in a paper by Bradley [START_REF] Bradley | On quantiles and the central limi question for strongly mixing sequences[END_REF]. The fact that, for ergodic sequences, this functional central

Figure 1 :

 1 Figure 1: The graph of a GPM map, with d = 4

  near 1, for some C > 0 and b > 1, then (3.4) holds.

. 5 )

 5 is sufficient. For the two simple examples above, this would give the constraint b > 1/2 instead of b > 1.

. 23 ) 3 . 7 .

 2337 Remark In the m-dependent case, Inequality (3.23) becomes W1(µn, µ) p p 1 n p-1 X0 p p . (3.24)

  t) -yi(t)| m(dt) .

Theorem 4 . 4 .F

 44 Assume that ∞ k=0 X k |F 0 (t) -F (t) 1 m(dt) < ∞ . (4.25) Then {n -1/2 S [ns] , s ∈ [0, 1]} converges in distribution in the space D L 1 (m) ([0, 1]) to an L1(m)valued Wiener process W . Moreover the explicit form of the covariance operator of W is obtained via equation (4.10) of Theorem 4.1 by taking Y k (t) = 1 X k ≤t -F (t).

Proposition 4 . 2 .

 42 Let B(t) = F (t)(1 -F (t)). The condition ∞ k=0 min{α 1,X (k), B(t)} m(dt) < ∞ (4.26)implies the condition (4.25), and hence the conclusion of Theorem 4.4. Moreover, the covariance operator Λ of W can be expressed as follows: for any f, g in L∞(m),Λ(f, g) = k∈Z E f (t)g(s)(1 X 0 ≤t -F (t))(1 X k ≤s -F (s)) m(dt)m(ds) .

  (

By Remark 4 . 3 ,( 1 X 2 ,

 4312 we know that, for any f in L ∞ (m),lim n→∞ 1 n E (f (Sn)) 2 = E (f (W1)) 2 = Λ(f, f ) . (4.28)Now, if we can prove that, for any f, g in L ∞ (m),k∈Z |Cov(f (Y0), g(Y k ))| < ∞ , (4.29) then the series Λ(f, g) = k∈Z Cov(f (Y0), g(Y k ))is well defined, andlim n→∞ 1 n E (f (Sn)) 2 = Λ(f, f ) . (4.30)From (4.28) and (4.30), we infer that, for anyf in L ∞ (m), Λ(f, f ) = Λ(f, f ).Applying this equality to f , g, and f + g it follows that, for any f, g in L ∞ (m),Λ(f, g) = Λ(f, g) ,which is the desired result. To prove (4.29), we first note that|Cov(f (Y0), g(Y k ))| ≤ f ∞ g ∞ (1 X 0 ≤t -F (t))E0(1 X k ≤s -F (s)) 1 m(dt)m(ds) . Now (1 X 0 ≤t -F (t))E0(1 X k ≤s -F (s)) 1 ≤ min{α 1,X (k), 2B(t), 2B(s)} .Hencek=0 0 ≤t -F (t))E0(1 X k ≤s -F (s)) 1 ≤ 2 ∞ k=0 min{α 1,X (k), B(t)} ∞ k=0 min{α 1,X (k), B(s)} . This implies that k∈Z |Cov(f (Y0), g(Y k ))| ≤ 2 f ∞ g ∞and (4.29) follows from (4.26). This completes the proof of Proposition 4.2.

  )and for any (s1, . . . , s d ) in [0, 1] d , the random vector n -1/2 (S [ns 1 ] , . . . , S [ns d ] ) t converges in distribution in (L 1 (m)) d to the Gaussian random vector (Ws 1 , . . . , Ws d ), where W is the L1(m)-valued Wiener process W with covariance operator Λ defined by: for any f, g in L∞(m),

	Proposition 4.1. Assume that, for m almost every t,
		Y0(t) = D0(t) + Z(t) -Z(t) • T ,	(4.11)
	where D0(t) is an integrable random variable such that E(D0(t)|F-1) = 0 almost surely. Let then
	Mn(t) = n k=1 D0(t) • T k . If
			Z(t) 1 m(dt) < ∞ ,	(4.12)
	then (4.7) holds. If moreover (4.9) holds, then the conclusion of Theorem 4.1 holds.
	Before proving Proposition 4.1, let us continue the proof of Theorem 4.1. Note first that, if
	(4.5) is satisfied, then (4.11) holds, with
	then
	lim n→∞	Sn(t) √ n	-	Mn(t) √ n 1	m(dt) = 0 .	(4.7)
	If moreover, for m almost every t,
	C(t) = lim inf n→∞	1 √ n	E(|Sn(t)|) < ∞ and	C(t) m(dt) < ∞ ,	(4.8)
	then
			D0(t) 2 m(dt) < ∞ ,	(4.9)
	Λ(f, g) = E	f (t)g(s)D0(t)D0(s) m(dt)m(ds) .	(4.10)
	Proof of Theorem 4.1. We first state the following intermediate result:

  is defined as above, the conditions (4.6) and (4.12) are the same. Hence, it follows from Proposition 4.1 that (4.7) holds as soon as (4.12) is satified. The second part of Theorem 4.1 will follow from Proposition 4.1 if we prove that (4.8) implies (4.9). By (4.6) it follows that be a positive integer, and let f be a separately Lipschitz function from (L 1 (m)) d to R. This means that there exists non-negative constants c1, . . . , c d such that |f (x1, . . . , x d ) -f (y1, . . . , y d )| ≤

	lim n→∞	Z(t) 1 √ n	= 0 for m-almost every t.	(4.13)
	Since Sn(t) = Mn(t) + Z(t) -Z(t) • T n , we infer from (4.13) that, for m almost every t,
	lim inf n→∞	Mn(t) 1 √ n	= lim inf n→∞	Sn(t) 1 √ n	.	(4.14)
	From (4.14) and (4.8), it follows that, for m almost every t,
		C(t) = lim inf n→∞	Mn(t) 1 √ n	< ∞ .
	Now, applying Theorem 1 and Remark 1.1 in Esseen and Janson [14], we deduce that, for m
	almost every t,
				D0(t) 2 =	π 2	C(t) ,
	so that (4.8) implies (4.9). This completes the proof of Theorem 4.1.
	Proof of Proposition 4.1. Since Sn(t) = Mn(t) + Z(t) -Z(t) • T n , it follows that
	Sn(t) √ n	-	Mn(t) √ n 1	m(dt) ≤	2 √ n	Z(t) 1 m(dt) ,
	and (4.7) follows from (4.12).
	Now, let d

  1/2 M [nt] , t ∈ [0, 1]} converges in distribution in the space D L 1 (m) ([0, 1]) to an L1(m)-valued Wiener process W , with covariance operator Λ given by (4.10). Together with (4.15), this completes the proof of Proposition 4.1.4.3 An invariance principle in L 1 (m) for adapted sequencesIn this subsection, we assume that the random variable Y0 is F0-measurable. Theorem 4.2. Assume that, for m-almost every t, the series U (t) defined in (4.5) converges in probability. Assume also that, for m-almost every t, the series

		n
			Y0(t)E0(Y k (t))		(4.16)
		k=0
	converge in L 1 , and let
			n
	L(t) = sup	Y0(t)E0(Y k (t))	.	(4.17)
		n≥0	k=0	1
	If moreover U (t) As an immediate consequence of Theorem 4.2, the following corollary holds:
	Corollary 4.1. Assume that
	max{1, |Y0(t)|}|E0(Y k (t))| 1 m(dt) < ∞ .	(4.19)
	k≥0
	Then the conclusion of Theorem 4.2 holds.
	Remark 4.3. Under the assumptions of Theorem 4.2, we shall prove that the sequence
	Tn =	1 n	max

1 m(dt) < ∞ and

L(t) m(dt) < ∞ ,

(4.18

)

then {n -1/2 S [nt] , t ∈ [0, 1]} converges in distribution in the space D L 1 (m) ([0, 1]) to an L1(m)-

valued Wiener process W , with covariance operator Λ defined by (4.10). 1≤k≤n |S k (t)|m(dt) 2 (4.20) is uniformly integrable (see Lemma 4.1 below). By standard arguments, this implies the following extension of Theorem 4.2: let ψ be any continuous function from (D L 1

  Proof of Theorem 4.2. Note first that, in this adapted case, all the conditions of Theorem 4.1 are satisfied. Indeed, since for m almost every t the series (4.16) converge in L 1 , it follows that the series ∞ k=0 Cov(Y0(t), Y k (t)) converge, and then

	lim n→∞	Sn(t) 2 2 n	= Var(Y0(t)) + 2	∞ k=1	Cov(Y0(t), Y k (t)) .
	Now, by definition of L(t),
						∞
		Var(Y0(t)) + 2	Cov(Y0(t), Y k (t)) ≤ 2L(t) .
						k=1
	Hence the condition (4.8) follows from (4.18) and the fact that
							∞
	C(t) ≤ Var(Y0(t)) + 2		Cov(Y0(t), Y k (t)) ≤ 2L(t) .
							k=1
	So, the conclusion of Theorem 4.1 holds with the the covariance function defined by (4.10).
	As usual it remains to prove the tightness, which reduces through Ascoli's theorem to: for any
	ε > 0,
	lim δ→0	lim sup n→∞	1 δ	P	max 1≤k≤[nδ]	|S k (t)| m(dt) >	√ nε = 0.
	But this follows straightforwardly from Lemma 4.1 below by applying Markov inequality at order
	2. The proof of Theorem 4.2 is complete.

2

. Lemma 4.1. Assume that, for m-almost every t, the series defined in (4.16) converges in L 1 . Assume moreover that the function L defined in (4.17) satisfies

(4.18)

. Then the sequence (Tn) n≥1 defined in (4.20) is uniformly integrable.

  It remains to prove(4.23). In fact this follows quite easily from Proposition 1 in[START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF]. Indeed, since for m-almost every t, the series defined in (4.16) converges in L 1 , it follows from this

	proposition that the sequence
				n 1	max 1≤k≤n	|S k (t)|	2
	is uniformly integrable for m-almost every t. Hence (4.23) holds as soon as
		lim M →∞	n→∞ lim sup	P(Tn > M ) = 0 .	(4.24)
	Now, applying (4.21) and (4.22),
	P(Tn > M ) ≤		E(Tn) M	≤	16 M	L(t) m(dt)	2	,
	and (4.24) follows. This completes the proof of Lemma 4.1.
									.22)
	Using (4.18), (4.21), (4.22) and the reverse Fatou Lemma, we infer that
		lim M →∞	n→∞ lim sup	E (Tn1T n>M ) = 0
	as soon as, for m-almost every t,
	lim M →∞	lim sup n→∞	1 √ n	max 1≤k≤n	|S k (t)| 1T n >M	2	= 0 .	(4.23)

  4.27) Proof of Theorem 4.4 and of Proposition 4.2. Theorem 4.4 is a direct consequence of Corollary 4.1 applied to the random variables Y k (t) = 1 X k ≤t -F (t). More precisely, since |Y0(t)| ≤ 1, the criterion (4.19) is exactly the criterion (4.25). It remains to prove Proposition 4.2. We first quote that condition (4.26) implies (4.25): this follows easily from the two upper bounds given in

  (see the condition (4.4) of Subsection 4.1) we know that the conclusion of Theorem 4.4 holds as soon asF X k |F 0 (t) -F (t) 2 m(dt) < ∞ . (5.1) Moreover, it follows from [6] that the condition (5.1) also implies the strong invariance principle. Let B(t) = F (t)(1 -F (t)). As quoted by Dédé (2009), the condition (5.1) is implied by The conditions (4.26) of Proposition 4.2 and the condition (5.2) are not easy to compare. However, if either m has finite mass or X0 is bounded, then (4.26) is equivalent to

	∞ k=0	1 k + 1 √
	∞ k=0	√	1 k + 1	min{α 1,X (k), B(t)} m(dt) .	(5.2)
				∞
					α 1,X (k) < ∞	(5.3)
				k=1
	and (5.2) is equivalent to
				∞ k=1	α 1,X (k) k	< ∞ .	(5.4)

limit theorem remains true with the much weaker coefficients α 1,X (k) is a consequence of a result by Dedecker and Rio [START_REF] Dedecker | On the functional central limit theorem for stationary processes[END_REF].

Concerning these three quantile conditions, our first result is Proposition 5.1 below.

Proposition 5.1. The following equivalences hold 1. The condition (5.6) is equivalent to

min{α 1,X (k), H(t)} dt < ∞ .

(5.9)

2. The condition (5.7) is equivalent to (3.1).

3. The condition (5.8) is equivalent to (5.5).

The hierarchy of these quantile conditions is given in Proposition 5.2 below.

Proposition 5.2. The following implications hold: (5.8) ⇒ (5.7) ⇒ (5.6).

Remark 5.1. At this point, it should be noticed that these three conditions are in fact very close. Indeed, by a simple application of Cauchy-Schwarz inequality, for any b > 1/2, (5.6)

and the condition on right hand is a slight reinforcement of (5.8).

Proof of Proposition 5.1. Assume that k≥0 α 1,X (k) < ∞. Then the function S defined on

is finite and non-increasing.

Proof of Item 1. By a simple change of variables, we see that the condition (5.9) is equivalent to

it follows that

which concludes the proof of Item 1.

Proof of Item 2. Starting from (5.11), it follows that

From (5.12), we infer that

Making the change of variables u = G -1 α (v), the result follows.

). Hence, there exists two positive constants A and B such that

Making the change of variables v = u 2 , the result follows.

Proof of Proposition 5.2. Since the function α -1 is non-increasing, one has

which proves that (5.8) implies (5.7).

It remains to prove that (5.7) implies (5.6). By Proposition 5.1, it is equivalent to prove that (3.1) implies (5.9). If (3.1) holds, then the function S defined on R + by (5.10) is finite and non-increasing. Hence, using again (3.1),

Consequently tS(t) ≤ C S(t), proving that (3.1) implies (5.9).

Sufficient conditions

In this subsection, we give some simple conditions on α 1,X (k) and H under which (5.7) (and hence (3.1)) is satisfied.

Proposition 5.3. The following conditions imply (5.7):

1.

4.

5.

∞ 0

Proof of Proposition 5.3. Proof of Item 1. Since (5.8) implies (5.7), it suffices to prove that Item 1 implies (5.8). Applying Cauchy Schwarz, we obtain that

.

Since E(|X0| p ) < ∞, the first integral on right hand is finite. It remains to prove that

The last condition means exactly that

< ∞ , which is equivalent to the condition of Item 1.

Proof of Item 2. Again, it suffices to prove that Item 2 implies (5.8). Now, the condition

). Hence, the condition (5.8) holds as soon as

By definition of α -1 , the last condition means exactly that

< ∞ , which is equivalent to the condition of Item 2. Proofs of Item 3, 4 and 5. For the proof of these points, we start from condition (3.1) which is equivalent to (5.7). Since we can control the behavior of α 1,X (k), we can give upper bounds for the function S defined by (5.10).