JérômeDedecker FlorenceMerlevède EmmanuelRio Ratesofconvergenceforminimaldistancesinthecentrallimittheoremunderprojectivecriteria

El e c t ro nic

Journ a l of

Pr

ob a b il i t y

Vol. 14 (2009), Paper no. 35, pages 978–1011.

Journal URL

http://www.math.washington.edu/~ejpecp/

Rates of convergence for minimal distances in the central limit theorem under projective criteria

Jérôme Dedecker^∗ Florence Merlevède^† Emmanuel Rio^‡

Abstract

In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale differ- ence sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given.

Key words:Minimal and ideal distances, rates of convergence, Martingale difference sequences, stationary sequences, projective criteria, weak dependence, uniform mixing.

AMS 2000 Subject Classification:Primary 60F05.

Submitted to EJP on November 5, 2007, final version accepted June 6, 2008.

∗Université Paris 6, LSTA, 175 rue du Chevaleret, 75013 Paris, FRANCE. E-mail: jerome.dedecker@upmc.fr

†Université Paris Est, Laboratoire de mathématiques, UMR 8050 CNRS, Bâtiment Copernic, 5 Boulevard Descartes, 77435 Champs-Sur-Marne, FRANCE. E-mail: florence.merlevede@univ-mlv.fr

‡Université de Versailles, Laboratoire de mathématiques, UMR 8100 CNRS, Bâtiment Fermat, 45 Avenue des Etats- Unis, 78035 Versailles, FRANCE. E-mail: rio@math.uvsq.fr

1 Introduction and Notations

LetX₁,X₂, . . . be a strictly stationary sequence of real-valued random variables (r.v.) with mean zero and finite variance. Set S_n = X₁+X₂+· · ·+X_n. By P_n−1/2S_n we denote the law of n^−1/2S_n and by G_σ² the normal distribution N(0,σ²). In this paper, we shall give quantitative estimates of the approximation ofP_n−1/2S_n byG_σ² in terms of minimal or ideal metrics.

LetL(µ,ν) be the set of the probability laws onR² with marginalsµ andν. Let us consider the following minimal distances (sometimes called Wasserstein distances of orderr)

W_r(µ,ν) =





 infnZ

|x−y|^rP(d x,d y):P∈ L(µ,ν)o

if 0<r<1 infnZ

|x−y|^rP(d x,d y)1/r

:P∈ L(µ,ν)o

ifr≥1 .

It is well known that for two probability measuresµandν onRwith respective distributions func- tions (d.f.) F andG,

W_r(µ,ν) =Z 1 0

|F⁻¹(u)−G⁻¹(u)|^rdu1/r

for anyr≥1. (1.1)

We consider also the following ideal distances of order r (Zolotarev distances of order r). For two probability measuresµandν, andr a positive real, let

ζ_r(µ,ν) =supnZ

f dµ− Z

f dν: f ∈Λ_ro ,

whereΛ_r is defined as follows: denoting byl the natural integer such thatl < r ≤l+1,Λ_r is the class of real functions f which arel-times continuously differentiable and such that

|f^(l)(x)−f^(l)(y)| ≤ |x−y|^r−l for any(x,y)∈R×R. (1.2) It follows from the Kantorovich-Rubinstein theorem (1958) that for any 0<r≤1,

W_r(µ,ν) =ζ_r(µ,ν). (1.3)

For probability laws on the real line, Rio (1998) proved that for anyr>1, W_r(µ,ν)≤c_r ζ_r(µ,ν)1/r

, (1.4)

wherec_ris a constant depending only onr.

For independent random variables, Ibragimov (1966) established that if X₁ ∈ L^p for p ∈]2, 3], then W₁(P_n−1/2S_n,G_σ²) = O(n^1−p/2) (see his Theorem 4.3). Still in the case of independent r.v.’s, Zolotarev (1976) obtained the following upper bound for the ideal distance: ifX₁∈L^pforp∈]2, 3], thenζ_p(P_n−1/2Sn,G_σ²) =O(n^1−p/2). From (1.4), the result of Zolotarev entails that, for p ∈]2, 3], W_p(P_n−1/2S_n,G_σ²) =O(n^1/p−1/2)(which was obtained by Sakhanenko (1985) for any p>2). From (1.1) and Hölder’s inequality, we easily get that for independent random variables inL^p withp ∈ ]2, 3],

W_r(P_n−1/2S_n,G_σ²) =O(n^{−(p−2)/2r}) for any 1≤r≤p. (1.5)

In this paper, we are interested in extensions of (1.5) to sequences of dependent random variables.

More precisely, for X₁ ∈ L^p and p in ]2, 3] we shall give L^p-projective criteria under which: for r∈[p−2,p]and(r,p)6= (1, 3),

W_r(P_n−1/2S_n,G_σ²) =O(n^−(p−2)/2 max(1,r)). (1.6) As we shall see in Remark 2.4, (1.6) applied tor =p−2 provides the rate of convergenceO(n⁻

p−2 2(p−1)) in the Berry-Esseen theorem.

When (r,p) = (1, 3), Dedecker and Rio (2008) obtained that W₁(P_n−1/2S_n,G_σ²) = O(n^−1/2) for stationary sequences of random variables in L³ satisfying L¹ projective criteria or weak depen- dence assumptions (a similar result was obtained by Pène (2005) in the case where the vari- ables are bounded). In this particular case our approach provides a new criterion under which W₁(P_n−1/2Sn,G_σ²) =O(n^−1/2logn).

Our paper is organized as follows. In Section 2, we give projective conditions for stationary martin- gales differences sequences to satisfy (1.6) in the case(r,p)6= (1, 3). To be more precise, let(X_i)_i∈Z

be a stationary sequence of martingale differences with respect to some σ-algebras (Fi)_i∈Z (see Section 1.1 below for the definition of(Fi)_i∈Z). As a consequence of our Theorem 2.1, we obtain that if(X_i)_i∈Zis inL^pwithp∈]2, 3]and satisfies

X∞

n=1

1 n^2−p/2

E

S²_n n F0

−σ²

p/2<∞, (1.7)

then the upper bound (1.6) holds provided that(r,p)6= (1, 3). In the case r = 1 and p = 3, we obtain the upper boundW₁(P_n−1/2Sn,G_σ²) =O(n^−1/2logn).

In Section 3, starting from the coboundary decomposition going back to Gordin (1969), and using the results of Section 2, we obtain L^p-projective criteria ensuring (1.6) (if (r,p) 6= (1, 3)). For instance, if(X_i)_i∈Zis a stationary sequence ofL^p random variables adapted to(Fi)_i∈Z, we obtain (1.6) for any p ∈]2, 3[ and any r ∈[p−2,p]provided that (1.7) holds and the series E(S_n|F0) converge inL^p. In the case wherep=3, this last condition has to be strengthened. Our approach makes also possible to treat the case of non-adapted sequences.

Section 4 is devoted to applications. In particular, we give sufficient conditions for some functions of Harris recurrent Markov chains and for functions of linear processes to satisfy the bound (1.6) in the case (r,p) 6= (1, 3) and the rate O(n^−1/2logn) when r = 1 and p = 3. Since projective criteria are verified under weak dependence assumptions, we give an application to functions of φ-dependent sequences in the sense of Dedecker and Prieur (2007). These conditions apply to unbounded functions of uniformly expanding maps.

1.1 Preliminary notations

Throughout the paper, Y is aN(0, 1)-distributed random variable. We shall also use the following notations. Let(Ω,A,P)be a probability space, andT :Ω7→Ωbe a bijective bimeasurable transfor- mation preserving the probabilityP. For aσ-algebraF0 satisfyingF0 ⊆ T⁻¹(F0), we define the nondecreasing filtration(Fi)_i∈Z byFi = T⁻ⁱ(F0). LetF_−∞ =T

k∈ZFk andF_∞=W

k∈ZFk. We shall denote sometimes byE_i the conditional expectation with respect toFi. LetX₀be a zero mean random variable with finite variance, and define the stationary sequence(X_i)_i∈ZbyX_i=X₀◦Tⁱ.

2 Stationary sequences of martingale differences.

In this section we give bounds for the ideal distance of order r in the central limit theorem for stationary martingale differences sequences(X_i)_i∈Zunder projective conditions.

Notation 2.1. For anyp>2, define the envelope normk.k1,Φ,p by kXk1,Φ,p=

Z 1

0

(1∨Φ⁻¹(1−u/2))^p−2Q_X(u)du

whereΦdenotes the d.f. of theN(0, 1)law, andQ_X denotes the quantile function of|X|, that is the cadlag inverse of the tail function x →P(|X|>x).

Theorem 2.1. Let(X_i)_i∈Zbe a stationary martingale differences sequence with respect to(Fi)_i∈Z. Let σdenote the standard deviation of X₀. Let p∈]2, 3]. Assume thatE|X₀|^p<∞and that

X∞

n=1

1 n^2−p/2

E

S_n² n F0

−σ²

1,Φ,p<∞, (2.1)

and ∞

X

n=1

1 n^2/p

E

S²_n n F0

−σ²

p/2<∞. (2.2)

Then, for any r ∈ [p−2,p] with (r,p) 6= (1, 3), ζ_r(P_n−1/2Sn,G_σ²) = O(n^1−p/2), and for p = 3, ζ₁(P_n−1/2S_n,G_σ²) =O(n^−1/2logn).

Remark 2.1. Leta>1 and p>2. Applying Hölder’s inequality, we see that there exists a positive constant C(p,a) such thatkXk1,Φ,p ≤ C(p,a)kXka. Consequently, if p ∈]2, 3], the two conditions (2.1) and (2.2) are implied by the condition (1.7) given in the introduction.

Remark 2.2. Under the assumptions of Theorem 2.1, ζ_r(P_n−1/2Sn,G_σ²) = O(n^−r/2) if r < p−2.

Indeed, let p^′ =r+2. Since p^′< p, if the conditions (2.1) and (2.2) are satisfied for p, they also hold forp^′. Hence Theorem 2.1 applies withp^′.

From (1.3) and (1.4), the following result holds for the Wasserstein distances of orderr.

Corollary 2.1. Under the conditions of Theorem 2.1, W_r(P_n−1/2S_n,G_σ²) =O(n^−(p−2)/2 max(1,r))for any r in[p−2,p], provided that(r,p)6= (1, 3).

Remark 2.3. Forpin]2, 3],W_p(P_n−1/2S_n,G_σ²) =O(n^{−(p−2)/2p}). This bound was obtained by Sakha- nenko (1985) in the independent case. For p < 3, we have W₁(P_n−1/2S_n,G_σ²) = O(n^1−p/2). This bound was obtained by Ibragimov (1966) in the independent case.

Remark 2.4. Recall that for two real valued random variables X,Y, the Ky Fan metricα(X,Y) is defined byα(X,Y) =inf{ǫ >0 :P(|X−Y|> ǫ)≤ǫ}. LetΠ(µ,ν)be the Prokhorov distance between µandν. By Theorem 11.3.5 in Dudley (1989) and Markov inequality, one has, for anyr>0,

Π(P_X,P_Y)≤α(X,Y)≤(E(|X−Y|^r))^1/(r+1).

Taking the minimum over the random couples (X,Y) with law L(µ,ν), we obtain that, for any 0< r ≤1, Π(µ,ν)≤(W_r(µ,ν))^1/(r+1). Hence, if Π_n is the Prokhorov distance between the law of n^−1/2S_n and the normal distributionN(0,σ²),

Π_n≤(W_r(P_n−1/2S_n,G_σ²))^1/(r+1)for any 0<r≤1 . Taking r=p−2, it follows that under the assumptions of Theorem 2.1,

Π_n=O(n⁻

p−2

2(p−1)) if p<3 andΠ_n=O(n^−1/4p

logn) if p=3. (2.3) For p in]2, 4], under (2.2), we have thatkPn

i=1E(X²_i −σ²|F_i−1)kp/2 =O(n^2/p)(apply Theorem 2 in Wu and Zhao (2006)). Applying then the result in Heyde and Brown (1970), we get that if (X_i)_i∈Zis a stationary martingale difference sequence inL^psuch that (2.2) is satisfied then

kF_n−Φ_σk∞=O n⁻

p−2 2(p+1)

.

whereF_nis the distribution function ofn^−1/2S_nandΦ_σis the d.f. of G_σ². Now kF_n−Φ_σk_∞≤ 1+σ⁻¹(2π)^−1/2

Π_n.

Consequently the bounds obtained in (2.3) improve the one given in Heyde and Brown (1970), provided that (2.1) holds.

Remark 2.5. If (X_i)_i∈Zis a stationary martingale difference sequence inL³ such thatE(X₀²) =σ²

and X

k>0

k^−1/2kE(X_k²|F0)−σ²k3/2<∞, (2.4) then, according to Remark 2.1, the conditions (2.1) and (2.2) hold forp=3. Consequently, if (2.4) holds, then Remark 2.4 giveskF_n−Φ_σk∞=O n^−1/4p

logn

. This result has to be compared with Theorem 6 in Jan (2001), which states thatkF_n−Φ_σk_∞=O(n^−1/4) if P

k>0kE(X_k²|F0)−σ²k3/2<

∞.

Remark 2.6. Notice that if(X_i)_i∈Zis a stationary martingale differences sequence, then the condi- tions (2.1) and (2.2) are respectively equivalent to

X

j≥0

2^j(p/2−1)k2^−jE(S₂²_j|F0)−σ²k1,Φ,p<∞, and X

j≥0

2^j(1−2/p)k2^−jE(S₂²_j|F0)−σ²kp/2<∞. To see this, letA_n=kE(S_n²|F0

−E(S_n²)k1,Φ,pandB_n=kE(S_n²|F0

−E(S_n²)kp/2. We first show that A_nandB_nare subadditive sequences. Indeed, by the martingale property and the stationarity of the sequence, for all positiveiand j

A_i+j = kE(S²_i + (S_i+j−S_i)²|F0

−E(S_i²+ (S_i+j−S_i)²)k1,Φ,p

≤ A_i+kE (S_i+j−S_i)²−E(S²_j)|F0 k1,Φ,p.

Proceeding as in the proof of (4.6), p. 65 in Rio (2000), one can prove that, for anyσ-fieldA and any integrable random variableX,kE(X|A)k1,Φ,p≤ kXk1,Φ,p. Hence

kE (S_i+j−S_i)²−E(S²_j)|F0

k1,Φ,p≤ kE((S_i+j−S_i)²−E(S²_j)|Fi

k1,Φ,p.

By stationarity, it follows thatA_i+j≤A_i+A_j. SimilarlyB_i+j≤B_i+B_j. The proof of the equivalences then follows by using the same arguments as in the proof of Lemma 2.7 in Peligrad and Utev (2005).

3 Rates of convergence for stationary sequences

In this section, we give estimates for the ideal distances of order r for stationary sequences which are not necessarily adapted toFi.

Theorem 3.1. Let(X_i)_i∈Zbe a stationary sequence of centered random variables inL^p with p∈]2, 3[, and letσ²_n=n⁻¹E(S²_n). Assume that

X

n>0

E(X_n|F0)and X

n>0

(X_−n−E(X_−n|F0))converge inL^p, (3.1)

and X

n≥1

n^−2+p/2kn⁻¹E(S²_n|F0)−σ²_nkp/2<∞. (3.2) Then the seriesP

k∈ZCov(X₀,X_k)converges to some nonnegativeσ², and 1. ζ_r(P_n−1/2Sn,G_σ²) =O(n^1−p/2)for r∈[p−2, 2],

2. ζ_r(P_n−1/2S_n,G_σ²

n) =O(n^1−p/2)for r∈]2,p].

Remark 3.1. According to the bound (5.40), we infer that, under the assumptions of Theorem 3.1, the condition (3.2) is equivalent to

X

n≥1

n^−2+p/2kn⁻¹E(S²_n|F0)−σ²kp/2<∞. (3.3) The same remark applies to the next theorem withp=3.

Remark 3.2. The result of item 1 is valid withσ_n instead ofσ. On the contrary, the result of item 2 is no longer true ifσ_n is replaced byσ, because for r ∈]2, 3], a necessary condition forζ_r(µ,ν) to be finite is that the two first moments ofν and µ are equal. Note that under the assumptions of Theorem 3.1, both W_r(P_n−1/2S_n,G_σ²) and W_r(P_n−1/2S_n,G_σ²

n) are of the order of n^−(p−2)/2 max(1,r). Indeed, in the case wherer∈]2,p], one has that

W_r(P_n−1/2S_n,G_σ²)≤W_r(P_n−1/2S_n,G_σ²

n) +W_r(G_σ2 n,G_σ²), and the second term is of order|σ−σ_n|=O(n^−1/2).

In the case wherep=3, the condition (3.1) has to be strengthened.

Theorem 3.2. Let(X_i)_i∈Z be a stationary sequence of centered random variables inL³, and letσ²_n = n⁻¹E(S²_n). Assume that (3.1) holds for p=3and that

X

n≥1

1 n

X

k≥n

E(X_k|F0)

3<∞ and X

n≥1

1 n

X

k≥n

(X_−k−E(X_−k|F0))

3<∞. (3.4) Assume in addition that

X

n≥1

n^−1/2kn⁻¹E(S²_n|F0)−σ²_nk3/2<∞. (3.5) Then the seriesP

k∈ZCov(X₀,X_k)converges to some nonnegativeσ²and

1. ζ₁(P_n−1/2S_n,G_σ²) =O(n^−1/2logn),

2. ζ_r(P_n−1/2Sn,G_σ²) =O(n^−1/2)for r∈]1, 2], 3. ζ_r(P_n−1/2S_n,G_σ²

n) =O(n^−1/2)for r∈]2, 3].

4 Applications

4.1 Martingale differences sequences and functions of Markov chains

Recall that the strong mixing coefficient of Rosenblatt (1956) between twoσ-algebras A and B is defined by α(A,B) =sup{|P(A∩B)−P(A)P(B)| : (A,B)∈ A × B }. For a strictly stationary sequence(X_i)_i∈Z, letFi=σ(X_k,k≤i). Define the mixing coefficientsα₁(n)of the sequence(X_i)_i∈Z

by

α₁(n) =α(F0,σ(X_n)).

For the sake of brevity, letQ=Q_X0 (see Notation 2.1 for the definition). According to the results of Section 2, the following proposition holds.

Proposition 4.1. Let (X_i)_i∈Z be a stationary martingale difference sequence in L^p with p ∈]2, 3].

Assume moreover that the series X

k≥1

1 k^2−p/2

Z α₁(k)

0

(1∨log(1/u))^(p−2)/2Q²(u)du and X

k≥1

1 k^2/p

Z α₁(k) 0

Q^p(u)du2/p

(4.1) are convergent.Then the conclusions of Theorem 2.1 hold.

Remark 4.1. From Theorem 2.1(b) in Dedecker and Rio (2008), a sufficient condition to get W₁(P_n−1/2Sn,G_σ²) =O(n^−1/2logn)is

X

k≥0

Z α₁(n)

0

Q³(u)du<∞.

This condition is always strictly stronger than the condition (4.1) whenp=3.

We now give an example. Consider the homogeneous Markov chain(Y_i)_i∈Zwith state spaceZde- scribed at page 320 in Davydov (1973). The transition probabilities are given byp_n,n+1=p_{−n,−n−1}= a_n forn≥0, p_n,0=p_−n,0=1−a_n forn>0, p_0,0=0,a₀=1/2 and 1/2≤a_n<1 for n≥1. This chain is irreducible and aperiodic. It is Harris positively recurrent as soon asP

n≥2Πⁿ_k=1⁻¹a_k<∞. In that case the stationary chain is strongly mixing in the sense of Rosenblatt (1956).

Denote by K the Markov kernel of the chain (Y_i)_i∈Z. The functions f such that K(f) =0 almost everywhere are obtained by linear combinations of the two functions f₁ and f₂ given by f₁(1) =1, f₁(−1) =−1 and f₁(n) = f₁(−n) =0 ifn6=1, and f₂(0) =1, f₂(1) = f₂(−1) =0 and f₂(n+1) = f₂(−n−1) =1−a_n⁻¹ifn>0. Hence the functions f such thatK(f) =0 are bounded.

If(X_i)_i∈Zis defined byX_i= f(Y_i), with K(f) =0, then Proposition 4.1 applies if

α₁(n) =O(n^1−p/2(logn)^−p/2−ε) for someε >0, (4.2)

which holds as soon as P₀(τ = n) = O(n^−1−p/2(logn)^−p/2−ε), where P₀ is the probability of the chain starting from 0, and τ= inf{n> 0,X_n = 0}. Now P₀(τ = n) = (1−a_n)Πⁿ_i=1⁻¹a_i for n≥ 2.

Consequently, if

a_i=1− p 2i

1+ 1+ε logi

forilarge enough , the condition (4.2) is satisfied and the conclusion of Theorem 2.1 holds.

Remark 4.2. If f is bounded and K(f) 6= 0, the central limit theorem may fail to hold forS_n = Pn

i=1(f(Y_i)−E(f(Y_i))). We refer to the Example 2, page 321, given by Davydov (1973), whereS_n properly normalized converges to a stable law with exponent strictly less than 2.

Proof of Proposition 4.1. Let B^p(F0) be the set of F0-measurable random variables such that kZkp≤1. We first notice that

kE(X²_k|F0)−σ²kp/2= sup

Z∈B^p/(p−2)(F0)

Cov(Z,X_k²). Applying Rio’s covariance inequality (1993), we get that

kE(X_k²|F0)−σ²kp/2≤2Z α₁(k) 0

Q^p(u)du2/p

,

which shows that the convergence of the second series in (4.1) implies (2.2). Now, from Fréchet (1957), we have that

kE(X²_k|F0)−σ²k1,Φ,p=supE((1∨ |Z|^p−2)|E(X_k²|F0)−σ²|),Z F0-measurable, Z∼ N(0, 1) . Hence, settingǫ_k=sign(E(X_k²|F0)−σ²),

kE(X_k²|F0)−σ²k1,Φ,p=sup

Cov(ǫ_k(1∨ |Z|^p−2),X_k²),ZF0-measurable,Z∼ N(0, 1) . Applying again Rio’s covariance inequality (1993), we get that

kE(X²_k|F0)−σ²k1,Φ,p≤CZ α₁(k) 0

(1∨log(u⁻¹))^(p−2)/2Q²(u)du , which shows that the convergence of the first series in (4.1) implies (2.1).

4.2 Linear processes and functions of linear processes In what follows we say that the series P

i∈Za_i converges if the two series P

i≥0a_i and P

i<0a_i converge.

Theorem 4.1. Let(a_i)_i∈Z be a sequence of real numbers in ℓ² such that P

i∈Za_i converges to some real A. Let (ǫ_i)_i∈Z be a stationary sequence of martingale differences in L^p for p ∈]2, 3]. Let X_k = P

j∈Za_jǫ_k−j, andσ²_n=n⁻¹E(S²_n). Let b₀=a₀−A and b_j=a_j for j6=0. Let A_n=P

j∈Z(Pn

k=1b_k−j)². If A_n=o(n), thenσ²_n converges toσ²=A²E(ǫ₀²). If moreover

X∞

n=1

1 n^2−p/2

E

1 n

Xⁿ

j=1

ǫ_j2 F0

−E(ǫ²₀)

p/2<∞, (4.3)

then we have

1. If A_n=O(1), thenζ₁(P_n−1/2S_n,G_σ²) =O(n^−1/2log(n)), for p=3,

2. If A_n=O(n^(r+2−p)/r), thenζ_r(P_n−1/2S_n,G_σ²) =O(n^1−p/2), for r∈[p−2, 1]and p6=3, 3. If A_n=O(n^3−p), thenζ_r(P_n−1/2S_n,G_σ²) =O(n^1−p/2), for r∈]1, 2],

4. If A_n=O(n^3−p), thenζ_r(P_n−1/2S_n,G_σ²

n) =O(n^1−p/2), for r∈]2,p].

Remark 4.3. If the condition given by Heyde (1975) holds, that is X∞

n=1

X

k≥n

a_k2

<∞ and X∞

n=1

X

k≤−n

a_k2

<∞, (4.4)

thenA_n=O(1), so that it satisfies all the conditions of items 1-4.

Remark 4.4. Under the additional assumptionP

i∈Z|a_i|<∞, one has the bound A_n≤4B_n, where B_n=

Xn

k=1

X

j≥k

|a_j|2

+ X

j≤−k

|a_j|2

. (4.5)

Proof of Theorem 4.1.We start with the following decomposition:

S_n=A

n

X

j=1

ǫ_j+ X∞

j=−∞

Xⁿ

k=1

b_k−j

ǫ_j. (4.6)

LetR_n =P_∞

j=−∞(Pn

k=1b_k−j)ǫ_j. SincekR_nk²₂ =A_nkǫ₀k²₂ and since|σ_n−σ| ≤ n^−1/2kR_nk2, the fact that A_n = o(n) implies that σ_n converges to σ. We now give an upper bound for kR_nkp. From Burkholder’s inequality, there exists a constantC such that

kR_nkp≤Cn

X∞

j=−∞

Xⁿ

k=1

b_k−j2

ǫ²_{j p/2}

o1/2

≤Ckǫ₀kp

pA_n. (4.7)

According to Remark 2.1, since (4.3) holds, the two conditions (2.1) and (2.2) of Theorem 2.1 are satisfied by the martingale M_n = APn

k=1ǫ_k. To conclude the proof, we use Lemma 5.2 given in Section 5.2, with the upper bound (4.7). ƒ

Proof of Remarks 4.3 and 4.4. To prove Remark 4.3, note first that A_n=

Xn

j=1

X^−j

l=−∞

a_l+ X∞

l=n+1−j

a_l2

+ X∞

i=1

ⁿ⁺ⁱX⁻¹

l=i

a_l2

+ X∞

i=1

X⁻ⁱ

l=−i−n+1

a_l2

.

It follows easily thatA_n=O(1)under (4.4). To prove the bound (4.5), note first that A_n≤3B_n+

X∞

i=n+1

ⁿ⁺ⁱX⁻¹

l=i

|a_l|2

+ X∞

i=n+1

X⁻ⁱ

l=−i−n+1

|a_l|2

.

LetT_i=P_∞

l=i|a_l|andQ_i =P₋i

l=−∞|a_l|. We have that X∞

i=n+1

ⁿ⁺ⁱ⁻¹X

l=i

|a_l|2

≤ T_n+1 X∞

i=n+1

(T_i−T_n+i)≤nT_n+1² X∞

i=n+1

X⁻ⁱ

l=−i−n+1

|a_l|2

≤ Q_n+1 X∞

i=n+1

(Q_i−Q_n+i)≤nQ²_n+1.

Sincen(T_n+1² +Q²_n+1)≤B_n, (4.5) follows.ƒ

In the next result, we shall focus on functions of real-valued linear processes X_k=h X

i∈Z

a_iǫ_k−i

−E

h X

i∈Z

a_iǫ_k−i

, (4.8)

where(ǫ_i)_i∈Zis a sequence of iid random variables. Denote by w_h(.,M)the modulus of continuity of the functionhon the interval[−M,M], that is

w_h(t,M) =sup{|h(x)−h(y)|,|x−y| ≤t,|x| ≤M,|y| ≤M}.

Theorem 4.2. Let(a_i)_i∈Zbe a sequence of real numbers inℓ² and(ǫ_i)_i∈Zbe a sequence of iid random variables inL². Let X_k be defined as in (4.8) andσ²_n= n⁻¹E(S_n²). Assume that h isγ-Hölder on any compact set, with w_h(t,M)≤C t^γM^α, for some C>0,γ∈]0, 1]andα≥0. If for some p∈]2, 3],

E(|ǫ₀|^2∨(α+γ)p)<∞ and X

i≥1

i^p/2−1 X

|j|≥i

a²_jγ/2

<∞, (4.9)

then the seriesP

k∈ZCov(X₀,X_k)converges to some nonnegativeσ², and 1. ζ₁(P_n−1/2S_n,G_σ²) =O(n^−1/2logn), for p=3,

2. ζ_r(P_n−1/2S_n,G_σ²) =O(n^1−p/2)for r∈[p−2, 2]and(r,p)6= (1, 3), 3. ζ_r(P_n−1/2S_n,G_σ²

n) =O(n^1−p/2)for r∈]2,p].

Proof of Theorem 4.2.Theorem 4.2 is a consequence of the following proposition:

Proposition 4.2. Let(a_i)_i∈Z,(ǫ_i)_i∈Zand(X_i)_i∈Zbe as in Theorem 4.2. Let(ǫ_i^′)_i∈Zbe an independent copy of(ǫ_i)_i∈Z. Let V₀=P

i∈Za_iǫ_−i and M_1,i=|V₀| ∨

X

j<i

a_jǫ_−j+X

j≥i

a_jǫ₋^′ _j

and M_2,i=|V₀| ∨

X

j<i

a_jǫ₋^′_j+X

j≥i

a_jǫ_−j . If for some p∈]2, 3],

X

i≥1

i^p/2−1 w_h

X

j≥i

a_jǫ_−j

,M_1,i

_p<∞ and X

i≥1

i^p/2−1 w_h

X

j<−i

a_jǫ_−j

,M_{2,−i p}<∞,

(4.10) then the conclusions of Theorem 4.2 hold.

To prove Theorem 4.2, it remains to check (4.10). We only check the first condition. Since w_h(t,M)≤C t^γM^α and the random variablesǫ_i are iid, we have

w_h

X

j≥i

a_jǫ_−j

,M_1,i

_p ≤ C

X

j≥i

a_jǫ_−j

γ

|V₀|^α _p+C

X

j≥i

a_jǫ_−j

γ

_pk|V₀|^αkp, so that

w_h

X

j≥i

a_jǫ_−j

,M_1,i p

≤C 2^α

X

j≥i

a_jǫ_−j

α+γ p+

X

j≥i

a_jǫ_−j

γ p

k|V₀|^αkp+2^α

X

j<i

a_jǫ_−j

α p

. From Burkholder’s inequality, for anyβ >0,

X

j≥i

a_jǫ_−j

β p=

X

j≥i

a_jǫ_−j

β

βp≤K X

j≥i

a²_jβ/2

kǫ₀k^β_2∨βp.

Applying this inequality withβ=γorβ =α+γ, we infer that the first part of (4.10) holds under (4.9). The second part can be handled in the same way. ƒ

Proof of Proposition 4.2. Let Fi = σ(ǫ_k,k ≤ i). We shall first prove that the condition (3.2) of Theorem 3.1 holds. We write

kE(S²_n|F0) − E(S²_n)kp/2≤2 Xn

i=1

Xn−i

k=0

kE(X_iX_k+i|F0)−E(X_iX_k+i)kp/2

≤ 4 Xn

i=1

Xn

k=i

kE(X_iX_k+i|F0)kp/2+2 Xn

i=1

Xi

k=1

kE(X_iX_k+i|F0)−E(X_iX_k+i)kp/2. We first control the second term. Let ǫ^′ be an independent copy of ǫ, and denote by E_ǫ(·) the conditional expectation with respect toǫ. Define

Y_i=X

j<i

a_jǫ_i−j, Y_i^′=X

j<i

a_jǫ^′_i−j,Z_i=X

j≥i

a_jǫ_i−j, and Z_i^′=X

j≥i

a_jǫ^′_i−j.

TakingFℓ=σ(ǫ_i,i≤ℓ), and settingh₀=h−E(h(P

i∈Za_iǫ_i)), we have kE(X_iX_k+i|F0)−E(X_iX_k+i)kp/2

= E_ǫ

h₀(Y_i^′+Z_i)h₀(Y_k+i^′ +Z_k+i)

−E_ǫ

h₀(Y_i^′+Z_i^′)h₀(Y_k+i^′ +Z_k+i^′ ) _p/2. Applying first the triangle inequality, and next Hölder’s inequality, we get that

kE(X_iX_k+i|F0)−E(X_iX_k+i)kp/2 ≤ kh₀(Y_k+i^′ +Z_k+i)kpkh₀(Y_i^′+Z_i)−h₀(Y_i^′+Z_i^′)kp

+ kh₀(Y_i^′+Z_i^′)kpkh₀(Y_k+i^′ +Z_k+i)−h₀(Y_k+i^′ +Z_k+i^′ )kp.