On mean central limit theorems for stationary sequences

www.imstat.org/aihp 2008, Vol. 44, No. 4, 693–726

DOI: 10.1214/07-AIHP117

© Association des Publications de l’Institut Henri Poincaré, 2008

On mean central limit theorems for stationary sequences

Jérôme Dedecker

and Emmanuel Rio

aLaboratoire de Statistique Théorique et Appliquée, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France.

E-mail: [email protected]

bLaboratoire de mathématiques, UMR 8100 CNRS, Bâtiment Fermat, 45 Avenue des Etats-Unis, 78035 VERSAILLES Cedex, France.

E-mail: [email protected]

Received 26 June 2006; revised 4 April 2007; accepted 7 April 2007

Abstract. In this paper, we give estimates of the minimalL¹distance between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary sequences satisfying projective criteria in the style of Gordin or weak dependence conditions.

Résumé. Dans cet article, nous donnons des majorations de la distance minimaleL¹entre la loi de la somme normalisée et sa loi limite gaussienne pour des suites stationnaires satisfaisant des critères projectifs à la Gordin ou des conditions de dépendance faible.

MSC:60F05

Keywords:Mean central limit theorem; Wasserstein distance; Minimal distance; Martingale difference sequences; Strong mixing; Stationary sequences; Weak dependence; Rates of convergence; Projective criteria

1. Introduction

LetX1, X2, . . . ,be a sequence of real-valued random variables (r.v.) with mean zero and finite variance. LetSn= X1+X2+ · · · +Xn. ByFnwe denote the distribution function (d.f.) ofn^−1/2Sn. LetΦσ be the d.f. of theN(0, σ²)- distribution. For independent and identically distributed (i.i.d.) r.v.’s, according to the central limit theorem (CLT), F_n(x)converges to Φ_σ(x)uniformly for x inR, where σ is the standard deviation ofX₁. Agnew [1] proved that the convergence also holds inL^r(R)for r >1/2. Agnew’s result is called mean CLT in the case r=1. Let then ρ_n^(r)= Fn−Φσr. Forr=1 andr=2 and i.i.d. random variables with finite absolute third moment, Esseen [11]

proved thatn^1/2ρ_n^(r)converges to some explicit constantA_r(F )depending only on the distribution functionF ofX₁ (Theorems 3.2 and 4.2 in [11]). In particular, Esseen’s results imply that

ρ_n^(r)=O n^−1/2

asn→ ∞. (1.1)

Next Zolotarev [29] obtained the upper boundA1(F )≤E(|X1|³)/(2σ²). The proofs of these results are based on the method of characteristic functions (cf. [18] for more details).

The caser=1 is of special interest, sinceρn⁽¹⁾ is exactly the minimal distance betweenn^−1/2Sn and a r.v. with distributionN(0, σ²)inL¹(cf. [10], Section 11.8, Problems 1 and 2). Now let

d1(X, Y )= sup

f∈Λ1(R)

E

f (X)−f (Y )

, (1.2)

whereΛ₁(R)is the set of 1-Lipschitzian functions fromRtoR. Applying the Kantorovich–Rubinstein theorem we also have thatρ_n⁽¹⁾=d₁(n^−1/2S_n, σ Y )ifY is aN(0,1)-distributed random variable.

In this paper we are interested in extensions of (1.1) forr=1 to sequences of dependent random variables. This subject was studied by Sunklodas [28] in the case of uniformly mixing (in the sense of Ibragimov) stationary sequences of real-valued random variables. Using the Stein method, he reached the rate of convergence O(n^−1/2(logn)²)in (1.1) for geometrically mixing sequences of random variables with finite eight moments. A different approach to get rates of convergence in the CLT is Bergström’s [2] inductive proof of the Berry–Esseen theorem, based on the Lindeberg method. Starting from Bergström’s recursion argument, Bolthausen [4] obtained exact rates of uniform convergence for martingale difference arrays. Rio [25] adapted Bergström’s method to weakly dependent sequences and obtained the Berry–Esseen theorem for stationary and uniformly bounded sequences of real-valued r.v.’s satisfying the condition

kkϕ(k) <∞, where(ϕ(k))k denotes the sequence of uniform mixing coefficients of the sequence (Xi)_i∈N, in the sense of Ibragimov (confer [18] for an exact definition of these coefficients). This result was extended to the multivariate case by Jan [19], Theorem 9. Jan also weakened the notion of weak dependence involved in Rio’s paper (cf. Theorem 1 in [20] for more details). However the dependence coefficients in [19] are too restrictive for the applications to some dynamical systems, such as Sinai’s billiard. Pène [22] noticed that the inductive proof of Jan [19] can be adapted to get the rate of convergence O(n^−1/2)for the minimalL¹-distance in the multivariate CLT for stationary sequences satisfying some dependence conditions. In particular her result applies to sums of bounded r.v.’s defined from dynamical systems (such as Sinai’s billiard) or strongly mixing sequences in the sense of Rosenblatt.

For example, Pène’s result yields (1.1) (withr=1) for stationary sequences of bounded random variables(X_i)_i∈N satisfying the condition

kkα(k) <∞, where(α(k))_kdenotes the sequence of strong mixing coefficients of(X_i)_i∈Z in the sense of Rosenblatt (confer [18] for a definition of these coefficients).

We now describe the contents of our paper. Our aim is to provide rates of convergence in the mean CLT for station- ary sequences of real-valued r.v.’s satisfying either projective criteria in the style of Gordin [14] or weak dependence conditions.

In Section 2, we give bounds in the stationary case involvingL^p-norms of conditional expectations. Let(Xi)_i∈Zbe a stationary sequence of real-valued random variables,Mk=σ (X_i: i≤k)andE_kdenote the conditional expectation with respect toMk. In Section 2.1, we obtain in Theorem 2.1 the rate of convergence O(n^−1/2logn)in the mean CLT for stationary and ergodic martingale differences sequences(X_i)_i∈Zwith finite absolute third moments satisfying the projective conditions

sup

m>0

m k=1

E0

X²_k−σ²

1

<∞ and sup

m>0

m k=1

X0E0

X_k²−σ²

1

<∞, (1.3)

whereσ²=VarX₀. In Section 2.2, we generalize Theorem 2.1 to ergodic stationary sequences satisfying projec- tive criteria. In Section 2.3 we give some applications to bounded sequences. For example, assuming that the series

k>0E₀(X_k)converges inL¹, Theorem 2.3 provides rates of convergence in the mean CLT as soon asE₀(S_m²/m) converges toσ²inL¹. This condition appears in the conditional CLT of Dedecker and Merlevède [6] and is rather mild. For example the rate of convergence O(n^−1/2logn)is obtained under the projective conditions

m>0

k≥m

E0(Xk) 1

<∞ and sup

m>0

E0

S_m² −mσ²

1<∞. (1.4)

Again the proofs are based on the Lindeberg method at order three.

In Section 3, we give projective conditions or weak dependence conditions implying (1.1) forr=1. Conditions (1.3) and (1.4) involve conditional second moments. It seems difficult to get the optimal rate of convergence O(n^−1/2) under second-order conditions (at least for the Berry–Esseen theorem: cf. [25] and [4], Theorem 4). Therefore our re- sults hold under projective conditions on the monoms of degree three. For example, (1.1) holds for stationary bounded martingale difference sequences under the projective conditions

k>0

E0

X_k²

−σ²

1<∞ and

k>0

sup

j≥k

E0

X_kX_j²

−E X_kX²_j

1<∞. (1.5)

For stationary sequences, one needs to strengthen (1.5): we obtain (1.1) for stationary sequences of bounded r.v.’s under the projective conditions

k>0

k sup

i≥j≥k

E0

XkX^α_jX^β_i

−E

XkX_j^αX^β_i

1<∞, with(α, β)∈ {0,1}², (1.6) which can also be deduced from Theorem 1.1 in [22]. It is worth noticing that the Berry–Esseen type Theorem 9 in [19] requiresL^∞-norms instead ofL¹-norms in (1.6). The proofs of these results are based on the Lindeberg method at order four. Therefore, in the unbounded case, the results hold for sequences of random variables with finite fourth moments (cf. Theorems 3.1(a) and 3.2(a) for detailed conditions). For example, Theorem 3.1(a) applied to strongly mixing and stationary sequences yields the rate of convergence O(n^−1/2)in the mean CLT if there exists somep >1 such that

k>0

k^{(p+1)/(p−1)}α(k) <∞ and E

|X₀|^ap

<∞, (1.7)

wherea=4. By contrast, the Berry–Esseen type theorem for functionals of stationary discrete Markov chains due to Bolthausen [3] holds under condition (1.7) witha=3. In order to improve Theorems 3.1(a) and 3.2(a) in the case of strongly mixing sequences we adapt the truncation method in [24] to our context. We then get the rate O(n^−1/2)in the mean CLT under the strong mixing condition

k>0

k^b _α(k)

0

Q³_|X

0|(u)du <∞, (1.8)

whereQ_|X0| denotes the quantile function of|X0|andb=1. This condition is implied by (1.7) witha=3, so that our result holds under Bolthausen’s [3] condition. Moreover, for stationary strongly mixing martingale difference sequences, we prove that (1.1) holds forp=1 under condition (1.8) withb=0. In Section 5 we give two classical examples of non irreducible Markov chains to which our results apply.

2. Projective criteria for stationary sequences

Throughout the paper,Y is aN(0,1)-distributed random variable.

We shall use the following notations. Let(Ω,A,P)be a probability space, andT:Ω→Ω be a bijective bimea- surable transformation preserving the probabilityP. An elementAis said to be invariant ifT (A)=A. We denote by Itheσ-algebra of all invariant sets. LetM0be a sub-σ-algebra ofAsatisfyingM0⊆T⁻¹(M0)and define the non- decreasing filtration(Mi)_i∈ZbyMi =T⁻ⁱ(M0). LetM∞=

i∈ZMi. Denote byEi the conditional expectation with respect toMi.

LetX₀be aM0-measurable and centered random variable. Throughout the sequel, the sequenceX=(X_i)_i∈Zis defined byX_i=X0◦Tⁱ. From the definition the sequence(X_i)_i∈Zis adapted to the filtration(Mi)_i∈Z.

2.1. Martingale difference sequences

In this section we obtain rates of convergence of the order ofn^−1/2lognin the mean CLT for stationary martingale difference sequences. In order to obtain these rates of convergence, we will just need a projective condition on the variablesX²_l, as in [19]. We first recall Jan’s results concerning the rates of convergence for the uniform distance between the distribution functions.

Assume that(Xi)i∈Zis a stationary martingale difference sequence inL³such thatE(X₀²)=σ²and

l>0

E₀

X²_l −σ²

3/2<∞. (2.1)

Then, by Theorem 6 in [19], ifY isN(0,1)-distributed, sup

t∈R

P

n^−1/2Sn≤t

−P(σ Y≤t ) =O n^−1/4

. (2.2)

Under projective conditions related to (2.1), the rate of convergence in the mean central limit theorem is at least O(n^−1/2logn)as shown in Theorem 2.1 below.

Theorem 2.1. Let(X_i)_i∈Zbe a stationary martingale difference sequence inL³,such thatE(X₀²|I)=E(X²₀)=σ² almost surely.LetΛ=σ⁻²E|X0|³andU_m=E0(X²₁+ · · · +X²_m)−mσ².Then

(a) d1(S_n, σ√

nY )≤^13σ₆ +^Λ₆log(1+2n)+_[^√2n]

m=1 X₀U_m1+2σU_m1

mσ² .

(b) Ifsup_m>0(X₀U_m1+ U_m1) <∞,thend₁(S_n, σ Y√

n)=O(logn).

Remark 2.1. From the ergodic theorem,(Um/m)converges a.s.and in L¹to0asmtends to∞.SinceX0∈L³,it follows that the sequence(X₀U_m/m)_mis uniformly integrable.Hence,under the assumptions of Theorem2.1,

m→∞lim m⁻¹

X₀U_m1+ U_m1

=0.

Therefore Theorem2.1(a) provides a rate of convergence in the mean CLT.For example,if X0E0(X²_l −σ²)1= O(l^−δ)andE₀(X²_l −σ²)1=O(l^−δ)for someδin]0,1[,then the rate of convergence in the mean CLT is of the order ofn^−δ/2.If Jan’s condition(2.1)holds,then(b)yields the rate of convergenceO(n^−1/2logn)in the mean CLT. For bounded random variables(b)holds as soon as the series

l>0E0(X_l²−σ²)converges inL¹.

Proof of Theorem 2.1. We prove Theorem 2.1 in the caseσ=1. The general case follows by dividing the r.v.’s byσ. Let(Yi)_i∈Nbe a sequence of independent random variables with normal distributionN(0,1). Suppose furthermore that the sequence(Yi)i∈Nis independent of(Xi)i∈N. LetY be aN(0,1)-distributed random variable, independent of the above defined sequences. LetT_n=Y₁+Y₂+ · · · +Y_n. For any 1-Lipschitzian functionf, letΔ(f )=E(f (S_n)− f (T_n)). From (1.2), we have to boundΔ(f ). Clearly

Δ(f )=E

f (S_n)−f (T_n)

≤E

f (S_n+Y )−f (T_n+Y )

+2E|Y|. (2.3)

In order to bound up the term on right-hand side, we apply the Lindeberg method.

Notation 2.1. Set f_k(x)=E(f (x +Y +T_n−T_k)). Let S0 =0, and, for k > 0, let Δ_k =f_k(S_k−1 +X_k)− fk(Sk−1+Yk).

Since the sequence(Y_i)_i∈Nis independent of the sequence(X_i)_i∈N, E

f (S_n+Y )−f (T_n+Y )

= n k=1

E(Δ_k). (2.4)

Next the functionsf_kareC^∞. Consequently, from the Taylor integral formula at orders three and four, Δ_k=f_k(S_k−1)(X_k−Y_k)+1

2f_k(S_k−1)

X_k²−Y_k²

−1

6f_k⁽³⁾(S_k−1)Y_k³+R_k, with

Rk≤1 6f_k⁽³⁾

∞|Xk|³+ 1 24f_k⁽⁴⁾

∞Y_k⁴. (2.5)

Consequently, for any 1-Lipschitzian functionf, Δ(f )≤2E|Y| +

n k=1

E(Rk)+ n k=1

E

f_k(Sk−1)Xk+1

2f_k(Sk−1)

X_k²−1

. (2.6)

The termsE(f_k(S_k−1)X_k)vanish under the martingale assumption. To bound up the other terms appearing in (2.6), we need to bound up the derivatives offk. This will be done via the lemma below.

Lemma 2.1. Letf be a1-Lipschitzian function,Y be a standard normal andB be a real-valued random variable, independent ofY.Then

dⁱ

dxⁱEf (x+t Y+B)

≤t¹⁻ⁱφ⁽ⁱ⁻¹⁾

1 for anyt >0 and any positive integeri,whereφdenotes the density ofY. Proof of Lemma 2.1. Letφ_tbe the density oft Y. Then

Ef (x+t Y+B)=E

f ∗φ_t(x+B) .

Sincefis 1-Lipschitzian, the Stieltjes measure dfoff is absolutely continuous with respect to the Lebesgue measure λandf=df/dλbelongs to[−1,1]. Next(f∗φ_t)⁽ⁱ⁾=f∗φ⁽ⁱ_t ⁻¹⁾, and consequently

dⁱ

dxⁱEf ∗φt(x+B) ≤f

∞φ_t⁽ⁱ⁻¹⁾

1.

Sinceφ_t⁽ⁱ⁻¹⁾(x)=t⁻ⁱφ⁽ⁱ⁻¹⁾(x/t ), it implies Lemma 2.1.

Noting that φ

1= 2

π≤4

5, φ

1= 8

πe≤1, φ⁽³⁾

1= 2

π+ 32

πe³≤ 8

5, (2.7)

and applying Lemma 2.1 witht=√

n−k+1, we infer from (2.5) that E(R_k)≤

Λ 6

(n−k+1)⁻¹+ 1

5

(n−k+1)^−3/2. (2.8)

Summing onk, we infer from (2.8) that n

k=1

E(R_k)+2E|Y| ≤ρ(n) withρ(n)=13 6 +Λ

6 log(1+2n). (2.9)

The control of the main term in (2.6) is derived from the lemma below.

Lemma 2.2. LetZ0be an integrable random variable with zero mean.SetZk=Z0◦T^kand letWm=_m

l=1E0(Zl).

Then,fors=2ors=3, n

k=1

E

f_k^(s)(S_k−1)Z_k

≤^[

√2n]

m=1

2m^1−s

X₀W_m1+2W_m1

.

Proof of Lemma 2.2. We first divide[1, n]into blocks of nonincreasing length.

Notation 2.2. Define the decreasing sequence of integers(n_i)_i≥0byn₀=nandn_i=max(0, ni−1−i)fori >0.Let pbe the first integer such thatn_p=0.Setm_i=ifori < pandm_p=n_p−1.

Next fixiin[1, p]. Let thenkbe any integer in]n_i, n_i−1]. Writing f_k^(s)(Sk−1)=f_n^(s)

i+1(Sn_i)+

k−1

j=ni+1

f_j^(s)₊₁(Sj)−f_j^(s)(Sj−1)

, (2.10)

we get that

ni−1

k=n_i+1

E

f_k^(s)(S_k−1)Z_k

=D_i+

ni−1−1 j=n_i+1

D_i,j, (2.11)

where D_i=E

f_n^(s)

i+1(S_ni)

ni−1

k=n_i+1

E_ni(Z_k)

D_i,j=E

f_j^(s)₊₁(S_j)−f_j^(s)(S_j−1) ⁿⁱ⁻¹

k=j+1

E_j(Z_k)

.

By definition of the sequence(Z_k)_k, for any integerj and any positivem, E_j(Z_j+1+Z_j+2+ · · · +Z_j+m)=W_m◦T^j.

Hence, from Lemma 2.1 applied witht=(n−n_i)^1/2andB=0, for anyi < p,

D_i≤(n−n_i)^(1−s)/2W_i1. (2.12)

MoreoverDp=0 from the centering assumption on the random variablesZk. Now, by definition,n−ni=i(i+1)/2 fori < p. Hence, from (2.12),

p−1

i=1

D_i ≤2

p−1 i=1

i^1−sW_i1, where −1+√

2n < p <1+√

2n. (2.13)

Next we bound upD_i,j. From the elementary equality f_j+^(s)₁(S_j)−f_j^(s)(S_j−1)=E_j

f_j+^(s)₁(S_j−1+X_j)−f_j+^(s)₁(S_j−1+Y_j) we get that

f_j^(s)₊₁(S_j)−f_j^(s)(S_j−1) ≤f_j^(s₊⁺₁¹⁾

∞E_j|X_j−Y_j| ≤(n−j )^−s/2

|X_j| +1 ,

whence

Di,j≤(n−j )^−s/2E

|X0| +1

|Wn_i−1−j|

. (2.14)

Fixn_i−1−j=m. Thenm_i > m >0 and 2(n−j )=i(i−1)+2m≥(i−1/2)². Hence, from the above inequality (recall thatmi=ifori < p)

p i=1

ni−1−1 j=n_i+1

D_i,j≤

p−1

m=1

X₀W_m1+ W_m1

^p

i=m+1

2^s/2

i−1 2

₋s

.

Now, from the convexity ofx^−s on]0,+∞[, p

i=m+1

i−1

2 ₋s

≤ _p

m

x^−sds≤ 1 s−1m^1−s

whence p i=1

n_i−1−1 j=ni+1

Di,j≤

p−1

m=1

2m^1−s

X0Wm1+ Wm1

. (2.15)

From (2.11), (2.13) and (2.15), we get Lemma 2.2.

Theorem 2.1(a) follows from both (2.6), (2.9) and Lemma 2.2 applied toZ0=X₀²−1. Theorem 2.1(b) is a conse-

quence of (a).

2.2. Projective criteria

In this section we give estimates of the rates of convergence in the mean CLT for stationary sequences satisfying projectiveL¹-criteria in the style of Gordin [14]. Our main result is Theorem 2.2 below.

Theorem 2.2. Let (X_i)_i∈Zbe a stationary sequence of centered random variables in L³ such thatE(X₀X_k|I)= E(X0Xk)a.s.for any integerk.Suppose furthermore that the sequenceX0E0(Sn)converges inL¹.Then the series E(X₀²)+2_∞

k=1E(X₀X_k)is convergent to some nonnegative realσ².Let Z₀=X₀²−σ²+2 lim

n X₀E₀(S_n), Z_l=Z₀◦T^l and W_m=E₀(Z₁+Z₂+ · · · +Z_m).

Suppose thatσ²>0.LetΛ=σ⁻²E|X0|³.Then d₁

S_n, σ√ nY

≤13σ

6 +Λ

6 log(1+2n)+^[

√2n]

m=1

X0W_m1+2σW_m1

mσ² +D,

where D=

n m=1

1 σ√

m

l≥m

X0E0(Xl) 1

+ n m=1

1

2m1+σ⁻²X₀²

E0(Sm)

1.

Remark 2.2. By Theorem1 in[8], the convergence in L¹ ofX0E0(Sn)implies the convergence in distribution of n^−1/2S_nto a mixture of Gaussian random variables.From[6]it also implies thatn^−1/2E₀(S_n)converges to0inL¹. Consequently,ifX²₀E0(Sn)converges inL¹ asn→ ∞,thenD=o(√

n).Moreover,from theL¹-ergodic theorem, (W_m/m)and(X₀W_m/m)converge to0inL¹under the above additional condition.In that case,Theorem2.2gives a rate of convergence in the mean CLT.

Proof of Theorem 2.2. Dividing the random variables byσ, we may assume thatσ =1. From (2.6) and and (2.9), for any 1-Lipschitzian functionf,

Δ(f )≤ n k=1

E

f_k(Sk−1)Xk+1

2f_k(Sk−1)

X_k²−1

+ρ(n), (2.16)

where the functionsf_kare defined in Notation 2.1 of Section 2.1. In order to bound up the terms of first order, we write f_k(S_k−1)=f₀(0)+

k−1

j=1

f_j+1(S_j)−f_j(S_j−1) .

Next

f_j+1(Sj)−f_j(Sj−1)=

f_j(Sj)−f_j(Sj−1)

−Ej

f_j+1(Sj+Y )−f_j+1(Sj)

=f_j(Sj−1)Xj+R_j, (2.17)

whereR_jis someMj-measurable random variable such that R_j ≤(2n−2j )⁻¹

X²_j+1

. (2.18)

SetUj,n=Ej(Sn−Sj). From (2.17) and (2.18) n

k=1

E

f_k(Sk−1)Xk

≤

n−1

j=1

E

f_j(Sj−1)XjUj,n

+(2n−2j )⁻¹1+X²_j Uj,n

1

.

Next E

f_j(S_j−1)X_jU_j,n

≤E

f_j(S_j−1)X_jU_j,∞

+(n−j+1)^−1/2

l>n

X_jE_j(X_l) 1

with the conventionXjUj,∞=limnXjUj,ninL¹. From the stationarity, (2.16) and the above inequalities we get that Δ(f )≤ρ(n)+1

2 n j=1

E

f_j(S_j−1)Z_j

+D₁+D₂, (2.19)

where D₁=

n m=1

m^−1/2

l≥m

X0E0(X_l) 1

and D₂ = n m=1

1

2m1+X₀²

E0(S_m)

1.

Theorem 2.2 follows then from (2.19) and Lemma 2.2 applied withs=2.

2.3. Applications to bounded random variables

Throughout this subsection we assume thatX₀ belongs to L^∞, and that E₀(S_n)converges in L¹. Then the series X0E0(S_n)converges inL¹and consequently Theorem 2.2 applies. Set

J₀= lim

n→∞E₀(S_n) and J_m=J₀◦T^m. (2.20)

We first provide a rate which involves the quantitiesE₀(m⁻¹S_m²)−σ²1appearing in the conditional CLT of [6].

Theorem 2.3. Let (X_i)_i∈Z be a stationary sequence of centered and bounded random variables such that E(X₀X_k|I)=E(X₀X_k)a.s.for any integerk.Suppose furthermore that the sequenceE₀(S_n)converges in L¹to J0.Then the seriesE(X₀²)+2_∞

k=1E(X0X_k)is convergent to some nonnegative realσ²andn⁻¹VarS_nconverges toσ².Suppose thatσ²>0and letL=σ⁻¹X0∞.

(a) IfS=

m≥0E0(Jm)1<∞,then d₁

S_n, σ√ nY

≤Clog(1+2n)+

[√ 2n]

m=1

2+L mσ

E₀(S_m²)−mσ²

1

for some constantCdepending only onX0∞,σ andS.

(b) IfE0(Jm)1≤Mδm^−δfor someδ∈ ]0,1[and some constantMδ,then

d₁ S_n, σ√

nY

≤C_δn^(1−δ)/2+

[√ 2n]

m=1

2+L mσ

E₀ S_m²

−mσ²

1

for some constantCδdepending onδ,Mδ,X0∞andσ.

Remark 2.3. The assumptions made in this section ensure that

nlim→∞E0

m⁻¹S²_m

−σ²

1=0,

which is the condition appearing in the conditional CLT of[6].Consequently Theorem2.3provides rates of conver- gence in the mean CLT.For example,if(a)holds andsup_m>0E₀(S_m²)−mσ²1<∞,thend₁(S_n, σ√

nY )=O(logn).

If (b)holds andE₀(S_m²)−mσ²1=O(m^1−δ)asm→ ∞,thend₁(S_n, σ√

nY )=O(n^(1−δ)/2).

Proof of Theorem 2.3. We first bound upD. LetM=sup_m>0E0(S_m)1. We have that D≤L

n m=1

m^−1/2

l≥m

E₀(X_l) 1

+1 2

1+L²

Mlog(1+2n).

Since

l≥mE₀(X_l)=E₀(

l≥mE_m−1(X_l))=E₀(J_m−1), we infer that D≤L

n−1

m=0

(m+1)^−1/2E₀(J_m)

1+1 2

1+L²

Mlog(1+2n). (2.21)

Next we bound up the r.v.’sW_m+mσ²−E0(S_m²)inL¹. By definition ofW_m, Wm+mσ²=E0

S_m² +2

m l=1

E0

Xl

k>m

El(Xk)

.

Therefore

W_m+mσ²−E₀ S_m²

1≤2 m l=1

X_lE_l(J_m)

1≤2X_0∞ m l=1

E₀(J_m−l)

1.

Hence

[√ 2n]

m=1

mσ²₋1

X0W_m1+σW_m1

≤

[√ 2n]

m=1

2+L mσ

E₀ S_m²

−mσ²

1+2X_0∞

m−1 l=0

E₀(J_l)

1

. (2.22)

Theorem 2.3 follows then easily from both Theorem 2.2, (2.21) and (2.22).

We now give an application of Theorem 2.3 to sequences satisfying projective criteria in the style of [13,14]. The proof, being elementary, is omitted.

Corollary 2.1. Let(Xi)i∈Zbe a stationary sequence of centered and bounded random variables.

(a) If_∞

m=0

_m

l=0E_−m(X₀X_l)−E(X₀X_l)1<∞and

m>0mE₀(X_m)1<∞,then the series of covariances converges toσ²andd₁(S_n, σ√

nY )=O(logn)asntends to∞,provided thatσ=0.

(b) If,for someδ∈]0,1[, supl∈[0,m]E_−m(X0X_l)−E(X0X_l)1=O(m^−1−δ)andE0(X_m)1=O(m^−1−δ),then the series of covariances converges toσ²andd₁(S_n, σ√

nY )=O(n^(1−δ)/2)asntends to∞provided thatσ=0.

Remark 2.4. For example,if the strong mixing coefficientsα2(k)of the sequence(X_i)_i∈Z(see(3.1)for the definition) satisfyα2(k)=O(k^−1−δ)then Corollary2.1(b)applies and provides the rate of convergenceO(n^−δ/2)in the mean CLT.