Hölderian invariance principle for hilbertian linear processes

DOI:10.1051/ps:2008011 www.esaim-ps.org

H ¨ OLDERIAN INVARIANCE PRINCIPLE FOR HILBERTIAN LINEAR PROCESSES

Alfredas Raˇ ckauskas

and Charles Suquet

Abstract. ^{Let (}ξn)_n≥1 be the polygonal partial sums processes built on the linear processes Xn=

i≥0ai(n−i), n ≥ 1, where (i)_i∈Z are i.i.d., centered random elements in some separable Hilbert space H and the ai’s are bounded linear operators H → H, with

i≥0ai < ∞. We investigate functional central limit theorem forξnin the H¨older spaces H^o_ρ(H) of functionsx: [0,1]→Hsuch that x(t+h)−x(t)=o(ρ(h)) uniformly int, whereρ(h) =h^αL(1/h), 0≤h≤1 with 0< α≤1/2 and Lslowly varying at infinity. We obtain the H^o_ρ(H) weak convergence ofξnto someHvalued Brownian motion under the optimal assumption that for anyc >0,tP(0> ct^1/2ρ(1/t)) =o(1) whenttends to infinity, subject to some mild restriction on Lin the boundary caseα= 1/2. Our result holds in particular with the weight functionsρ(h) =h^1/2ln^β(1/h),β >1/2.

R´esum´e. ^{Soit (ξ}n)_n≥1 le processus polygonal de sommes partielles bˆati sur le processus lin´eaire Xn =

i≥0ai(n−i), n ≥ 1, les (i)_i∈Z ´etant des ´el´ements al´eatoires i.i.d., centr´es d’un espace de Hilbert s´eparableHet lesai’s des op´erateurs lin´eaires born´esH→H, v´erifiant

i≥0ai<∞. Nous

´

etudions le th´eor`eme limite central fonctionnel pourξndans les espaces de H¨older H<sup>o</sup><sub>ρ</sub>(H) de fonctions x: [0,1]→H v´erifiantx(t+h)−x(t)=o(ρ(h)) uniform´ement ent, o`uρ(h) =h^αL(1/h), 0≤h≤1 avec 0 < α ≤ 1/2 et L `a variation lente. Nous prouvons la convergence en loi dans H^o_ρ(H) de ξn

vers un mouvement brownien `a valeurs dans H, sous la condition optimale que pour tout c > 0, tP(0> ct<sup>1/2</sup>ρ(1/t)) =o(1) quandttend vers l’infini, au prix dans le cas limiteα= 1/2 d’une l´eg`ere restriction surL. Notre r´esultat s’applique en particulier au casρ(h) =h^1/2ln^β(1/h),β >1/2.

Mathematics Subject Classification. 60F17, 60B12.

Received December 14, 2007. Revised March 11, 2008.

Introduction

Let us denote by C[0,1] = C([0,1],R) the space of continuous functions x: [0,1] →R, endowed with the supremum norm. The classical Donsker-Prohorov invariance principle states the C[0,1]-weak convergence to

Keywords and phrases.Central limit theorem in Banach spaces, H¨older space, functional central limit theorem, linear process, partial sums process.

∗Research supported by a French-Lithuanian cooperation agreement “PHC Egide Gilibert”.

1 Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania;

[email protected]

2 Institute of Mathematics and Informatics, Akademijos str. 4, 08663 Vilnius, Lithuania

3 Laboratoire P. Painlev´e, UMR 8524 CNRS, Universit´e Lille I, Bˆat. M2, Cit´e Scientifique, 59655 Villeneuve d’Ascq Cedex, France;[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2009

some Brownian motion W of the polygonal line ξ_n built on the partial sums of a centered, square integrable, i.i.d. sequence (X_i)_i≥1of real random variables. This result has a lot of applications, especially in statistics, and continues to receive many extensions. Our current contribution involves three directions of extension, dealing with:

• infinite dimensionalX_i’s;

• other topological frameworks than C[0,1] for the weak convergence ofξ_n;

• dependentX_i’s.

When the X_i’s are i.i.d. random elements in some separable Banach space (B, ), we say that X1 satisfies the central limit theorem in B, denoted by X1 ∈ CLT(B), if n^−1/2S_n := n^−1/2(X1+· · ·+X_n) converges in distribution inB (the limit is then necessarily some Gaussian random element inB). It is well-known that the central limit theorem inB is not a direct extension of the finite dimensional case. Depending on the geometry of the spaceB, one can even find some bounded random elementX1which does not satisfy the CLT, seee.g.[8].

In the nice case whereB is a Hilbert space,X1 ∈CLT(B) is equivalent toEX1 = 0 andEX1² <∞. The invariance principle inBinherits the geometric pathologies of the CLT inBin the following sense. Denoting by ξ_nthe polygonal line with vertices (k/n, S_k), we say thatX1∈FCLT(B) ifn^−1/2ξ_n converges in distribution to someB-valued Brownian motion in the space C([0,1], B) of continuous functions [0,1]→B. In 1973, Kuelbs [6]

proved thatX1∈FCLT(B) if and only ifX1∈CLT(B).

Replacing C([0,1],R) or C([0,1], B)’s topology by a stronger one, leads to invariance principles of a wider scope than in the classical setting. Indeed this provides more continuous functionals of the paths of ξ_n. For instance, invariance principles in H¨older spaces have statistical applications to the detection of a changed segment in data [13,15]. Let us recall the first H¨olderian invariance principle, obtained by Lamperti [7]. For 0 < α < 1, let H^o_α[0,1] = H^o_α([0,1],R) be the vector space of continuous functions x : [0,1]→ R such that

δ→lim0ω_α(x, δ) = 0, where

ω_α(x, δ) = sup

s,t∈[0,1], 0<t−s<δ

|x(t)−x(s)|

|t−s|^α · H^o_α[0,1] is a separable Banach space when endowed with the norm

x_α:=|x(0)|+ω_α(x,1).

Lamperti proved that if 0< α <1/2 andE|X1|^p<∞, wherep > p(α) := 1/(1/2−α), thenn^−1/2ξ_n converges in distribution toW in the space H^o_α[0,1]. This result was completed and extended by the authors [12] to the case of Banach space valuedX_i’s in the following way. Putρ(h) =h^αL(1/h), 0≤h≤1 with 0< α≤1/2 and L slowly varying at infinity. Let H^o_ρ(B) = H^o_ρ([0,1], B) be the H¨older space of functions x: [0,1]→ B, such that ||x(t+h)−x(t)|| =o(ρ(h)), uniformly int (the relevant H¨older norm and the technical assumptions on L are explicited below in Section 1). Then n^−1/2ξ_n converges weakly to some B-valued Brownian motion in the space H^o_ρ(B)if and only if X1∈CLT(B) and for every positive c, lim_t→∞tP{X1> cθ(t)} = 0, where θ(t) :=t^1/2ρ(1/t). In the familiar case whereρ(h) =h^α, the second condition is equivalent toP{X1> t}= o(t^−p(α)).

In view of statistical applications, there is an obvious interest in extending the invariance principles beyond the classical case of i.i.d. observations. A recent survey of invariance principles in C[0,1] for stationary se- quences is [10]. For invariance principles under various weak dependence conditions, let us also mention [3].

Hamadouche [4] gives some H¨olderian invariance principles for real valued α-mixing or associated X_i’s. In a recent contribution, Juodis et al. [5] investigate the invariance principle in H^o_α for some linear processes X_i=

j≥0a_ji−j, where (_j)_j∈Zare i.i.d., centered and square integrable random variables with

j≥0a²_j <∞.

When

i≥0|ai| <∞, they show that n^−1/2ξ_n converges weakly in H^o_α[0,1] to some Brownian motion under the optimal assumption thatP{|0| ≥t}=o(t^−p(α)).

A natural extension of linear process to infinite dimensional spaces is linear process in the separable Hilbert space H, obtained by replacing the constants a_j by continuous linear operatorsH → H, acting on the i.i.d.

random elements_j’s inH(for a formal definition, see (1.5) below). Merlev`edeet al. [9] obtained optimal central limit theorem for such processes and this result was completed by Dedecker and Merlev`ede [2] who established the corresponding functional central limit theorem in C([0,1],H). The present contribution investigates the functional central limit theorem for such linear processes in the space H^o_ρ(H). Our main result extends the short memory case in [5] both with infinite dimensional setting and general H¨olderian weights functionsρ.

The paper is organized as follows. The main result together with all the preliminary material is presented in Section 1. Proofs are given in Section 2 which starts by a general methodology to prove invariance principles in H^o_ρ(B) (Theorem2.1 which may be of independent interest). Technical lemmas and tools are gathered in Section 3.

1. Results 1.1. Notations

Let (B, ) be a separable Banach space. A B-valued Brownian motion W_Q with covariance operator Q is a Gaussian process indexed by [0,1], with independent increments such that W_Q(t)−W_Q(s) has the same distribution as|t−s|^1/2Y, whereY is a centered Gaussian random element inB with covariance operatorQ.

We write C(B) for the Banach space of continuous functions x : [0,1] → B endowed with the supremum norm x∞ := sup{x(t); t∈[0,1]}. Letρbe a real valued non decreasing function on [0,1], null and right continuous at 0, positive on (0,1]. Put

ω_ρ(x, δ) := sup

s,t∈[0,1], 0<t−s<δ

x(t)−x(s) ρ(t−s) ·

We associate toρthe H¨older space

H^o_ρ(B) :={x∈C(B); lim

δ→0ω_ρ(x, δ) = 0}, equipped with the norm

xρ:=x(0)+ω_ρ(x,1).

To discard triviality, we may assume that ρ(h) ≥ ch for some positive constant c. Then H^o_ρ(B) inherits the separability ofB (see [11]). As in [12], we shall restrict our study to the case of weight functionsρin the class Rdefined below. For anyρin R, the space H^o_ρ(B) supports anyB-valued Brownian motion.

Definition 1.1. Let R be the class of non decreasing functions ρ : [0,1] → R, positive on (0,1], such that ρ(0) = 0 and satisfying

i) for some 0< α≤1/2, and some positive functionLwhich is normalized slowly varying at infinity,

ρ(h) =h^αL(1/h), 0< h≤1; (1.1)

ii) θ(t) =t^1/2ρ(1/t) isC¹ on [1,∞);

iii) there is aβ >1/2 and somea >1, such thatθ(t) ln^−β(t) is non decreasing on [a,∞).

We say that a function is ultimately decreasing or increasing or non decreasing or non increasing if the corresponding monotonicity holds on some interval [c,∞).

Remark 1.2. Clearly L(t) ln^−β(t) is normalized slowly varying for any β >0, so when α <1/2,t^1/2−αL(t) ln^−β(t) is ultimately non decreasing and iii) is automatically satisfied.

The assumption ii) ofC¹regularity forθ is not a real restriction, since the functionρ(1/t) being α-regularly varying at infinity (that isρ(1/t) =t^−αL(t), t≥1) is asymptotically equivalent to aC^∞α-regularly varying function ˜ρ(1/t) (see [1]). Then the corresponding H¨olderian norms are equivalent.

Put b := inf_t≥1θ(t). Since by iii), the function θ(t) is ultimately increasing and lim_t→∞θ(t) =∞, we can define its generalized inverseϕon [b,∞) by

ϕ(u) := sup{t≥1; θ(t)≤u}. (1.2)

With this definition, we haveθ(ϕ(u)) =uforu≥bandϕ(θ(t)) =tfort≥a.

The following notation is convenient for the various weak convergences considered in the paper. Let X be some separable Banach space and (Y_n)_n≥1be a sequence of random elements in X. We write

Y_n−−−−→^X

n→∞ Y,

for the weak convergence of (Y_n)_n≥1in the spaceXto the random elementY,i.e. Ef(Y_n) converges toEf(Y) for any continuous and boundedf :X→R.

For the sequence (X_n)_n≥1 of random elements in the separable Banach spaceB, put S0:= 0, S_n:=

n i=1

X_i (1.3)

and define the partial sums processξ_n by

ξ_n(t) :=S[nt]+ (nt−[nt])X[nt]+1, t∈[0,1], (1.4) where [nt] denotes the integer part ofnt. As polygonal lines, the paths ofξ_n belong to H^o_ρ(B) for everyρinR since thenρ(h)≥chfor some constantc >0.

In this paper we consider the case where (X_k)_k≥0 is a linear process with values in the separable Hilbert spaceHof the form

X_k= ∞ i=0

a_i(_k−i), k= 0,1, . . . , (1.5)

where (a_i, i∈Z) is a given sequence of continuous linear operatorsH→Hwitha_i= 0 fori <0 and (_i, i∈Z) is a sequence of independent identically distributed random elements inHwithE0= 0 andE0²<∞. We shall abbreviate the notationa_i(_k−i) ina_ik−i. In the same spirit, we use the same notation for the norm in Hand the operator norm on the space of continuous linear operatorsH→H. If we assume that

i∈Zai<∞ then the series in (1.5) converges almost surely in the strong topology ofHand its sumX_kis a random element ofH. This follows by Itˆo-Nisio theorem (seee.g.[8], p. 151), since E||

ia_ik−i||² ≤E²₀

i||ai||².Moreover (X_k)_k≥0 is stationary.

1.2. Main result

Theorem 1.3. Let (X_k)_k≥0 be the linear process defined by (1.5) and assume that (a_i)_i≥0 satisfies:

∞ i=0

ai<∞. (1.6)

Define the continuous linear operator

A:=

∞ i=0

a_i, (1.7)

assume that A = 0 and denote by A^∗ its adjoint operator. Write K for the covariance operator of the square integrable random element0 inH. LetS_nandξ_n be the partial sums and partial sums process built on(X_k)_k≥0, defined by (1.3) and (1.4). Then for everyρ∈ R,

n^−1/2ξ_n ^H

oρ(H)

−−−−→

n→∞ W_Q, (1.8)

whereW_Q is aHvalued Brownian motion with covariance operatorQ=AKA^∗, if for every c >0,

t→∞lim tP{0> cθ(t)}= 0. (1.9)

Condition (1.9) is optimal because the class of linear processes considered includes the special case whereX_k =_k and it is known from [12] that in this case (1.9) is necessary for the weak-H^o_ρ(H) convergence ofn^−1/2ξ_n toW_Q. It is easily seen that ifα <1/2 in (1.1), then we can drop the requirement “for everyc >0” in (1.9) and simply takec= 1. But this requirement cannot be dropped ifα= 1/2, see Remark 12 in [12].

To illustrate Theorem 1.3, it seems worth focusing on the cases ρ(h) = h^α, 0 < α < 1/2 and ρ(h) = h^1/2ln^β(b/h) whereβ >1/2 and bis some positive constant chosen so thatρincreases on [0,1].

Corollary 1.4. If ρ(h) =h^α,0< α <1/2in Theorem 1.3, the convergence (1.8) holds if (1.9) is replaced by

t→∞lim t^pP{0> t}= 0, (1.10)

wherep= (1/2−α)⁻¹.

Corollary 1.5. Ifρ(h) =h^1/2ln^β(b/h),β >1/2in Theorem1.3, the convergence (1.8) holds if (1.9) is replaced by

E exp

d0^1/β

<∞, for eachd >0. (1.11)

2. Proofs 2.1. General reduction

We describe here a general method to establish the weak-H^o_ρ(B) convergence of the partial sums process v_n⁻¹ξ_n built on random elementsX_n of the separable Banach spaceB. This may be of independent interest to prove invariance principles under various kind of dependence of the underlying sequence (X_n)_n≥1.

The functionρis assumed to belong toRall along the paragraph. According to [14],v_n⁻¹ξ_n converges weakly toξ in H^o_ρ(B) if and only if

a) the “finite dimensional” distributions of v⁻_n¹ξ_n converge to those ofξ;

b) the sequence (v⁻_n¹ξ_n)_n≥1is tight in H^o_ρ(B).

The convergence of “finite dimensional distributions” in a) means namely v⁻_n¹

ξ_n(s1), . . . , ξ_n(s_m) _B^m

−−−−→

n→∞

ξ(s1), . . . , ξ(s_m) ,

form≥1 and 0≤s1<· · ·< s_m≤1. This terminology is used here as a convenient analogy with the classical caseB=R. But it should not conceal that in general, problems of infinite dimensional weak convergence may appear already at this stage in connection with some central limit theorem in B, involving the geometry of B. Of course if v_n⁻¹ξ_n satisfies already a functional central limit theorem in the space C(B), condition a) is automatically fulfilled.

Let us discuss now the tightness problem. General conditions implying the tightness of a sequence of random elements in H^o_ρ(B) may be found in [14] (Th. 2 and Rem. 1). To translate this result in the setting of partial sums processξ_n, write for simplicity

t_k=t_j,k=k2^−j, k= 0,1, . . . ,2^j, j= 1,2, . . . Then the tightness of (v⁻_n¹ξ_n)_n≥1in H^o_ρ(B) is easily seen to hold under the conditions:

i) for everyt∈[0,1], (v_n⁻¹ξ_n(t))_n≥1 is tight onB;

ii) for every positiveε,

J→∞lim lim sup

n→∞ P

supj≥J

1

v_nρ(2^−j) max

0≤k<2^jξ_n(t_k+1)−ξ_n(t_k) ≥ε = 0.

The following theorem provides a practical way to reduce the checking of ii). It is worth noticing thatnothing is assumed about the dependence structure of (X_n)_n≥1 in its statement. Here and throughout the paper, logn stands for the logarithm with basis 2, so that 2^logn=n.

Theorem 2.1. Letξ_n be the partial sums process built on(X_k)_k≥0, defined by (1.4). Then (v⁻_n¹ξ_n)_n≥1 is tight in H^o_ρ(B)if:

(1) for every t∈[0,1],(v_n⁻¹ξ_n(t))_n≥1 is tight on B;

(2) 1

v_nρ(1/n) max

1≤i≤nX_i converges in probability to0;

(3) for every positiveε,

J→∞lim lim sup

n→∞ P

J≤j≤maxlogn

1

v_nρ(2^−j) max

0≤k<2^jS[ntk+1]−S[ntk]≥ε = 0.

If the X_i’s have identical distribution, then Condition 2 can be replaced by

∀ε >0, nP

X1 ≥εv_nρ(1/n)

−−−−→

n→∞ 0. (2.1)

Clearly under identical distribution of theX_i’s, (2.1) implies Condition 2. Moreover when (2.1) is sufficient for (v_n⁻¹ξ_n)_n≥1 to satisfy the invariance principle in C(B), then we can drop Condition 1 and concentrate on the verification of (2.1) and Condition 3 to prove the invariance principle in H^o_ρ(B).

Proof of Theorem 2.1. We have to check ii). Denote byP0=P0(J, n) the probability appearing in Condition ii).

ThenP0is bounded byP1+P2 where P1:=P

J≤j≤maxlogn

1

v_nρ(2^−j) max

0≤k<2^jξ_n(t_k+1)−ξ_n(t_k) ≥ε and

P2:=P

j>suplogn

1

v_nρ(2^−j) max

0≤k<2^jξ_n(t_k+1)−ξ_n(t_k) ≥ε .

Estimation ofP2. Asj >logn,t_k+1−t_k= 2^−j<1/nand then witht_k in say [l/n,(l+ 1)/n), eithert_k+1 is in (l/n,(l+ 1)/n] or belongs to

(l+ 1)/n,(l+ 2)/n

, where 1≤l≤n−2 depends onkandj.

In the first case, computingξ_n(t_k+1)−ξ_n(t_k) by linear interpolation ofξ_n betweenξ_n(l/n) andξ_n((l+ 1)/n), we obtain

ξn(t_k+1)−ξ_n(t_k)=nXl+12^−j≤2^−jn max

1≤i≤nXi.

Ift_k andt_k+1 are in consecutive intervals, then

ξn(t_k+1)−ξ_n(t_k) ≤ ξn(t_k)−ξ_n((l+ 1)/n)+ξn((l+ 1)/n)−ξ_n(t_k+1)

≤

(l+ 1)/n−t_k+t_k+1−(l+ 1)/n n max

1≤i≤nXi

= 2^−jn max

1≤i≤nX_i.

Recalling that θ(t) =t^1/2ρ(1/t) is ultimately non decreasing, this estimate ofξn(t_k+1)−ξ_n(t_k) leads to P2≤P

j>suplogn

1

v_nρ(2^−j)n2^−j max

1≤i≤nXi ≥ε

=P n^1/2

v_n max

1≤i≤nX_i sup

j>logn

1

2^j/2ρ(2^−j)(n2^−j)^1/2≥ε

≤P n^1/2

v_nθ(n) max

1≤i≤nX_i ≥ε

=P 1

v_nρ(1/n) max

1≤i≤nX_i ≥ε , fornlarge enough, whence by Condition 2, lim_n→∞P2= 0.

Estimation ofP1. Letu_k = [nt_k]. Thenu_k ≤nt_k≤1 +u_k and 1 +u_k≤u_k+1≤nt_k+1≤1 +u_k+1. Therefore ξ_n(t_k+1)−ξ_n(t_k) ≤ ξ_n(t_k+1)−S_uk+1+S_uk+1−S_uk+S_uk−ξ_n(t_k).

SinceSuk−ξ_n(t_k) ≤ X1+uk andξn(t_k+1)−S_uk+1 ≤ X1+uk+1 we obtainP1≤P1,1+ 2P1,2, where P1,1:=P

J≤j≤maxlogn

1

v_nρ(2^−j) max

1≤k≤2^jS_uk+1−S_uk ≥ ε 2 P1,2:=P

J≤j≤maxlogn

1

v_nρ(2^−j) max

1≤i≤nXi ≥ ε 4 .

InP1,2, the maximum overj is realized forj= [logn], so lim_n→∞P1,2= 0 by Condition 2.

Gathering all the estimates, we finally obtain

J→∞lim lim sup

n→∞ P0= lim

J→∞lim sup

n→∞ P1,1= 0,

by Condition 3.

2.2. Proof of Theorem 1.3

We need to check the convergence of finite dimensional distributions and tightness.

The invariance principle inC(H) is established under (1.6) by Dedecker and Merlev`ede [2] as a special case of their Theorem 5 (see also in [10] Prop. 17 and the discussion p. 21). From thisC(H) invariance principle, we already have the convergence of finite dimensional distributions ofn^−1/2ξ_n and Condition 1 of our Theorem2.1 is satisfied. So it remains to check (2.1) and Condition 3.

First we note that our assumption (1.9) implies via Lemma3.7below that for every positive constantc,

t→∞lim tP

X0 ≥cθ(t)

= 0.

So it remains only to check Condition 3, that is lim_J→∞lim sup_n→∞P1(J, n, ε) = 0, with P1(J, n, ε) =P

J≤j≤maxlogn

1

n^1/2ρ(2^−j) max

0≤k<2^jS_uk+1−S_uk≥ε , (2.2) whereu_k= [nt_k] = [nk2^−j]. It is useful to note here that asj≤logn,

1≤u_k+1−u_k ≤n2^−j+ 1≤2n2^−j, 0≤k <2^j. (2.3) Let us fix an arbitraryδ >0 and define

_l:=_l1{l ≤δθ(n)} −E_l1{l ≤δθ(n)}, (2.4)

_l:=_l1{l> δθ(n)} −E_l1{l> δθ(n)}. (2.5) SinceE_l= 0,_l=_l+_land we have

S_uk+1−S_uk= ∞ l=−∞

b_k,ll=Z_j,k⁽¹⁾+Z_j,k⁽²⁾,

where

Z_j,k⁽¹⁾= ∞ l=−∞

b_k,ll, Z_j,k⁽²⁾= ∞ l=−∞

b_k,ll (2.6)

and

b_k,l:=

uk+1

i=uk+1

a_i−l. (2.7)

Hence, we have

P1(J, n, ε)≤P₁⁽¹⁾(J, n, ε, δ) +P₁⁽²⁾(J, n, ε, δ), (2.8) where fori= 1,2,

P₁⁽ⁱ⁾(J, n, ε, δ) :=P

J≤j≤maxlogn

1

ρ(2^−j) max

0≤k<2^jZ_j,k⁽ⁱ⁾> ε 2n^1/2

.

Estimation ofP₁⁽²⁾(J, n, ε, δ). First we apply Chebyshev inequality to obtain P₁⁽²⁾(J, n, ε, δ)≤

J≤j≤logn

4 ε²nρ(2^−j)²

0≤k<2^j

EZ_j,k⁽²⁾². (2.9)

Next observe that from Lemma3.1below, there is some constantc0 such that for any positive integerm ∞

l=−∞

^m

i=1

a_i−l

²≤c0m. (2.10)

In view of the Hilbertian structure ofHand recalling (2.3), we get then EZ_j,k⁽²⁾²=

∞ l=−∞

Eb_k,ll²

≤ ∞ l=−∞

bk,l²E_l²

≤c0(u_k+1−u_k)E0²

≤2n2^−jc0E0².

Going back to (2.9) with the estimates provided by Lemmas3.2and 3.3below, we obtain forn≥J ≥J0, P₁⁽²⁾(J, n, ε, δ)≤ Kδ²

ε² sup

t≥ntP

0 ≥δθ(t) ,

where the integerJ0depends onρonly while the constantK depends onρand on the sequence (a_i)_i≥0. Thus (1.8) gives for every positiveδ

n→∞lim P₁⁽²⁾(J, n, ε, δ) = 0. (2.11)

Estimation ofP₁⁽¹⁾(J, n, ε, δ). Using (2.3), we get P₁⁽¹⁾(J, n, ε, δ)≤

J≤j≤logn

P

1

(2n2^−j)^1/2 max

0≤k<2^jZ_j,k⁽¹⁾≥ ε 2√

2θ(2^j)

≤

J≤j≤logn

P

0≤k<max2^j

Z_j,k⁽¹⁾

(u_k+1−u_k)^1/2 ≥ ε 2√

2θ(2^j)

≤

J≤j≤logn

0≤k<2^j

P

Z_j,k⁽¹⁾

(u_k+1−u_k)^1/2 ≥ ε 2√

2θ(2^j)

. (2.12)

In order to use an exponential inequality forZ_j,k⁽¹⁾, we need an upper bound for some Orlicz norm (see Section3 for the relevant material). According to Lemma 3.5below, for 1< γ≤2, we have

Z_j,k⁽¹⁾

ψγ ≤K_γEZ_j,k⁽¹⁾+K_γ

l∈Z

b_k,l^γ

1/γ δθ(n) ln^1/γ{ϕ

δθ(n)

}, (2.13)

where _γ¹+_γ¹ = 1 andb_k,lis defined by (2.7).

To control the first term in the bound (2.13), we get by independence of theb_ll’s, Hilbertian structure ofH and (2.10),

EZ_j,k⁽¹⁾≤

EZ_j,k⁽¹⁾²1/2

=

l∈Z

Ebk,l_l² 1/2

≤

l∈Z

bk,l² 1/2

E0²1/2

≤2

E0²1/2

c0(u_k+1−u_k)1/2

.

Next, asγ≥2, we get withM :=

i∈Za_i,

l∈Z

b_k,l^γ ≤M^γ−2c0(u_k+1−u_k).

Implanting these estimates into (2.13), we obtain

Z_j,k⁽¹⁾ (u_k+1−u_k)^1/2

ψγ

≤K

1 +

n2^−j1/2−1/γ δθ(n) ln^1/γϕ

δθ(n)

, (2.14)

with a constantK depending onγ, on (a_i)_i∈Zand on the distribution ofX1.

From this point, the remainder part of a detailed proof would be an exact reproduction of the corresponding part in the proof of Theorem 8 in [12], pp. 235–237. So we shall content ourselves with providing some explanation on the role of the parameter γ. Using iii) in the definition of the class R we can choose some β >1/2 such thatθ(t) ln^−β(t) is ultimately non decreasing. Then we require that 1/2<1/γ < β. Then going back to (2.12) with the exponential inequality resulting from (2.14) and (3.10) leads after some work to

P₁⁽¹⁾(J, n, ε, δ)≤ 2e^−J

1−e⁻¹+ 4nexp

−clnϕ δθ(n) δ^γ

,

forn≥J ≥j0, provided thatγ <(1−α)⁻¹, wherej0is defined as in [12], p.237. It is easily seen that there is always a choice ofβ making compatible both conditions imposed onγ. It is important to note here that neither j0, nor the constantcdepend onδ. Next forδ <1 andnlarge enough we have from [12] or Lemma3.6below:

lnϕ δθ(n)

δ^γlnn ≥δ^1/β−γ, hence

4nexp

−clnϕ δθ(n) δ^γ

≤4 exp

(1−cδ^1/β−γ) lnn .

As 1/β < γ, we can finally chooseδsmall enough to make 1−cδ^1/β−γ negative. This leads to lim sup

n→∞ P₁⁽¹⁾(J, n, ε, δ)≤ 2e^−J 1−e⁻¹·

Recalling (2.8) and (2.11), the same upper bound holds for lim sup_n→∞P1(J, n, ε), so lettingJ tend to infinity ends the proof.

3. Tools and auxiliary results

The following lemma extends with a more elementary proof Lemma 1 in [9].

Lemma 3.1. If the sequence a= (a_l)_l∈Z in the Banach space (B, )satisfies a¹(B)=

l∈Z

a_l<∞, (3.1)

then

Q²_n(a) := 1 n

l∈Z

^n−l

i=1−l

a_i

²−−−−→

n→∞

i∈Z

a_i

². (3.2)

Proof. First it is easily seen by combining the triangle inequalities in B and in ²(R), that Q_n satisfies the triangle inequality in¹(B). Next an elementary computation provides

Q²_n(a)≤ a²1(B).

Both properties enable us to reduce the problem via a classical 3εargument in checking the convergence (3.2) for anyain the dense subspace¹₀(B) of sequences with finite support. Ifa∈¹₀(B), thena_i= 0 for|i|> i0, so

Q²_n(a) = 1 n

l=n+i0

l=1−i0

^n−l

i=1−l

a_i ². In the above sums, the n−2i0 blocks _n−l

i=1−la_i indexed by l = 1 +i0, . . . , n−i0 are complete, i.e. equal to A =

|i|≤i0a_i =

i∈Za_i. As it remains 4i0 incomplete blocks, each bounded in norm by a²1(B), the

convergence (3.2) follows.

Lemma 3.2. There is an integer J0 depending only onρ(h) =h^αL(1/h), such that for n≥J ≥J0, T_J,n:=

J≤j≤logn

1

ρ(2^−j)² ≤ 2 2^α−1

n

θ(n)²· (3.3)

Proof. Let us denote by m the integer part of logn and put n := 2^m, so that n/2< n ≤n. Recalling that θ(t) =t^1/2ρ(1/t), we have

T_J,n= m j=J

2^j θ(2^j)² =

m−J

l=0

n2^−l

θ(n2^−l)² = n θ(n)²

m−J

l=0

v_n,l,

withv_n,l:= 2^−lθ(n)²θ(n2^−l)⁻². To estimate the ratiov_n,l+1/v_n,l, we note that θ(2s)²

θ(s)² = 2^1−2αL(2s)² L(s)² ·

AsLis slowly varying, there is somes0depending onLandαsuch that fors≥s0,L(2s)²L(s)⁻²≤2^α, whence θ(2s)²

θ(s)² ≤2^1−α, s≥s0. (3.4)

If 2^J≥2s0, we obtainv_n,l+1/v_n,l≤2^−αfor 0≤l < m−J, therefore T_J,n≤ n

θ(n)² 1 1−2^−α·

Now, let us fix J0 large enough such that 2^J0 ≥2s0 and θ is non decreasing on [2^J0,∞). To obtain (3.3), it remains to note thatn/2< n≤nand that by (3.4),θ(n)²≥θ(n/2)²≥2^α−1θ(n)². Lemma 3.3. If lim_t→∞tP

0 ≥δθ(t)

= 0, then the random element 0 defined by (2.5) satisfies E0²≤Cδ²θ(n)²

n sup

t≥ntP

0 ≥δθ(t)

, (3.5)

where the positive constant C depends only onρ.

Proof. To get rid of the centering term in0, we use successively triangular inequality, (a+b)²≤2a²+ 2b²and (EY)²≤EY² to obtain

E0²≤4E01{0> δθ(n)}². Now we note that

E01{0> δθ(n)}²= _∞

0 2tP

01{0> δθ(n)}> t dt

=δ²θ(n)²P

0> δθ(n)

+I_n,δ, (3.6)

where

I_n,δ :=

_∞

δθ(n)2tP

0> t dt.

The substitutiont=δθ(s) gives I_n,δ=δ²

_∞

n P

0> δθ(s)

2θ(s)θ(s) ds

≤δ²sup

u≥nuP

0 ≥δθ(u)

∞ n

2θ(s)θ(s)

s ds. (3.7)

Integrating by parts and noting thatθ(s)²/svanishes at infinity, we obtain _∞

n

2θ(s)θ(s)

s ds=−θ(n)²

n +

_∞

n

ρ(1/s)² s ds≤

1/n 0

ρ(u)² u du.

The weight function ρsatisfies (see (8) in [12]) _h

0

ρ(u)

u du≤c2ρ(h), 0< h≤1.

Asρis non decreasing, this leads to _∞

n

2θ(s)θ(s)

s ds≤c2ρ(1/n)²=c2θ(n)²

n · (3.8)

Reporting the estimates (3.7) and (3.8) in (3.6) leads to the inequality (3.5) withC= 1 +c2. Let us give now some hints about the Orlicz norms used in the paper. Set forγ≥1, andXa random element in the Banach space (B, ),

X_ψγ:= inf{c >0; E exp(X/c^γ)≤2}. (3.9)

Then X_ψγ defines a norm on the space of random elements inB satisfyingE exp(X/c^γ)<∞ for somec and it is easily seen that

P{X ≥x} ≤2 exp

− x^γ X^γ_ψγ

, x >0. (3.10)

The following result provides an useful bound for the ψ_γ Orlicz norm of a finite sum of independent random elements in B.

Theorem 3.4 (Talagrand [16, Th. 4]). Let (Y_i)_i∈N be a sequence of independent mean zero random elements in the Banach space(B, ). Then for1< γ≤2, and any finite subset I of N,

i∈I

Y_i

ψγ≤K_γ

E

i∈I

Y_i+

i∈I

Yi^γ_ψγ 1/γ

,