Kahane-Khintchine inequalities and functional central limit theorem for random fields

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Kahane-Khintchine inequalities and functional central limit theorem for random fields

Mohamed El Machkouri

To cite this version:

Mohamed El Machkouri. Kahane-Khintchine inequalities and functional central limit theorem for random fields. Stochastic Processes and their Applications, Elsevier, 2002, 102 (2), pp.285-299. �hal- 00488685�

entral limit theorem for random elds

Mohamed EL MACHKOURI,

Laboratoire de Mathématiques Raphaël Salem

UMR CNRS 6085, Université de Rouen

mohamed.elmahkouriuniv-rouen.fr

01 Juin 2002

Abstrat

WeestablishnewKahane-KhinthineinequalitiesinOrlizspaesindued by

exponentialYoungfuntionsfor stationaryrealrandomeldswhiharebounded

or satisfy some nite exponential moment ondition. Next, we give suient

onditions for partial sum proesses indexed by lasses of sets satisfying some

metri entropy ondition to onverge indistribution to a set-indexed Brownian

motion. Moreover, the lass of random elds that we study inludes φ^-mixing

andmartingaledierene random elds.

AMSClassiations (2000) : 60 F05,60 F17,60 G60

Keywordsandphrases: Kahane-Khinthineinequalities,funtionalentrallimit

theorem,invarianepriniple,martingaledierenerandomelds,mixingrandom

elds,Orliz spaes,metri entropy.

1 Introdution

Let (Xk)_k∈^{Zd be a stationary eld of zero mean} real-valued random variables. If A

is a olletion of Borel subsets of [0,1]^{d, dene the smoothed partial sum proess} {Sn(A) ;A ∈ A} ^by

Sn(A) = X

i∈{1,...,n}^d

λ(nA∩Ri)Xi ⁽¹⁾

whereRi =]i1−1, i1]×...×]id−1, id] ^{is the unit ubewith upper ornerat} i ^and λ ^is

the Lebesgue measure on R^d.

The main aim of this paper is to study the asymptoti behaviour of the sequene of

proesses{S_n(A) ; A∈ A} ^{intermsof the validity ofthe funtionalentrallimittheo-}

rem(FCLT) using newKahane-Khinthine inequalities(f. setion3). More preisely,

we derive the following property: the sequene {n^−d/2Sn(A) ;A ∈ A} ^{onverges in}

distribution to a mixture of Brownian motions in the spae C(A) ^{of ontinuous real}

funtionson A ^{equipped with the metri of uniform onvergene.}

To measure the size of the olletionA ^{one usually onsiders the metri entropy with}

respet tothe Lebesgue measure. Dudley[9℄proved the existeneofa standard Brow-

nianmotion with sample pathsin the spae C(A) ^if A ^{has nite entropy integral(i.e.}

Condition (5^{) insetion 4 holds).}

The rst weak onvergene results for Q^{d-indexed partial sum proesses were estab-}

lishedintheiidasefortheolletionQ^{doflower-leftquadrantsin}[0,1]^{d,thatistosay}

theolletion{[0, t₁]×. . .×[0, t_d] ; (t₁, . . . , t_d)∈[0,1]^d}^{. TheywereprovedbyWihura}

[26℄ under a nite variane ondition and earlier by Kuelbs [17℄ under additionalmo-

mentrestritions. When thedimension d^{is reduedtoone, theseresults oinidewith}

the original invariane priniple of Donsker [7℄. In 1983, Pyke [22℄ derived a weak

onvergene result for the proess {Sn(A) ; A ∈ A} ^{in the iid ase provided that the}

olletionA ^{satises anentropy ondition with inlusion(i.e. Condition (}6^{) insetion}

4holds). However,thisFCLTrequiredmomentonditionswhihbeomemorestritas

the size of A ^{inreases. Bass [2℄ and} simultaneously Alexander and Pyke [1℄ extended Pyke's result toiid randomelds with nite variane.

Foruniformφ^-mixingandβ^{-mixingrandomelds,GoldieandGreenwood[12℄adapted}

Bass'sapproah whih ismainlybased onBernstein's inequalityfor iidrandom elds.

In1991, Chen[3℄proved a FCLTforQd^{-indexed partialsum of}non-uniformφ^-mixing

randomelds(thenon-uniformφ^{-mixingoeientswasintroduedbyDobrushinand}

Nahapetian [6℄). Reently,Dedeker[5℄gaveanL^{∞-projetive riterionforthe partial}

sumproess{n^−d/2Sn(A) ; A∈ A}^{toonvergetoan}A^{-indexedBrownianmotionwhen}

the olletionA ^{satises only the entropy ondition (}5^{)of Dudley. This new riterion}

is valid for martingale dierene bounded randomelds and provides a new riterion

for non-uniform φ^{-mixing bounded random elds. In the unbounded ase, Dedeker}

gaveanL^p-version(p >1^{)of his}L^{∞-projetiveriterionfor}Q^{d-indexedpartialsum of}

random elds with moments stritly greater than 2. Next, for non-uniform φ^-mixing

random elds, using the haining method of Bass [2℄ and establishing Bernstein type

inequalities, Dedeker proved the FCLT for the partial sum proess {Sn(A) ; A ∈ A}

provided that the olletion A ^{satises the more strit entropy ondition with inlu-}

sion (6^{) and under both nite fourth moments and an algebrai deay of the mixing}

oeients.

Inapreviouswork(see[10℄),itisshownthattheFCLTmaybenotvalidforp-integrable (0≤p <+∞^{)martingaledierenerandomelds. More preisely,thefollowingresult}

isproved.

Theorem(El Mahkouri, Volný) Let (Ω,F, µ, T) ^{be an ergodi dynamial system}

with positive entropy where Ω ^{is a Lebesgue spae,} µ ^{is a} probability measure and T ^is

a Z^d-ation. For any nonnegative real p^{, there exist a real funtion} f ∈ L^p(Ω) ^{and a}

olletion A ^{of regular Borel subsets of} [0,1]^{d suh that}

1) For any k ⁱⁿ Z^d, E f ◦T^k|σ(f ◦Tⁱ; i6=k)

= 0^{. We say that the random eld} (f ◦T^k)_k∈Zd ^{isa strong martingale dierene random eld.}

2) The olletion A ^{satises the entropy ondition with inlusion} (6)^.

3) The partial sum proess {n^−d/2Sn(f, A) ; A∈ A} ^{is not tight in the spae} C(A)

where

Sn(f, A) := X

i∈{1,...,n}^d

λ(nA∩Ri)f ◦Tⁱ.

The above theorem shows that not only Dedeker's FCLT for bounded random elds

(see [5℄) annot be extended to p-integrable (0 ≤ p < +∞^{) random elds but also it}

lays emphasis on that Bass, Alexander and Pyke's result (see [1℄, [2℄) for iid random

eldsannot hold for martingale dierenerandomelds.

In the present work, undera projetive ondition similarto Dedeker's one, we estab-

lishsome so-alledKahane-Khinthine inequalitiesforstationaryreal randomelds in

Orliz spaes indued by exponential Young funtions (f. setion 3). We require the

randomeldtobebounded ortosatisfysomeniteexponentialmomentondition (f.

Assumption(2^{)in setion3). These} inequalitiesextend previous ones forsequenes of iid bounded random variables (see for example [14℄, [15℄, [20℄). With the help of the

aboveinequalities,weareinpositiontoprovethetightnessofthesequene ofproesses

{n^−d/2S_n(A) ; A ∈ A} ^{in the spae} C(A) ^{when the olletion} A ^{satises an entropy}

ondition related to the moments of the random eld (i.e. Condition (8^{) in setion 4}

holds). The onvergene of the nite-dimensional laws is a simple onsequene of a

[4℄, Theorem 2.2).

Before presenting our results in more details, let us explain the main dierene of

our approah in tightness's proof of the sequene of proesses {n^−d/2S_n(A) ; A ∈ A}

with Dedeker's one. In fat, Dedeker's proof is based on an exponential inequal-

ity of Hoeding type derived from a Marinkiewiz-Zygmund type inequality for p^-

integrablereal randomelds(f. Inequality (11^{)insetion5)by optimizingin}p^{. That}

isthereasonwhy theboundedness onditionisneessary. Ourapproahombinesthis

Marinkiewiz-Zygmund type inequality with a property of the norm in Orliz spaes

indued by exponentialYoung funtions (f. Lemma 1^{) whih allows us to derive the}

announed Kahane-Khinthine inequalities under only the assumption of some nite

exponential moment.

2 Notations

Byastationary real randomeldwe mean any family(Xk)_k∈Z^{d of} real-valuedrandom variables dened on a probability spae (Ω,F,P) ^{suh that for any} (k, n) ∈ Z^d×N^∗

andany (i1, ..., in)∈Z^nd,the random vetors (Xi1, ..., Xin) ^and (Xi1+k, ..., Xin+k)^have

the same law.

Letµ^{bethelawofthestationaryrealrandomeld}(Xk)_k∈^{Zdandonsidertheprojetion} f ^fromR^Zd toRdened byf(ω) = ω0 ^{andthe familyof}translationoperators(T^k)_k∈^Zd

fromR^Zd toR^Zd dened by (T^k(ω))i =ωi+k ^{for any} k∈Z^d and any ω ⁱⁿR^Zd. Denote byB^theBorelσ^-algebraofR. Therandomeld(f◦T^k)_k∈Zd ^denedontheprobability spae(R^Zd,B^Zd, µ)^{isstationarywiththesamelawas}(X_k)_k∈Z^d. Consequently,without lossof generality,one an suppose that

(Ω,F,P) = (R^Zd,B^Zd, µ) ^and Xk=f◦T^k, k∈Z^d.

Anelement A ^of F ^{is said tobe invariant if}T^k(A) =A ^{for any} k ∈Z^d. We denoteby

I ^the σ^{-algebra of allmeasurable invariantsets.}

On the lattie Z^d we dene the lexiographi order as follows: if i = (i1, ..., id) ^and j = (j1, ..., jd) ^{are distint elements of} Z^d, the notation i <lex j ^{means that either} i1 < j1 ^{or for some} p ⁱⁿ {2,3, ..., d}^, ip < jp ^and iq = jq ^for 1 ≤ q < p^{. Let the sets} {V_i^k; i∈Z^d, k ∈N^∗} ^{be dened as follows:}

V_i¹ ={j ∈Z^d; j <lexi},

and fork ≥2

V_i^k=V_i¹∩ {j ∈Z^d; |i−j| ≥k} ^where |i−j|= max

1≤k≤d|i_k−j_k|.

Forany subset Γ ^of Z^d, deneF^Γ =σ(Xi; i∈Γ)^{. If} Xi ^{belongs to}L¹(P)^{, set} Ek(Xi) =E(Xi|FV_i^k).

Mixing coef f icients f or random f ields. ^{Given two} σ^-algebras U ^and V ^of F^{, dif-}

ferent measures of their dependene have been onsidered in the literature. We are

interested by one ofthem. Theφ^{-mixingoeient has been introduedby Ibragimov}

[13℄ and an be dened by

φ(U,V) = sup{kP(V|U)−P(V)k∞, V ∈ V}.

Now,let(Xk)_k∈Z^{d bearealrandomeldanddenoteby}|Γ|^{theardinalityofanysubset} Γ ^of Z^d. In the sequel, we shall use the following non-uniform φ^{-mixing oeients}

dened for any (k, l, n) ⁱⁿ (N^∗∪ {∞})²×N by

φk,l(n) = sup{φ(F^Γ1,F^Γ2), |Γ1| ≤k, |Γ2| ≤l, d(Γ1,Γ2)≥n},

where the distane d ^{is dened by} d(Γ₁,Γ₂) = min{|i−j|, i ∈ Γ₁, j ∈ Γ₂}^{. We say}

that the random eld (X_k)_k∈Zd ^is φ^{-mixingif there exists apair} (k, l) ⁱⁿ (N^∗ ∪ {∞})²

suh that limn→+∞φk,l(n) = 0^.

Formore about mixingoeients one an refer toDoukhan [8℄orRio [23℄.

Y oung f unctions and Orlicz spaces. ^{Reall that a Young funtion} ψ ^{is a real on-}

vex nondereasing funtiondened onR⁺ whih satises

t→+∞lim ψ(t) = +∞ ^and ψ(0) = 0.

We dene the Orliz spae Lψ ^{as the spae of real random variables} Z ^{dened on}

the probability spae (Ω,F,P) ^{suh that} E[ψ(|Z|/c)] < +∞ ^{for some} c > 0^{. The}

OrlizspaeLψ ^{equipped withthe so-alledLuxemburgnorm}k.k^{ψ dened foranyreal}

randomvariable Z ^by

kZk^ψ = inf{c >0 ; E[ψ(|Z|/c)]≤1}

isaBanahspae. Formore aboutYoung funtionsand Orlizspaesone an referto

Krasnosel'skiiand Rutikii [16℄.

A real random eld (X_k)_k∈Zd ^{is said to be a martingale dierene random eld if it}

satisesthe following ondition: forany m ⁱⁿZ^d

E(X_m|σ(X_k; k <_lexm) ) = 0 ^a.s.

Letβ >0^{. We denote by} ψ_β ^{the Young funtiondened for any} x∈R⁺ by

ψβ(x) = exp((x+hβ)^β)−exp(h^β_β) ^where hβ = ((1−β)/β)^1/β11{0<β<1}.

We are interested inKahane-Khinthine inequalitiesfora large lass ofrandom elds.

Infat,weshallgiveaprojetiveondition(thatistosayaonditionexpressedinterms

of a series of onditional expetations) omparable to that introdued by Dedeker to

prove a entral limit theorem for stationary square-integrable random elds (see [4℄)

and a funtionalentral limittheorem for stationary bounded random elds (see [5℄).

Consider the followingassumption :

∃q ∈]0,2[ ∃θ >0 E[exp(θ|X0|^β(q))]<+∞ ⁽²⁾

whereβ(q) = 2q/(2−q) ^forany 0< q <2^{. Our rst resultis the following.}

Theorem 1 Let (Xi)_i∈Z^{d be a zero mean stationary real random eld whih satises}

the assumption (2) ^{for some} 0 < q < 2^{. There exists a positive universal onstant} M1(q) ^{depending only on} q ^{suh that for any family} (ai)_i∈^{Zd of real numbers and any}

nite subset Γ ^of Z^d,

X

i∈Γ

aiXi

ψq

≤M1(q) X

i∈Γ

|ai|bi,q(X)

!1/2

(3)

where

bi,q(X) :=|ai| kX0k²ψβ(q)+ X

k∈V₀¹

|ak+i|

q

|XkE|k|(X0)|

2 ψβ(q)

.

If(Xi)_i∈Z^{d is bounded thenfor any} 0< q ≤2^{, there existsa universalpositiveonstant} M2(q) ^{depending only on} q ^{suh that for any family} (ai)_i∈Z^{d of real numbers and any}

nite subset Γ ^of Z^d,

X

i∈Γ

aiXi

ψq

≤M2(q) X

i∈Γ

|ai|bi,∞(X)

!1/2

(4)

where

bi,∞(X) :=|ai| kX0k²∞+ X

k∈V₀¹

|ak+i| kXkE|k|(X0)k^∞.

Remark 1 If(Xi)i∈Zd ^{isa martingale dierenerandom eld then}

b_i,q(X) =|a_i| kX₀k²ψ_β(q) ^and b_i,∞(X) =|a_i| kX₀k²∞.

Thus, the inequalities (3^{) and (}4^{) extend previous ones} established for sequenes of bounded i.i.d. randomvariables(see [14℄, [15℄, [20℄).

UsingSering's inequality (see [19℄ or [24℄), we dedue from Theorem 1 ^{the following}

resultfor stationary φ^{-mixingreal random elds.}

Corollary 1 Let (X_i)_i∈Z^{d be a zero mean stationary real random eld whih satises}

the assumption (2) ^{for some} 0 < q < 2^{. For any family} (a_i)_i∈Z^{d of real numbers and}

any nite subset Γ ^of Z^d,

X

i∈Γ

aiXi

ψq

≤M1(q)kX0k^ψβ(q)

X

i∈Γ

|ai|ebi,q(X)

!1/2

where

ebi,q(X) :=|ai|+C(q) X

k∈V₀¹

|ak+i| q

φ∞,1(|k|),

M1(q)^{isthe positiveonstantintroduedin Theorem}1 ^andC(q) ^{isapositive universal}

onstantdepending only on q^.

If (Xi)_i∈Z^{d is bounded then for any} 0< q≤2^{, any family} (ai)_i∈Z^{d of real numbers and}

any nite subset Γ ^of Z^d,

X

i∈Γ

a_iX_i ψq

≤M₂(q)kX₀k∞

X

i∈Γ

|a_i|eb_i,∞(X)

!1/2

where

ebi,∞(X) := |ai|+ 2 X

k∈V₀¹

|ak+i|φ∞,1(|k|),

One an notie that in the unbounded ase we were able to give Kahane-Khinthine

inequalitiesonlyinOrlizspaesLψq ^when0< q <2^{butfor bounded randomeld we}

establishedtheseinequalitieseven inthespae Lψ2^{. That isthe reasonwhy weannot}

givea proofof the FCLTforrandom eldswith niteexponentialmoments (Theorem

2^{)underDudley's entropyondition (}5^{)unlikeasinthease ofbounded randomelds}

(see [5℄).

LetA ^{bea olletionof Borelsubsets of} [0,1]^{d. We fous onthe sequene of proesses} {Sn(A) ;A ∈ A} ^{dened by (}1^{). As a funtion of} A^{, this proess is ontinuous with}

respet tothe pseudo-metri ρ(A, B) =p

λ(A∆B)^.

To measure the size of A ^{one onsiders the metri entropy: denote by} H(A, ρ, ǫ) ^the

logarithm of the smallest number of open balls of radius ǫ ^{with respet to} ρ ^whih

form a overing of A^{. The funtion} H(A, ρ, .) ^{is the entropy of the lass} A^{. A more}

strittoolisthe metrientropy with inlusion: assume thatA ^{istotallyboundedwith}

inlusioni.e. for eah positive ǫ ^{thereexists anite olletion}A(ǫ)^{of Borel subsets of} [0,1]^{dsuhthatforany} A∈ A^,thereexistA^{− and} A⁺ⁱⁿA(ǫ)^with A⁻⊆A⊆A^{+ and} ρ(A⁻, A⁺) ≤ ǫ^{. Denote by} H(A, ρ, ǫ) ^{the logarithm of the ardinality of the smallest}

olletion A(ǫ)^{. The funtion} H(A, ρ, .) ^{is the entropy with inlusion (or braketing}

entropy) of the lass A^{. Let} C(A) ^{be the spae of ontinuous real funtions on} A^,

equipped with the norm k.k^{A dened by} kfk^A= sup

A∈A|f(A)|.

A standard Brownian motion indexed by A ^{is a mean zero Gaussianproess} W ^with

sample paths in C(A) ^and Cov(W(A),W(B))= λ(A∩B)^{. From Dudley [9℄ we know}

that suh a proess exists if

Z 1

0

pH(A, ρ, ǫ)dǫ <+∞. ⁽⁵⁾

SineH(A, ρ, .)≤H(A, ρ, .)^{,the standard Brownian motion}W ^{is well dened if} Z 1

0

pH(A, ρ, ǫ)dǫ <+∞. ⁽⁶⁾

We say that the sequene {n^−d/2Sn(A) ; A ∈ A} ^{satises the funtional entral limit}

theorem (FCLT) if it onverges in distribution to a mixture of A^{-indexed Brownian}

motionsinthe spae C(A)^{(whih meansthat the limitingproess is ofthe form}η W^,

whereW ^{isastandard Brownianmotionindexedby} A ^andη ^isanonnegativerandom variableindependent of W^).

In the sequel, we shall give a projetive riterion whih implies the tightness of the

sequene{n^−d/2S_n(A) ; A∈ A}ⁱⁿC(A)^{undertheassumption(}2^)ofniteexponential moments and provided that the lass A ^{satises an entropy ondition related to the}

momentsof the randomeld (i.e. Condition (8^{) holds). The ase of bounded station-}

aryreal random elds wasstudied by Dedeker in[5℄where he proved that the FCLT

holds underthe L^{∞-projetive riterion} X

k∈V₀¹

kXkE|k|(X0)k^∞<+∞

andfor any olletionA ^{satisfyingonlyDudley's entropy ondition (}5^{). Forany Borel}

set A ⁱⁿ[0,1]^{d, let}∂A ^{be the boundaryof} A^{. Wesay that} A ^{isregular if} λ(∂A) = 0^.

Theorem 2 Let (Xi)_i∈^{Zd be a zero mean stationary real random eld whih satises}

the assumption (2) ^{for some} 0< q <2 ^{and assume that} X

k∈V₀¹

q

|XkE|k|(X0)|

2

ψ_β(q) <+∞. ⁽⁷⁾

Let A ^{be a olletion of regular Borel subsets of} [0,1]^{d satisfying the following entropy}

ondition

Z 1

0

(H(A, ρ, ǫ))^1/qdǫ <+∞. ⁽⁸⁾

Then

1) For the σ^-algebra I ^{of invariant sets dened in setion 2, we have} X

k∈Z^d

p

|E(X0Xk|I)| ²

ψ_β(q) <+∞. ⁽⁹⁾

Denote by η ^the nonnegative and I-measurable random variable

η= X

k∈Z^d

E(X0Xk|I).

2) The sequene of proesses {n^−d/2Sn(A) ;A ∈ A} ^{onverges in} distribution in

C(A) ^to √η W ^where W ^{is a standard Brownian motion indexedby} A ^{and inde-}

pendent of I^.

InTheorem2^{,oneanseethatweontrolthesizeofthelass}A^{viathelassialmetri}

entropy (without inlusion). In fat, all the earlier results we know (in partiular [1℄,

[2℄, [5℄) about the FCLTfor unbounded proesses indexed by large lasses of sets deal

with the more strit braketing entropy.

UsingSering'sinequality(see[19℄or[24℄),wederivefromTheorem2^thefollowing

resultfor stationary φ^{-mixingreal random elds.}