Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert space

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Berry-Esseen’s central limit theorem for non-causal linear processes in Hilbert space

Mohamed El Machkouri

To cite this version:

Mohamed El Machkouri. Berry-Esseen’s central limit theorem for non-causal linear processes in Hilbert space. 2010. �hal-00488676�

linear proesses in Hilbert spae

Mohamed EL MACHKOURI

June 2, 2010

Abstrat

LetH^{bearealseparableHilbertspaeand}(a_k)_k∈Z^{asequeneofboundedlinear}

operatorsfromH ^to H^{. Weonsiderthelinearproess}X ^{dened forany}kⁱⁿZ^by X_k=P

j∈Za_j(ε_k−j)^where(ε_k)_k∈Z^{isasequeneofi.i.d. entered}H^{-valuedrandom}

variables. WeinvestigatetherateofonvergeneintheCLTforX^{andinpartiular}

we obtain the usual Berry-Esseen's bound provided that

P

j∈Z|j|ka_jk_L(H) < +∞

andε₀ ^{belongs to} L^∞_H^.

Shorttitle: Berry-Esseen'sCLT for Hilbertian linear proesses.

Key words: Central limit theorem, Berry-Esseen bound, linear proess, Hilbert

spae.

1 Introdution and notations

Let(H,k.kH) ^{be a separable real Hilbert spae and} (L,k.kL(H)) ^{be the lass of bounded}

linear operators fromH ^to H ^{with its usual uniform norm. Consider a sequene} (εk)k∈Z

of i.i.d. entered random variables, dened on a probability spae (Ω,A,P)^{, with values}

inH^{. If}(a_k)_k∈Z ^{isasequene in}L^,wedenethe(non-ausal)linearproessX = (X_k)_k∈Z

inH ^by

Xk=X

j∈Z

aj(εk−j), k ∈Z. ⁽¹⁾

If

P

j∈ZkajkL(H) <∞ ^and Ekε0kH < +∞ ^{then the series in} (1) ^{onverges almost surely}

andinL¹_H(Ω,A,P)^{(see Bosq[2℄). Theondition} P

j∈ZkajkL(H) <∞^{isknowtobesharp}

for the

√n-normalized partial sums of X ^{to satises a CLT provided that} (εk)k∈Z ^are

i.i.d. entered having nite seond moments(see Merlevede et al. [6℄). In this work, we

investigate the rate of onvergene in the CLT for X ^{under the ondition}

X

j∈Z

|j|^τka_jkL(H) <∞ ⁽²⁾

withτ = 1^when(εk)k∈Z ^{areassumed tobei.i.d. enteredandsuhthat}ε0 ^belongstoL^∞_H

andτ = 1/2^when(ε_k)_k∈Z^{arei.i.d. enteredandsuhthat}ε₀^{belongstosomeOrlizspae} L_H,ψ ^{(see setion 2). This problem was previously studied(with} τ = 1 ^{in Condition} (2)⁾

by Bosq [3℄ for (ausal) Hilbert linear proesses but a mistake in his proof was pointed

out by V. Paulauskas [7℄. However, in the partiular ase of Hilbertian autoregressive

proesses of order 1, Bosq [4℄ obtained the usual Berry-Esseen inequality provided that

(εk)k∈Z ^{are i.i.d. entered with} ε0 ⁱⁿ L^∞_H^.

2 Main result

Inthe sequel,Cε0 ^istheautoovarianeoperatorofε0^,A:=P

j∈Zaj ^andA^{∗ istheadjoint}

of A^{. For any sequene} Z = (Zk)k∈Z ^{of randomvariableswith values in} H ^wedenote

∆n(Z) = sup

t∈R

P

√1 n

n

X

k=1

Zk

H

≤t

!

−P(kNkH ≤t)

whereN ∼ N(0, ACε0A^∗)^.

Forany j ∈Z^{, denote} cj,n=Pn

i=1bi−j ^where bi =ai ^{for any} i6= 0^and b0 =a0−A^.

Lemma 1 For any positive integer n^,

n

X

k=1

Xk =A

n

X

k=1

εk

!

+Qn+Rn

where Q_n=Pn k=1

P

|j|>na_k−j(ε_j) ^and R_n=P

|j|≤nc_j,n(ε_j)^.

Reall that a Young funtion ψ ^{is a real onvex} nondereasing funtion dened on R⁺

whih satises limt→+∞ψ(t) = +∞ ^and ψ(0) = 0^{. We dene the Orliz spae} LH,ψ ^as

thespaeof H^{-valuedrandomvariables}Z ^{dened onthe} probabilityspae(Ω,F,P)^suh

that E[ψ(kZkH/c)] < +∞ ^{for some} c > 0^{. The Orliz spae} LH,ψ ^{equipped with the}

so-alledLuxemburg norm k.kψ ^{dened for any} H^{-valued randomvariable}Z ^by kZk^ψ = inf{c >0 ;E[ψ(kZk^H/c)]≤1}

is a Banah spae. In the sequel, c(N) ^{denotes a bound of the density of} N(0, ACε0A^∗)

(see Davydov et al. [5℄). Our main result isthe following.

Theorem 1 Let (εk)k∈Z ^{be a sequene of i.i.d. entered} H^{-valued random variables and}

let X ^{be the Hilbertian linear proess dened by} (1)^.

i) If ε0 ^belongsto L^∞_H ^and P

j∈Z|j|kajkL(H) <∞ ^then

∆n(X)≤ c1

√n ⁽³⁾

where c1 = c2 + 14c(N)kε0k∞P

j∈Z|j|kajkL(H) ^and c2 ^{is a positive onstant whih}

depend only on the distribution of ε0^.

ii) If ψ ^{is a Young funtion then}

∆n(X)≤∆n(A(ε)) +ϕ

c(N)kQ_n+R_nkψ

√n

(4)

where ϕ(x) =xh⁻¹(1/x) ^and h(x) =xψ(x) ^{for any real} x >0^.

The inequality (4^{) ensures a rate of onvergene to zero for} ∆n(X) ^as n ^{goes to innity}

provided that ∆n(A(ε0)) ^{goes to zero as} n ^{goes to innity and a bound for} kQn+Rnk^ψ

exists. As just anillustration, wehave the followingorollary.

Corollary 1 Assume that(εk)k∈Z ^{arei.i.d. entered}H^{-valued random variablesandthat}

the ondition (2) ^{holds with} τ = 1/2^.

i) Ifε0^belongstoLH,ψ1 ^then∆n(X) =O

log√n n

whereψ1 ^{istheYoungfuntiondened}

byψ1(x) = exp(x)−1^.

ii) If ε0 ^belongsto L^r_H ^for r≥3 ^then ∆n(X) = O

n^−2(r+1)r

.

Proof of Lemma 1. For any positiveinteger n^{, we have} Rn =

n

X

j=−n

cj,n(εj) =

n

X

k=1 n

X

j=−n

bk−j(εj)

=

n

X

k=1

X

j∈[−n,n]\{k}

ak−j(εj) + (a0−A)

n

X

k=1

εk

!

=

n

X

k=1 n

X

j=−n

ak−j(εj)−A

n

X

k=1

εk

!

=−Q_n+

n

X

k=1

X_k−A

n

X

k=1

ε_k

! .

The proof of Lemma1 ^isomplete.

Proof of Theorem 1. Let λ > 0 ^and t > 0 ^{be xed and denote} U = A(Pn

k=1εk/√ n)

and V = (Qn+Rn)/√

n^{. So}U +V =Pn

k=1Xk/√ n ^and

P(kU+VkH ≤t)≤P(kUkH ≤t+λ) +P(kVkH ≥λ) ⁽⁵⁾

Forλ0 = 2kVk∞^{, we obtain}

P(kU +VkH ≤t)−P(kNkH ≤t)≤P(kUkH ≤t+λ0)−P(kNkH ≤t).

Ifc(N) ^{denotes abound for the density of} kNkH ^{(see Davydov et al. [5℄)then}

∆n(X)≤∆n(A(ε)) + 2c(N)kQ√_n+R_nk∞

n .

Noting that

Qn = X

j≥n+2

aj

−n−1

X

k=1−j

εk

! +X

j<0

aj n−j

X

k=n+1

εk

!

(6)

and

Rn=R^′_n+R^′′_n ⁽⁷⁾

where

R_n^′ =−

−1

X

j=−n

a_j

−j

X

k=1

ε_k

!

− X

j<−n

a_j

n

X

k=1

ε_k

!

−X

j>0

a_j

n

X

k=n−j+1

ε_k

!

R^′′_n =

n

X

j=1

aj 0

X

k=−j+1

εk

! +

2n

X

j=n+1

aj n−j

X

k=−n

εk

! ,

wederive that kQn+Rnk∞≤7kε0k∞P

j∈Z|j|kajkL(H) ^and onsequently

∆n(X)≤∆n(A(ε)) + 14c(N)kε0k∞P

j∈Z|j|kajkL(H)

√n .

Combining the last inequality with the Berry-Esseen inequality for i.i.d. entered H^-

valued randomvariables (see Yurinski [9℄or Bosq [2℄, Theorem 2.9) we obtain (3^).

In the other part, if ψ ^{is a Young funtion we have} P(kVkH ≥ λ) ≤ _ψ(^λ/k¹Vkψ) ^and

keeping in mindinequality (5)^{, we derive}

∆n(X)≤∆n(A(ε)) +c(N)λ+ 1 ψ(λ/kVk^ψ).

Noting that c(N)λ = ¹

ψ(^λ/kVkψ) ^{if and only if} λ = ^ϕ(^c(N)kVkψ)

c(N) ^where ϕ ^{is dened by} ϕ(x) =xh⁻¹(1/x)^and h ^by h(x) =xψ(x)^{, we onlude}

∆n(X)≤∆n(A(ε)) +ϕ

c(N)kQn+Rnkψ

√n

.

The proof of Theorem 1^{is omplete.}

Proof of Corollary 1. Assume that kε0kψ1 < ∞ ^where ψ1 ^{is the Young funtion}

dened by ψ1(x) = exp(x)−1^{. There exists} a > 0 ^{suh that} E(exp(akε0kH)) ≤ 2^{. So,}

thereexist (see Arak and Zaizsev [1℄) onstants B ^and L ^{suh that} Ekε0k^mH ≤ m!

2 B²L^m−2, m= 2,3,4, ...

Applying Pinelis-Sakhanenkoinequality (see Pinelis and Sakhanenko [?℄ or Bosq [2℄), we

obtain

P

q

X

k=p

εk

H

≥x

!

≤exp

− x²

2(q−p+ 1)B²+ 2xL

, x >0

andusingLemma2.2.10inVanDerVaartandWellner[?℄,thereexistsauniversalonstant

K ^{suh that}

q

X

k=p

ε_k ψ1

≤K

L+Bp

q−p+ 1

(8)

Combining (6^{), (}7^{) and (}8^{), we derive} kQn+Rnkψ1 ≤ CP

j∈Z

p|j|kajkL(H) ^{where the}

onstantC ^{doesnot depend on} n^{. Keeping inmind the} Berry-Esseen's entrallimitthe- oremfori.i.d. entered H^{-valuedrandomvariables(seeYurinski [9℄orBosq [2℄, Theorem}

2.9), we apply Theorem 1 ^{with the Young funtion} ψ₁^{. Sine the funtion} ϕ ^{dened by} ϕ(x) =xh⁻¹(1/x)^with h(x) =xψ₁(x) ^satises

limx→0

ϕ(x)

xlog(1 + _x¹) = 0,

wederive ∆n(X) = O

logn

√n

.

Now, assume that kε0kr < ∞ ^{for some} r ≥ 3^{. Applying Pinelis inequality (see Pinelis}

[8℄), thereexists a universal onstant K ^{suh that}

q

X

k=p

εk

r

≤K



r

q

X

k=p

Ekεkk^rH

!1/r

+√ r

q

X

k=p

Ekεkk²H

!1/2



and onsequently

q

X

k=p

εk

r

≤2Krkε0kr

pq−p+ 1. ⁽⁹⁾

Combining (6^{), (}7^{) and (}9^{), we derive} kQn +Rnk^r ≤ CP

j∈Z

p|j|kajkL(H) ^{where the}

onstant C ^{does not depend on} n^{. Again, applying} Berry-Esseen's CLT (see Yurinski [9℄ or Bosq [2℄, Theorem 2.9) and Theorem 1 ^{with the Young funtion} ψ(x) = x^{r and}

the funtion ϕ ^{given by} ϕ(x) = x^{r/(r+1), we obtain}∆n(X) = O

n^−2(r+1)r

. The proof of

Corollary1 ^isomplete.

Referenes

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random variables. Pro. Steklov Inst. Math., (1(174)), 1988. A translation of Trudy

Mat.Inst. Steklov. 174(1986).

[2℄ Denis Bosq. Linear proesses in funtion spaes, volume 149 of Leture Notes in

Statistis. Springer-Verlag, New York,2000. Theory and appliations.

[3℄ Denis Bosq. Berry-Esseen inequality for linear proesses in Hilbert spaes. Statist.

Probab. Lett., 63(3):243247,2003.

essesinHilbertspaes [Statist.Probab.Lett.63(2003),no.3,243247;mr1986323℄.

Statist. Probab. Lett., 70(2):171174,2004.

[5℄ Yu.A.Davydov,M.A.Lifshits,andN.V.Smorodina.Loalpropertiesofdistributions

of stohasti funtionals, volume 173 of Translations of Mathematial Monographs.

Amerian Mathematial Soiety, Providene, RI, 1998. Translated from the 1995

Russian originalby V. E. Nazakinskiand M. A. Shishkova.

[6℄ FloreneMerlevède, MagdaPeligrad,andSergey Utev. Sharponditions forthe CLT

of linear proesses ina Hilbert spae. J. Theoret. Probab., 10(3):681693,1997.

[7℄ V. Paulauskas. Personal ommuniation, 2004.

[8℄ Iosif Pinelis. Optimum bounds for the distributions of martingales inBanah spaes.

Ann. Probab., 22(4):16791706,1994.

[9℄ V.V.Yurinski. Ontheaurayofnormalapproximationoftheprobabilityofhitting

a ball. Teor. Veroyatnost. i Primenen., 27(2):270278,1982.

MohamedEL MACHKOURI

Laboratoirede Mathématiques Raphaël Salem

UMRCNRS 6085, Université de Rouen

Avenue de l'université

76801Saint-Etienne du Rouvray

mohamed.elmahkouriuniv-rouen.fr