Improved results in almost sure central limit theorems for the maxima and partial sums for Gaussian sequences

(1)

R E S E A R C H

Open Access

Improved results in almost sure central limit

theorems for the maxima and partial sums for

Gaussian sequences

Qunying Wu

*

*_{Correspondence:} [email protected]

College of Science, Guilin University of Technology, Guilin, 541004, P.R. China

Abstract

LetX,X1,X2,. . .be a standardized Gaussian sequence. The universal results in almost sure central limit theorems for the maximaMnand partial sums and maxima (Sn/σn,Mn) are established, respectively, whereSn=

n

i=1Xi,

σ

n2= VarSn, and Mn= max1≤i≤nXi.

MSC: 60F15

Keywords: standardized Gaussian sequence; maxima; partial sums and maxima; almost sure central limit theorem

1 Introduction

Starting with Brosamler [] and Schatte [], in the last two decades several authors inves-tigated the almost sure central limit theorem (ASCLT) dealing mostly with partial sums of random variables. Some ASCLT results for partial sums were obtained by Ibragimov and Lifshits [], Miao [], Berkes and Csáki [], Hörmann [], Wu [–], and Wu and Chen []. The concept has already started to have applications in many areas. Fahrner and Stadtmüller [] and Nadarajah and Mitov [] investigated ASCLT for the maxima of i.i.d. random variables. The ASCLT of Gaussian sequences has experienced new develop-ments in the recent past years. Signiﬁcant recent contributions can be found in Csáki and Gonchigdanzan [], Chen and Lin [], Tanet al.[], and Tan and Peng [], extending this principle by proving ASCLT for the maxima of a Gaussian sequence. Further, Peng et al.[–], Zhaoet al.[], and Tan and Wang [] studied the maximum and partial sums of a standardized nonstationary Gaussian sequence.

A standardized Gaussian sequence{Xn;n≥}is a sequence of standard normal ran-dom variables, and for any choice ofn, i, . . . ,in, the joint distribution ofXi, . . . ,Xin is

ann-dimensional normal distribution. Throughout this paper we assume{Xn;n≥}is a standardized Gaussian sequence with covarianceri,j:=Cov(Xi,Xj). For eachn≥, let Sn=

n

i=Xi, σn =VarSn,Mn=max≤i≤nXi. The symbolsSn/σnandMndenote partial sums and maxima, respectively. Let(·) andφ(·) denote the standard normal distribution function and its density function, respectively, andIdenote an indicator function.An∼Bn denoteslim_n_→∞An/Bn= , andAnBnmeans that there exists a constantc>  such that An≤cBnfor suﬃciently largen. The symbolcstands for a generic positive constant which

(2)

may diﬀer from one place to another. The normalizing constantsanandbnare deﬁned by

an= (lnn)/, bn=an–

ln lnn+ln(π) an

. ()

Chen and Lin [] obtained the following almost sure limit theorem for the maximum of a standardized nonstationary Gaussian sequence.

Theorem A Let{Xn;n≥}be a standardized nonstationary Gaussian sequence such that

|rij| ≤ρ|i–j|for i=j whereρn< for all n≥andρn _ln_n_{(ln ln} _n₎+ε.Let the numerical se-quence{uni; ≤i≤n,n≥}be such that n( –(λn))is bounded andλn=min≤i≤nuni≥ cln/n for some c> .Ifn_i_=( –(uni))→τ as n→ ∞for someτ≥,then

lim

n→∞



lnn n

k=

 kI

_k

i=

(Xi≤uki)

=exp(–τ) a.s.

Zhaoet al.[] obtained the following almost sure limit theorem for maximum and partial sums of standardized nonstationary Gaussian sequence.

Theorem B Let{Xn;n≥}be a standardized nonstationary Gaussian sequence.Suppose that there exists a numerical sequence{uni; ≤i≤n,n≥}such that

n

i=( –(uni))→τ for some <τ<∞and n(–(λn))is bounded,whereλn=min≤i≤nuni.Ifsupi= j|rij|=δ< ,

n

j=

j–

i=

|rij|=o(n), ()

sup

i≥

n

j= |rij|

ln/n

(ln lnn)+ε for someε> , ()

then

lim

n→∞



lnn n

k=

 kI

_k

i=

(Xi≤uki), Sk σk ≤

y

=exp(–τ)(y) a.s. for all y∈R ()

and

lim

n→∞ 

lnn n

k=

 kI

ak(Mk–bk)≤x, Sk σk

≤y

=exp –e–x(y) a.s. for all x,y∈R. ()

By the terminology of summation procedures (seee.g.Chandrasekharan and Minak-shisundaram [], p.) one shows that the larger the weight sequence in ASCLT is, the stronger the relation becomes. Based on this view, one should also expect to get stronger results if one uses larger weights. Moreover, it would be of considerable interest to deter-mine the optimal weights.

(3)

2 Main results

Set

dk=

exp(lnαk)

k , Dn= n

k=

dk for ≤α< /. ()

Our theorems are formulated in a more general setting.

Theorem . Let{Xn;n≥}be a standardized Gaussian sequence.Let the numerical se-quence{uni; ≤i≤n,n≥}be such that

n

i=( –(uni))→τ for some≤τ<∞and n( –(λn))is bounded,whereλn=min≤i≤nuni.Suppose thatρn< for all n≥such that

|rij| ≤ρ|i–j| for i=j, ρn 

lnn(lnDn)+ε

for someε> . ()

Then

lim

n→∞

 Dn

n

k=

dkI

_k

i=

(Xi≤uki)

=exp(–τ) a.s. ()

and

lim

n→∞  Dn

n

k=

dkI ak(Mk–bk)≤x

=exp –e–x a.s. for any x∈R, ()

where anand bnare deﬁned by().

Theorem . Let{Xn;n≥}be a standardized Gaussian sequence.Let the numerical se-quence{uni; ≤i≤n,n≥}be such that

n

i=( –(uni))→τ for some≤τ<∞and n( –(λn))is bounded,whereλn=min≤i≤nuni.Suppose thatsupi=j|rij|=δ< ,there exists a constant <c< /such that

≤i<j≤n rij

≤cn, ()

max ≤i≤n

n

j= |rij|

ln/n

lnDn

. ()

Then

lim

n→∞  Dn

n

k=

dkI

_k

i=

(Xi≤uki), Sk σk

≤y

=exp(–τ)(y) a.s. for any y∈R ()

and

lim

n→∞

 Dn

n

k=

dkI

ak(Mk–bk)≤x, Sk σk≤

y

=exp–e–x(y) a.s. for any x,y∈R, ()

(4)

Takinguki=ukfor ≤i≤kin Theorems . and ., we can immediately obtain the following corollaries.

Corollary . Let{Xn;n≥}be a standardized Gaussian sequence.Let the numerical sequence{un;n≥}be such that n( –(un))→τfor some≤τ<∞.Suppose that con-dition()is satisﬁed.Then()and

lim

n→∞  Dn

n

k=

dkI(Mk≤uk) =exp(–τ) a.s.

hold.

Corollary . Let{Xn;n≥} be a standardized Gaussian sequence.Let the numerical sequence {un;n≥} be such that n( –(un))→τ for some ≤τ <∞.Suppose that

sup_i₌_j|rij|=δ< ,there exists a constant <c< /such that conditions()and()are satisﬁed.Then()and

lim

n→∞  Dn

n

k=

dkI

Mk≤uk, Sk σk ≤

y

=exp(–τ)(y) a.s. for any y∈R

hold.

By the terminology of summation procedures (seee.g.Chandrasekharan and Minak-shisundaram [], p.), we have the following corollary.

Corollary . Theorems.and.,Corollaries.and.remain valid if we replace the weight sequence{dk;n≥}by any{dk∗;n≥}such that≤d∗k≤dk,

∞

k=d∗k=∞. Remark . Obviously, the condition () is signiﬁcantly weaker than the condition (),

and in particular takingα= ,i.e., the weightdk= e/k, we haveDn∼elnnandlnDn∼

ln lnn, in this case, the condition () is signiﬁcantly weaker than the condition (), and the conclusions () and () become () and (), respectively. Therefore, our Theorem . not only gives substantial improvements for the weight but also has greatly weakened re-strictions on the covariancerijin Theorem B obtained by Zhaoet al.[].

Remark . Theorem A obtained by Chen and Lin [] is a special case of Theorem . whenα= . When{Xn;n≥}is stationary,uni=un, ≤i≤n, andα= , Theorem . is Corollary . obtained by Csáki and Gonchigdanzan [].

Remark . Whether (), (), (), and () work also for some /≤α<  remains an open question.

3 Proofs

(5)

and challenges; to overcome the diﬃculties and challenges the following ﬁve lemmas play an important role. The proofs of Lemmas . to . are given in the Appendix.

Lemma .(Normal comparison lemma, Theorem .. in Leadbetteret al.[]) Suppose

ξ, . . . ,ξnare standard normal variables with covariance matrix= (ij),andη, . . . ,ηn similarly with covariance matrix_{= (}

ij),and letρij=max(|ij|,|ij|),maxi=jρij=δ< . Further,let u, . . . ,unbe real numbers.Then

_P(ξj≤ujfor j= , . . . ,n) –P(ηj≤ujfor j= , . . . ,n)

≤K

≤i<j≤n

_ij–_ijexp

– u



i +uj ( +ρij)

for some constant K,depending only onδ.

Lemma . Suppose that the conditions of Theorem.hold,then there exists a constant

γ > such that

sup ≤k≤l

k

i=

l

j=i+ |rij|exp

– u



ki+ulj ( +|rij|)



lγ +  (lnDl)+ε

, ()

EI

_l

i=

(Xi≤uli)

–I

_l

i=k+

(Xi≤uli)

k

l for≤k<l, ()

Cov

I

_k

i=

(Xi≤uki)

,I

_l

i=k+

(Xi≤uli)

 lγ +

 (lnDl)+ε

for≤k<l, ()

whereεis deﬁned by().

Lemma . Suppose that the conditions of Theorem.hold,then there exists a constant

γ > such that

EI

_l

i=

(Xi≤uli), Sl σl

≤y

–I

_l

i=k+

≤y

k l

γ

for≤k<l, ()

Cov

I

_k

i=

y

,I

_l

i=k+

(Xi≤uli), Sl σl ≤

y

k l

γ +k

/₍_ln_l)/

l/_ln_D

l

for≤k< l

lnl. ()

The following weak convergence results are the extended versions of Theorem .. of Leadbetteret al.[] to the nonstationary normal random variables.

Lemma . Suppose that the conditions of Theorem.hold,then

lim

n→∞P

_n

i=

(Xi≤uni)

(6)

Suppose that the conditions of Theorem.hold,then

lim

n→∞P

_n

i=

(Xi≤uni), Sn σn≤

y

= e–τ(y). ()

Lemma . Let{ξn;n≥}be a sequence of uniformly bounded random variables.If

Var _n

k=

dkξk

Dn

(lnDn)+ε

for someε> ,then

lim

n→∞  Dn

n

k=

dk(ξk–Eξk) =  a.s.,

where dnand Dnare deﬁned by().

Proof Similarly to the proof of Lemma . in Wu [], we can prove Lemma ..

Proof of Theorem. Using Lemma .,P(n_i_=(Xi≤uni))→exp(–τ), and hence by the Toeplitz lemma,

lim

n→∞  Dn

n

k=

dkP

_k

i=

(Xi≤uki)

=exp(–τ).

Therefore, in order to prove (), it suﬃces to prove that

lim

n→∞

 Dn

n

k=

dk

I

_k

i=

(Xi≤uki)

–P

_k

i=

(Xi≤uki)

=  a.s.,

which will be done by showing that

Var _n

k=

dkI

_k

i=

(Xi≤uki)

Dn

(lnDn)+ε

()

for someε>  from Lemma .. Letξk:=I(

k

i=(Xi≤uki)) –P(

k

i=(Xi≤uki)). ThenEξk=  and|ξk| ≤ for allk≥. Hence

Var _n

k=

dkI

_k

i=

(Xi≤uki)

= n

k=

d_kEξ_k+ 

≤k<l≤n

dkdlE(ξkξl)

:=T+T. ()

Since|ξk| ≤ andexp(lnβx) =exp(

x

 (lnu)β–

u du),β< , is a slowly varying function at inﬁnity, from Seneta [], it follows that

T≤

∞

k=

exp(lnα_k) k =c≤

D

n (lnDn)+ε

(7)

By Lemma ., for ≤k<l,

_E(ξkξl)≤

Cov

I

_k

i=

(Xi≤uki)

,I

_l

i=

(Xi≤uli)

–I

_l

i=k+

(Xi≤uli)

+

Cov

I

_k

i=

(Xi≤uki)

,I

_l

i=k+

(Xi≤uli)

EI

_l

i=

(Xi≤uli)

–I

_l

i=k+

(Xi≤uli)

+

Cov

I

_k

i=

(Xi≤uki)

,I

_l

i=k+

(Xi≤uli)

k l

γ + 

(lnDl)+ε

forγ=min(,γ) > . Hence,

T

n

l=

l

k=

dkdl

k l

γ +

n

l=

l

k=

dkdl  (lnDl)+ε

:=T+T. ()

By () in Wu [],

Dn∼  αln

–α

nexp lnαn, lnDn∼lnαn,

exp lnαn∼ αDn (lnDn)

–α α

forα> .

()

From this, combined with the fact that_xl(t) tβ dt∼

l(x)x–β

–β asx→ ∞forβ<  andl(x) is a slowly varying function at inﬁnity (see Proposition .. in Binghamet al.[]), we get

T≤

n

l=

dl lγ

l

k=

exp(lnα_k) k–γ

n

l=

dl lγl

γ_exp_lnα l

≤Dnexp lnαn

_D

n

(lnDn)(–α)/α, α> ,

Dn, α= 

≤ Dn

(lnDn)+ε

()

for  <ε< ( – α)/α.

Now, we estimateT. Forα> , by ()

T =

n

l=

dl (lnDl)+ε

Dl n

l=

exp(lnαl)(lnl)–α–αε l

∼ n

e

exp(lnαx)(lnx)–α–αε

x dx=

lnn



(8)

∼ lnn



expyαy–α–αε+ – α–αε α exp y

α y–α–αε

dy

=

lnn



(α)–exp yαy–α–αεdy

explnα_n₍_ln_n)–α–αε

Dn

(lnDn)+ε

. ()

Forα= , noting the fact thatDn∼lnn, similarly we get

T∼

n

l= lnl l(ln lnl)+ε∼

n



lnx x(ln lnx)+εdx

=

lnn

ln

y

(lny)+εdy

lnn (ln lnn)+ε∼

D_n (lnDn)+ε

. ()

Equations ()-(), ()-() together establish (), which concludes the proof of (). Next, takeuni=un=x/an+bn. Then we see that

n

i=( –(uni)) =n( –(un))→exp(–x) asn→ ∞(see Theorem .. in Leadbetteret al.[]) and hence () immediately follows from () withuni=x/an+bn.

This completes the proof of Theorem ..

Proof of Theorem . Using Lemma ., P(n_i_=(Xi ≤uni),Sn/σn≤y)→e–τ(y), and hence by the Toeplitz lemma,

lim

n→∞

 Dn

n

k=

dkP

_k

i=

y

= e–τ(y).

Therefore, in order to prove (), it suﬃces to prove that

lim

n→∞

 Dn

n

k=

dk

I

_k

i=

y

–P

_k

i=

y

=  a.s.,

which will be done by showing that

Var _n

k=

dkI

_k

i=

(Xi≤uki), Sk σk

≤y

Dn

(lnDn)+ε

()

for some ε>  from Lemma .. Letηk:=I(

k

i=(Xi≤uki),Sσkk ≤y) –P(

k

i=(Xi≤uki), Sk

σk ≤y). By Lemma ., for ≤k<l/lnl,

_E(ηkηl)≤

Cov

I

_k

i=

(Xi≤uki), Sk σk≤

y

,I

_l

i=

y

–I

_l

i=k+

(Xi≤uli), Sl σl≤

y

(9)

+

Cov

I

_k

i=

y

,I

_l

i=k+

y

≤ EI

_l

i=

y

–I

_l

i=k+

y

+

Cov

I

_k

i=

y

,I

_l

i=k+

y

k l

γ +k

/_ln/_l

l/_ln_D

l .

Hence,

Var _n

k=

dkI

_k

i=

y

= n

k=

d_kEη_k+ 

≤k<l≤n

dkdlE(ηkηl)

≤k<l≤n

dkdlE(ηkηl)

n

l=

≤k<l/lnl dkdl

k l

γ +

n

l=

≤k<l/lnl dkdl

k/_ln/_l

l/_ln_D

l

+ n

l=

l/lnl≤k≤l dkdl

:=T+T+T. ()

By the proof of (),

T

D_n (lnDn)+ε

for  <ε< ( – α)/α. ()

Now, we estimateT. Forα> , by ()

T≤

n

l=

dlln/l l/_ln_D

l l

k=

exp(lnαk) k/

n

l=

dlln/l l/_ln_D

l

l/exp lnαl

∼ n

e

exp(lnαx)(lnx)/–α

x dx=

lnn



exp yαy/–αdy

∼ lnn



expyαy/–α+ – α α exp y

α y/–α

dy

=

lnn



(α)–exp yαy/–αdy

explnαn(lnn)/–α D



n (lnDn)/α

Dn

(lnDn)+ε

()

(10)

Forα= ,

T

n

l= ln/l l/_{ln ln}_l

l

k=

 k/

n

l= ln/l lln lnl

∼ n



(lnx)/

xln lnxdx=

lnn

ln

y/ lnydy

(lnn)/ ln lnn ∼

D/_n (lnDn)

, ()

T≤

n

l=

dlexp lnαl

l/lnl≤k≤l  k

n

l=

dlexplnαl

ln lnl

Dnexp lnαn

ln lnn

_D

nln lnDn

(lnDn)(–α)/α, α> , DnlnDn, α= 

≤ Dn

(lnDn)+ε

. ()

Equations ()-() together establish () forε=min(ε,ε) > , which concludes the

proof of (). Next, takeuni=un=x/an+bn. Then we see that

n

i=( –(uni)) =n( – (un))→exp(–x) asn→ ∞(see Theorem .. in Leadbetteret al.[]) and hence () immediately follows from () withuni=x/an+bn.

This completes the proof of Theorem ..

Appendix

Proof of Lemma. By assumption (), we haveδ:=sup_i₌_j|rij|< . Deﬁneλsuch that  < λ< /( +δ) – , for ≤k≤l,

k

i=

l

j=i+ |rij|exp

– u



ki+ulj ( +|rij|)

= k

i=

i+≤j≤l,j–i≤lλ

|rij|exp

– u



ki+ulj ( +|rij|)

+ k

i=

i+≤j≤l,j–i>lλ

|rij|exp

– u



ki+ulj ( +|rij|)

:=H+H. ()

Sincen( –(λn)) is bounded, whereλnis the same as deﬁned in Theorem ., there exists a constantc>  such thatn( –(λn))≤c.vnis deﬁned to satisfyn( –(vn)) =c; then clearlyvn≤λn.

Since  –(x)∼φ(x)/xasx→ ∞, we have

exp

–v



n 

∼cvn

n, vn∼

√

(11)

By () and (),

H≤

k

i=

i+≤j≤l,j–i≤lλ ρj–iexp

–v



k+vl ( +δ)

≤

k

i=

≤s≤lλ ρsexp

–v



k+vl ( +δ)

≤klλ(lnk)

/(+δ)

k/(+δ)

(lnl)/(+δ)

l/(+δ) ≤

(lnl)/(+δ)

l/(+δ)––λ

≤ 

lγ ()

for  <γ< /( +δ) –  –λ.

Settingσj=supi≥jρi, by () and (),

σ_lλ=sup i≥lλ

ρisup i≥lλ



lni(lnDi)+ε

= 

lnlλ₍_ln_D lλ)+ε



lnl(lnDl)+ε

and

σ_lλvkvl  (lnDl)+ε

for all ≤k≤l.

This, combined with (), shows

H ≤

k

i=

i+≤j≤l,j–i>lλ ρj–iexp

– v



k+vl ( +ρj–i)

≤

k

i=

lλ_≤_s_≤_l ρsexp

–v



k+vl 

exp

σ_lλ(v_k+v_l) 

klσlλ vk

k vl

l  (lnDl)+ε

.

This, together with () and () implies that () holds.

It is well known thatP(B) –P(AB)≤P(A) for any sets¯ AandB, then using the condition thatn( –(λn)) is bounded, for ≤k<l, we get

EI

_l

i=

(Xi≤uli)

–I

_l

i=k+

(Xi≤uli)

=P

_l

i=k+

(Xi≤uli)

–P

_l

i=

(Xi≤uli)

≤P(Xi>ulifor some ≤i≤k)≤ k

i=

P(Xi>uli)

≤k  –(λl)

=k

ll  –(λl)

k

l.

(12)

Now we prove (). By (), applying the normal comparison lemma, Lemma ., for ≤k<l,

Cov

I

_k

i=

(Xi≤uki)

,I

_l

i=k+

(Xi≤uli)

=P(X≤uk, . . . ,Xk≤ukk,Xk+≤ul,k+, . . . ,Xl≤ull)

–P(X≤uk, . . . ,Xk≤ukk)P(Xk+≤ul,k+, . . . ,Xl≤ull)

k

i=

l

j=k+ |rij|exp

– u



ki+ulj ( +|rij|)



lγ + 

lnl(lnDl)+ε .

Hence, () holds.

Proof of Lemma. Notice, for ≤k<l,

EI

_l

i=

y

–I

_l

i=k+

(Xi≤uli), Sl σl≤

y

=P

_l

i=k+

y

–P

_l

i=

y

≤P

_l

i=

≤y

–P

_l

i=

(Xi≤uli)

P

Sl σl

≤y

+

P

_l

i=k+

≤y

–P

_l

i=k+

(Xi≤uli)

P

Sl σl

≤y

+P

Sl σl ≤

y

P

_l

i=k+

(Xi≤uli)

–P

_l

i=

(Xi≤uli)

:=H+H+H. ()

Usingl– |__≤_i_<_j_≤_lrij| ≤σl≤l+ |

≤i<j≤lrij|and (), there exist constantsci> , i= , , such that

cl≤σl≤cl. ()

Hence, using (), for ≤i≤l≤n,

Cov

Xi, Sl σl

≤ 

σl l

j= |rij|

ln/_l

l/_ln_D

l

()

and

Cov

Sk σk

,Sl σl

≤ 

σkσl k

i=

l

j= |rij|

k/

l/

(lnl)/ lnDl

(13)

Noting the fact thatlnlandlnDlare slowly varying functions at inﬁnity, () and () imply that there exists  <μ<  such that, for suﬃciently largel,

max ≤i≤l

Cov

Xi, Sl σl

≤μ

and

max ≤k<l/lnl

Cov

Sk σk

,Sl σl

≤μ.

Combining (), (), and the normal comparison lemma, Lemma ., fori= , ,

Hi l

i= Cov

Xi, Sl σl

exp

–u



li+y ( +μ)

≤

l

i= Cov

Xi, Sl σl

exp

– v



l ( +μ)

≤ (lnl)/+/(+μ)

l/(+μ)–/_ln_D

l ≤ 

lγ ()

for  <γ< /( +μ) – /.

By the proof of (), we haveHk/l. This, combined with () and (), implies that

() holds forγ =min(,γ) > .

Now we prove (). Again applying the normal comparison lemma, Lemma ., for ≤ k<l/lnl,

Cov

I

_k

i=

y

,I

_l

i=k+

y

=P

X≤uk, . . . ,Xk≤ukk, Sk σk

≤y,Xk+≤ul,k+, . . . ,Xl≤ull, Sl σl

≤y

–P

X≤uk, . . . ,Xk≤ukk, Sk σk ≤

y

×P

Xk+≤ul,k+, . . . ,Xl≤ull, Sl σl

≤y

k

i=

l

j=k+ |rij|exp

– u



ki+ulj ( +|rij|)

+ k

i= Cov

Xi, Sl σl

exp

– u



ki+y ( +|Cov(Xi,Sl/σl)|)

+ l

j=k+ Cov

Xj, Sk σk

exp

– u



lj+y ( +|Cov(Xj,Sk/σk)|)

+Cov

Sk σk

,Sl σl

exp

– y



 +|Cov(Sk/σk,Sl/σl)|

(14)

By (), () to (),

H≤

k

i=

l

j= |rij|exp

–v



k+vl ( +δ)

k(lnl)

/

lnDl

(lnk)/(+δ)

k/(+δ)

(lnl)/(+δ)

l/(+δ)

≤(lnl)/+/(+δ)

l/(+δ)–_ln_D

l

≤ 

lγ

for  <γ< /( +δ) – ,

H

k

i= Cov

Xi, Sl σl

exp

– v



k ( +μ)

k(lnl)

/

l/_ln_D

l

(lnk)/(+μ)

k/(+μ) ≤

(lnl)/+/(+μ)

l/(+μ)–/_ln_D

l

≤ 

lγ,

H≤

l

j=k+ Cov

Xj, Sk σk

exp

– v



l ( +μ)

≤ 

σk k

i=

l

j= |rij|exp

– v



l ( +μ)

k

k/

(lnl)/

lnDl

(lnl)/(+μ)

l/(+μ) ≤

 lγ

and

H≤ Cov

Sk σk

,Sl σl

k/

l/

(lnl)/

lnDl .

Hence () follows forγ =min(γ,γ) >  and thus () and () hold forγ =min(γ,

γ, ) > .

Proof of Lemma . On applying (), it follows from the normal comparison lemma, Lemma ., that

P

_n

i=

(Xi≤uni)

– n

i=

(uni)

≤i<j≤n

|rij|exp

– u



ni+unj ( +|rij|)



nγ +  (lnDn)+ε →

.

Hence, byn_i_=( –(uni))→τ, we get

lim

n→∞P

_n

i=

(Xi≤uni)

= lim

n→∞ n

i=

(uni) = lim n→∞exp

_n

i=

ln (uni)

= lim

n→∞exp

– n

i=

 –(uni)

(15)

That is, () holds. Using the proof ofHand (), () follows from

lim

n→∞P

_n

i=

(Xi≤uni), Sn σn

≤y

= lim

n→∞P

_n

i=

(Xi≤uni)

lim

n→∞P

Sn σn

≤y

= e–τ_(y).

Competing interests

The author declares to have no competing interests.

Author’s information

Qunying Wu is a professor, doctor, working in the ﬁeld of probability and statistics.

Acknowledgements

The author is very grateful to the referees and the Editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. The author was supported by the National Natural Science Foundation of China (11361019), project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ([2011] 47), and the Support Program of the Guangxi China Science Foundation (2013GXNSFDA019001).

Received: 18 October 2014 Accepted: 10 March 2015 References

1. Brosamler, GA: An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc.104, 561-574 (1988) 2. Schatte, P: On strong versions of the central limit theorem. Math. Nachr.137, 249-256 (1988)

3. Ibragimov, IA, Lifshits, M: On the convergence of generalized moments in almost sure central limit theorem. Stat. Probab. Lett.40, 343-351 (1998)

4. Miao, Y: Central limit theorem and almost sure central limit theorem for the product of some partial sums. Proc. Indian Acad. Sci. Math. Sci.118(2), 289-294 (2008)

5. Berkes, I, Csáki, E: A universal result in almost sure central limit theory. Stoch. Process. Appl.94, 105-134 (2001) 6. Hörmann, S: Critical behavior in almost sure central limit theory. J. Theor. Probab.20, 613-636 (2007) 7. Wu, QY: Almost sure limit theorems for stable distribution. Stat. Probab. Lett.81(6), 662-672 (2011)

8. Wu, QY: An almost sure central limit theorem for the weight function sequences of NA random variables. Proc. Indian Acad. Sci. Math. Sci.121(3), 369-377 (2011)

9. Wu, QY: A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law. J. Inequal. Appl.2012, 17 (2012). doi:10.1186/1029-242X-2012-17

10. Wu, QY, Chen, PY: An improved result in almost sure central limit theorem for self-normalized products of partial sums. J. Inequal. Appl.2013, 129 (2013). doi:10.1186/1029-242X-2013-129

11. Fahrner, I, Stadtmüller, U: On almost sure max-limit theorems. Stat. Probab. Lett.37, 229-236 (1998) 12. Nadarajah, S, Mitov, K: Asymptotics of maxima of discrete random variables. Extremes5, 287-294 (2002) 13. Csáki, E, Gonchigdanzan, K: Almost sure limit theorems for the maximum of stationary Gaussian sequences. Stat.

Probab. Lett.58, 195-203 (2002)

14. Chen, SQ, Lin, ZY: Almost sure max-limits for nonstationary Gaussian sequence. Stat. Probab. Lett.76, 1175-1184 (2006)

15. Tan, ZQ, Peng, ZX, Nadarajah, S: Almost sure convergence of sample range. Extremes10, 225-233 (2007) 16. Tan, ZQ, Peng, ZX: Almost sure convergence for non-stationary random sequences. Stat. Probab. Lett.79, 857-863

(2009)

17. Peng, ZX, Liu, MM, Nadarajah, S: Conditions based on conditional moments for max-stable limit laws. Extremes11, 329-337 (2008)

18. Peng, ZX, Wang, LL, Nadarajah, S: Almost sure central limit theorem for partial sums and maxima. Math. Nachr.282(4), 632-636 (2009)

19. Peng, ZX, Weng, ZC, Nadarajah, S: Almost sure limit theorems of extremes of complete and incomplete samples of stationary sequences. Extremes13, 463-480 (2010)

20. Zhao, SL, Peng, ZX, Wu, SL: Almost sure convergence for the maximum and the sum of nonstationary Gaussian sequences. J. Inequal. Appl.2010, Article ID 856495 (2010). doi:10.1155/2010/856495

21. Tan, ZQ, Wang, YB: Almost sure central limit theorem for the maxima and sums of stationary Gaussian sequences. J. Korean Stat. Soc.40, 347-355 (2011)

22. Chandrasekharan, K, Minakshisundaram, S: Typical Means. Oxford University Press, Oxford (1952)

23. Chuprunov, A, Fazekas, I: Almost sure versions of some analogues of the invariance principle. Publ. Math. (Debr.)

54(3-4), 457-471 (1999)

24. Chuprunov, A, Fazekas, I: Almost sure versions of some functional limit theorems. J. Math. Sci. (N.Y.)111(3), 3528-3536 (2002)

25. Fazekas, I, Rychlik, Z: Almost sure functional limit theorems. Ann. Univ. Mariae Curie-Skłodowska, Sect. A56(1), 1-18 (2002)

26. Leadbetter, MR, Lindgren, G, Rootzén, H: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983)