Wu Journal of Inequalities and Applications (2015) 2015:109
DOI 10.1186/s13660-015-0634-3
RESEARCH
Open Access
Improved results in almost sure central limit
theorems for the maxima and partial sums for
Gaussian sequences
Qunying Wu*
*
Correspondence:
wqy666@glut.edu.cn
College of Science, Guilin University
of Technology, Guilin, 541004,
P.R. China
Abstract
Let X, X1 , X2 , . . . be a standardized Gaussian sequence. The universal results in almost
sure central limit theorems for the maxima Mn and partial
sums and maxima
(Sn /σn , Mn ) are established, respectively, where Sn = ni=1 Xi , σn2 = Var Sn , and
Mn = max1≤i≤n Xi .
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 investigated 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 developments in the recent past years. Significant recent contributions can be found in Csáki and
Gonchigdanzan [], Chen and Lin [], Tan et al. [], and Tan and Peng [], extending
this principle by proving ASCLT for the maxima of a Gaussian sequence. Further, Peng
et al. [–], Zhao et 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 random variables, and for any choice of n, i , . . . , in , the joint distribution of Xi , . . . , Xin is
an n-dimensional normal distribution. Throughout this paper we assume {Xn ; n ≥ } is
a standardized Gaussian sequence with covariance ri,j := Cov(Xi , Xj ). For each n ≥ , let
Sn = ni= Xi , σn = Var Sn , Mn = max≤i≤n Xi . The symbols Sn /σn and Mn denote partial
sums and maxima, respectively. Let (·) and φ(·) denote the standard normal distribution
function and its density function, respectively, and I denote an indicator function. An ∼ Bn
denotes limn→∞ An /Bn = , and An Bn means that there exists a constant c >  such that
An ≤ cBn for sufficiently large n. The symbol c stands for a generic positive constant which
© 2015 Wu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly credited.
Wu Journal of Inequalities and Applications (2015) 2015:109
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may differ from one place to another. The normalizing constants an and bn are defined by
an = ( ln n)/ ,
bn = a n –
ln ln n + 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

. Let the numerical se|rij | ≤ ρ|i–j| for i = j where ρn <  for all n ≥  and ρn ln n(ln ln
n)+ε
quence {uni ;  ≤ i ≤ n, n ≥ } be such that n( – (λn )) is bounded and λn = min≤i≤n uni ≥
c ln/ n for some c > . If ni= ( – (uni )) → τ as n → ∞ for some τ ≥ , then
k
n
  lim
I
(Xi ≤ uki ) = exp(–τ ) a.s.
n→∞ ln n
k i=
k=
Zhao et 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 ni= ( – (uni )) → τ
for some  < τ < ∞ and n(–(λn )) is bounded, where λn = min≤i≤n uni . If supi=j |rij | = δ < ,
j–
n |rij | = o(n),
()
j= i=
sup
n
|rij | i≥ j=
ln/ n
(ln ln n)+ε
for some ε > ,
()
then
k
n
Sk
  lim
(Xi ≤ uki ),
≤ y = exp(–τ )(y) a.s. for all y ∈ R
I
n→∞ ln n
k i=
σk
()
k=
and
n
Sk
 
I ak (Mk – bk ) ≤ x,
≤ y = exp –e–x (y) a.s. for all x, y ∈ R. ()
n→∞ ln n
k
σk
lim
k=
By the terminology of summation procedures (see e.g. Chandrasekharan and Minakshisundaram [], 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 determine the optimal weights.
The purpose of this paper is to give substantial improvements for weight sequences and
to weaken greatly conditions () and () in Theorem B obtained by Zhao et al. []. We
will study and establish the ASCLT for maximum Mn and maximum and partial sums of
the standardized Gaussian sequences, and we will show that the ASCLT holds under a
fairly general growth condition on dk = k – exp(lnα k),  ≤ α < /.
Wu Journal of Inequalities and Applications (2015) 2015:109
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2 Main results
Set
dk =
exp(lnα k)
,
k
Dn =
n
dk
for  ≤ α < /.
()
k=
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 ni= ( – (uni )) → τ for some  ≤ τ < ∞ and
n( – (λn )) is bounded, where λn = min≤i≤n uni . Suppose that ρn <  for all n ≥  such that
|rij | ≤ ρ|i–j|
for i = j,
ρn 
ln n(ln Dn )+ε
for some ε > .
()
Then
k
n
 dk I
(Xi ≤ uki ) = exp(–τ ) a.s.
lim
n→∞ Dn
i=
()
k=
and
n
 dk I ak (Mk – bk ) ≤ x = exp –e–x
n→∞ Dn
lim
a.s. for any x ∈ R,
()
k=
where an and bn are defined by ().
Theorem . Let {Xn ; n ≥ } be a standardized Gaussian sequence. Let the numerical se
quence {uni ;  ≤ i ≤ n, n ≥ } be such that ni= ( – (uni )) → τ for some  ≤ τ < ∞ and
n( – (λn )) is bounded, where λn = min≤i≤n uni . Suppose that supi=j |rij | = δ < , there exists
a constant  < c < / such that
rij ≤ cn,
()
≤i<j≤n
max
≤i≤n
n
|rij | j=
ln/ n
.
ln Dn
()
Then
k
n
Sk
 dk I
(Xi ≤ uki ),
≤ y = exp(–τ )(y) a.s. for any y ∈ R
lim
n→∞ Dn
σk
i=
()
k=
and
n
Sk
 dk I ak (Mk – bk ) ≤ x,
≤y
lim
n→∞ Dn
σk
k=
= exp –e–x (y) a.s. for any x, y ∈ R,
where an and bn are defined by ().
()
Wu Journal of Inequalities and Applications (2015) 2015:109
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Taking uki = uk for  ≤ i ≤ k in 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 )) → τ for some  ≤ τ < ∞. Suppose that condition () is satisfied. Then () and
n
 dk I(Mk ≤ uk ) = exp(–τ ) a.s.
lim
n→∞ Dn
k=
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
supi=j |rij | = δ < , there exists a constant  < c < / such that conditions () and () are
satisfied. Then () and
n
Sk
 dk I Mk ≤ uk ,
≤ y = exp(–τ )(y) a.s. for any y ∈ R
n→∞ Dn
σk
lim
k=
hold.
By the terminology of summation procedures (see e.g. Chandrasekharan and Minakshisundaram [], 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  ≤ dk∗ ≤ dk , ∞
k= dk = ∞.
Remark . Obviously, the condition () is significantly weaker than the condition (),
and in particular taking α = , i.e., the weight dk = e/k, we have Dn ∼ e ln n and ln Dn ∼
ln ln n, 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 restrictions on the covariance rij in Theorem B obtained by Zhao et al. [].
Remark . Theorem A obtained by Chen and Lin [] is a special case of Theorem .
when α = . When {Xn ; n ≥ } is stationary, uni = un ,  ≤ i ≤ n, and α = , Theorem . is
Corollary . obtained by Csáki and Gonchigdanzan [].
Remark . Whether (), (), (), and () work also for some / ≤ α <  remains an
open question.
3 Proofs
The proof of our results follows a well-known scheme of the proof of an a.s. limit theorem, e.g. Berkes and Csáki [], Chuprunov and Fazekas [, ], and Fazekas and Rychlik
[]. We will point out that the weight from dk = /k is extended to dk = exp(lnα k)/k,
 ≤ α < /, and relaxed restrictions on the covariance rij encountered great difficulties
Wu Journal of Inequalities and Applications (2015) 2015:109
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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 Leadbetter et al. []) Suppose
ξ , . . . , ξn are standard normal variables with covariance matrix  = ( ij ), and η , . . . , ηn
similarly with covariance matrix  = ( ij ), and let ρij = max(| ij |, | ij |), maxi=j ρij = δ < .
Further, let u , . . . , un be real numbers. Then
P(ξj ≤ uj for j = , . . . , n) – P(ηj ≤ uj for j = , . . . , n)
ui + uj


exp
–
–
≤K
ij
ij
( + ρij )
≤i<j≤n
for some constant K , depending only on δ.
Lemma . Suppose that the conditions of Theorem . hold, then there exists a constant
γ >  such that
sup
uki + ulj


γ +
|rij | exp –
,
( + |rij |)
l
(ln Dl )+ε
j=i+
l
k ≤k≤l i=
EI
l
(Xi ≤ uli ) – I
i=
Cov I
l
(Xi ≤ uli )
i=k+
k
(Xi ≤ uki ) , I
i=
l
(Xi ≤ uli )
i=k+
k
l
()
for  ≤ k < l,


+
lγ (ln Dl )+ε
()
for  ≤ k < l,
()
where ε is defined by ().
Lemma . Suppose that the conditions of Theorem . hold, then there exists a constant
γ >  such that
l
l
γ
Sl
Sl
k
EI
(Xi ≤ uli ), ≤ y – I
(Xi ≤ uli ), ≤ y σ
σ
l
l
l
i=
i=k+
l
k
Sk
Sl
(Xi ≤ uki ),
≤ y ,I
(Xi ≤ uli ), ≤ y
Cov I
σk
σl
i=
i=k+
γ
k / (ln l)/
l
k
+ /
for  ≤ k <
.
l
l ln Dl
ln l
for  ≤ k < l, ()
()
The following weak convergence results are the extended versions of Theorem .. of
Leadbetter et al. [] to the nonstationary normal random variables.
Lemma . Suppose that the conditions of Theorem . hold, then
n
lim P
(Xi ≤ uni ) = e–τ .
n→∞
i=
()
Wu Journal of Inequalities and Applications (2015) 2015:109
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Suppose that the conditions of Theorem . hold, then
n
Sn
lim P
(Xi ≤ uni ),
≤ y = e–τ (y).
n→∞
σ
n
i=
()
Lemma . Let {ξn ; n ≥ } be a sequence of uniformly bounded random variables. If
Var
n
dk ξk k=
Dn
(ln Dn )+ε
for some ε > , then
n
 dk (ξk – Eξk ) =  a.s.,
n→∞ Dn
lim
k=
where dn and Dn are defined by ().
Proof Similarly to the proof of Lemma . in Wu [], we can prove Lemma ..
n
i= (Xi
Proof of Theorem . Using Lemma ., P(
Toeplitz lemma,
≤ uni )) → exp(–τ ), and hence by the
k
n
 dk P
(Xi ≤ uki ) = exp(–τ ).
lim
n→∞ Dn
i=
k=
Therefore, in order to prove (), it suffices to prove that
k
k
n
 dk I
(Xi ≤ uki ) – P
(Xi ≤ uki )
=  a.s.,
lim
n→∞ Dn
i=
i=
k=
which will be done by showing that
Var
n
k
dk I
(Xi ≤ uki )
i=
k=
Dn
(ln Dn )+ε
for some ε >  from Lemma .. Let ξk := I(
 and |ξk | ≤  for all k ≥ . Hence
Var
n
k=
k
dk I
(Xi ≤ uki )
i=
=
n
k
i= (Xi
k=
x (ln u)β–
u
Since |ξk | ≤  and exp( lnβ x) = exp( 
infinity, from Seneta [], it follows that
∞
exp( lnα k)
k=
k
=c≤
≤ uki )) – P(
dk Eξk + 
:= T + T .
T ≤
()
Dn
.
(ln Dn )+ε
k
i= (Xi
≤ uki )). Then Eξk =
dk dl E(ξk ξl )
≤k<l≤n
()
du), β < , is a slowly varying function at
()
Wu Journal of Inequalities and Applications (2015) 2015:109
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By Lemma ., for  ≤ k < l,
l
l
k
E(ξk ξl ) ≤ Cov I
(Xi ≤ uki ) , I
(Xi ≤ uli ) – I
(Xi ≤ uli )
i=
+ Cov I
E I
i=
k
l
(Xi ≤ uki ) , I
(Xi ≤ uli )
i=
l
i=k+
(Xi ≤ uli ) – I
i=
i=k+
l
(Xi ≤ uli )
i=k+
l
k
(Xi ≤ uki ) , I
(Xi ≤ uli )
+ Cov I
i=
i=k+
γ

k
+
l
(ln Dl )+ε
for γ = min(, γ ) > . Hence,
T n l
dk dl
l= k=
γ n l

k
+
dk dl
l
(ln Dl )+ε
l= k=
:= T + T .
()
By () in Wu [],
 –α
ln n exp lnα n ,
ln Dn ∼ lnα n,
α
αDn
for α > .
exp lnα n ∼
–α
(ln Dn ) α
Dn ∼
()
–β
x
From this, combined with the fact that  l(t)
dt ∼ l(x)x
as x → ∞ for β <  and l(x) is
–β
tβ
a slowly varying function at infinity (see Proposition .. in Bingham et al. []), we get
n
l
n
dl exp(lnα k)
dl γ
l exp lnα l
γ
–γ


l
k
l γ
l=
k=
l=
Dn
, α > ,
≤ Dn exp lnα n (ln Dn )(–α)/α
Dn ,
α=
T ≤
≤
Dn
(ln Dn )+ε
()
for  < ε < ( – α)/α.
Now, we estimate T . For α > , by ()
T =
n
l=
∼
e
n
exp( lnα l)(ln l)–α–αε
dl
D
l
(ln Dl )+ε
l
n
l=
α
exp( ln x)(ln x)–α–αε
dx =
x
ln n
exp yα y–α–αε dy

Wu Journal of Inequalities and Applications (2015) 2015:109
ln n ∼
exp yα y–α–αε +

Page 8 of 15
 – α – αε
exp yα y–α–αε dy
α
ln n
(α)– exp yα y–α–αε dy
=

exp  lnα n (ln n)–α–αε
Dn
.
(ln Dn )+ε
()
For α = , noting the fact that Dn ∼ ln n, similarly we get
T ∼
n
l=
ln l
∼
l(ln ln l)+ε
ln n
=
ln 

n
ln x
dx
x(ln ln x)+ε
ln n
Dn
y
dy
∼
.
(ln y)+ε
(ln ln n)+ε
(ln Dn )+ε
()
Equations ()-(), ()-() together establish (), which concludes the proof of ().
Next, take uni = un = x/an + bn . Then we see that ni= ( – (uni )) = n( – (un )) → exp(–x)
as n → ∞ (see Theorem .. in Leadbetter et al. []) and hence () immediately follows
from () with uni = x/an + bn .
This completes the proof of Theorem ..
Proof of Theorem . Using Lemma ., P(
hence by the Toeplitz lemma,
n
i= (Xi
≤ uni ), Sn /σn ≤ y) → e–τ (y), and
k
n
Sk
 dk P
(Xi ≤ uki ),
≤ y = e–τ (y).
lim
n→∞ Dn
σ
k
i=
k=
Therefore, in order to prove (), it suffices to prove that
k
k
n
Sk
Sk
 lim
dk I
(Xi ≤ uki ),
≤y –P
(Xi ≤ uki ),
≤y
=
n→∞ Dn
σk
σk
i=
i=
a.s.,
k=
which will be done by showing that
Var
n
k=
k
Sk
Dn
dk I
(Xi ≤ uki ),
≤y
σk
(ln Dn )+ε
i=
for some ε >  from Lemma .. Let ηk := I(
Sk
σk
≤ y). By Lemma ., for  ≤ k < l/ ln l,
k
i= (Xi
()
≤ uki ), σSk ≤ y) – P(
k
l
k
Sk
Sl
E(ηk ηl ) ≤ Cov I
(Xi ≤ uki ),
≤ y ,I
(Xi ≤ uli ), ≤ y
σk
σl
i=
i=
l
Sl
(Xi ≤ uli ), ≤ y
–I
σl
i=k+
k
i= (Xi
≤ uki ),
Wu Journal of Inequalities and Applications (2015) 2015:109
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l
k
Sk
Sl
+ Cov I
(Xi ≤ uki ),
≤ y ,I
(Xi ≤ uli ), ≤ y
σk
σl
i=
i=k+
l
l
Sl
Sl
(Xi ≤ uli ), ≤ y – I
(Xi ≤ uli ), ≤ y
≤EI
σl
σl
i=
i=k+
l
k
Sk
Sl
(Xi ≤ uki ),
≤ y ,I
(Xi ≤ uli ), ≤ y
+ Cov I
σk
σl
i=
i=k+
γ
k
k / ln/ l
+ /
.
l
l ln Dl
Hence,
Var
n
k=
=
n
k
Sk
dk I
(Xi ≤ uki ),
≤y
σk
i=
n
dk dl
l= ≤k<l/ ln l
+
dk dl E(ηk ηl ) ≤k<l≤n
k=
dk Eηk + 
dk dl E(ηk ηl )
≤k<l≤n
γ n
k
k / ln/ l
+
dk dl /
l
l ln Dl
l= ≤k<l/ ln l
n
dk dl
l= l/ ln l≤k≤l
:= T + T + T .
()
By the proof of (),
T Dn
(ln Dn )+ε
for  < ε < ( – α)/α.
()
Now, we estimate T . For α > , by ()
n
n
l
dl ln/ l exp(lnα k)
dl ln/ l /
l exp lnα l
T ≤
l/ ln Dl
k /
l/ ln Dl
l=
n
k=
α
l=
ln n
exp( ln x)(ln x)
dx =
exp yα y/–α dy
x
e

ln n 
–
α
exp yα y/–α +
exp yα y/–α dy
∼
α

ln n
=
(α)– exp yα y/–α dy
∼
/–α

exp  lnα n (ln n)/–α Dn
(ln Dn )+ε
for  < ε < /(α) – .
Dn
(ln Dn )/α
()
Wu Journal of Inequalities and Applications (2015) 2015:109
Page 10 of 15
For α = ,
T n
l
n
ln/ l 
ln/ l
/
/
l ln ln l
k
l ln ln l
l=
k=
n
(ln x)/
dx =
x ln ln x
∼

T ≤
/
l=
ln n
ln 
y/
dy
ln y
D/
n
(ln n)
∼
,
ln ln n
(ln Dn )
n
dl exp lnα l
l=
()

dl exp lnα l ln ln l
k
n
l/ ln l≤k≤l
l=
Dn exp lnα n ln ln n

Dn ln ln Dn
, α > ,
(ln Dn )(–α)/α
α=
Dn ln Dn ,
≤
Dn
.
(ln Dn )+ε
()
Equations ()-() together establish () for ε = min(ε , ε ) > , which concludes the
proof of (). Next, take uni = un = x/an + bn . Then we see that ni= ( – (uni )) = n( –
(un )) → exp(–x) as n → ∞ (see Theorem .. in Leadbetter et al. []) and hence ()
immediately follows from () with uni = x/an + bn .
This completes the proof of Theorem ..
Appendix
Proof of Lemma . By assumption (), we have δ := supi=j |rij | < . Define λ such that  <
λ < /( + δ) – , for  ≤ k ≤ l,
uki + ulj
|rij | exp –
( + |rij |)
j=i+
k l
i=
=
k
i= i+≤j≤l,j–i≤l
+
k
uki + ulj
|rij | exp –
( + |rij |)
λ
i= i+≤j≤l,j–i>l
uki + ulj
|rij | exp –
( + |rij |)
λ
:= H + H .
()
Since n( – (λn )) is bounded, where λn is the same as defined in Theorem ., there
exists a constant c >  such that n( – (λn )) ≤ c. vn is defined to satisfy n( – (vn )) = c;
then clearly vn ≤ λn .
Since  – (x) ∼ φ(x)/x as x → ∞, we have

vn
v
exp – n ∼ c ,

n
vn ∼
√
 ln n.
()
Wu Journal of Inequalities and Applications (2015) 2015:109
Page 11 of 15
By () and (),
k
H ≤
i= i+≤j≤l,j–i≤l
≤ klλ
   
k
vk + vl
v + vl
ρj–i exp –
ρs exp – k
≤
(
+
δ)
(
+ δ)
λ
λ
i=
≤s≤l
/(+δ)
(ln k)
k /(+δ)
/(+δ)
(ln l)
l/(+δ)
≤
/(+δ)
(ln l)
l/(+δ)––λ

lγ
≤
()
for  < γ < /( + δ) –  – λ.
Setting σj = supi≥j ρi , by () and (),
σlλ = sup ρi sup
i≥lλ
i≥lλ



=
ln i(ln Di )+ε ln lλ (ln Dlλ )+ε
ln l(ln Dl )+ε
and
σl λ v k v l 
(ln Dl )+ε
for all  ≤ k ≤ l.
This, combined with (), shows
k
H ≤
i= i+≤j≤l,j–i>l
k ≤
i= lλ ≤s≤l
klσlλ
v + vl
ρj–i exp – k
( + ρj–i )
λ
 
v + vl
σ λ (v + vl )
ρs exp – k
exp l k



vk vl
.
k l
(ln Dl )+ε
This, together with () and () implies that () holds.
It is well known that P(B) – P(AB) ≤ P(Ā) for any sets A and B, then using the condition
that n( – (λn )) is bounded, for  ≤ k < l, we get
EI
l
(Xi ≤ uli ) – I
i=
=P
l
l
(Xi ≤ uli )
i=k+
(Xi ≤ uli ) – P
i=k+
l
(Xi ≤ uli )
i=
≤ P(Xi > uli for some  ≤ i ≤ k) ≤
k
i=
k
≤ k  – (λl ) = l  – (λl )
l
k
.
l
Hence, () holds.
P(Xi > uli )
Wu Journal of Inequalities and Applications (2015) 2015:109
Page 12 of 15
Now we prove (). By (), applying the normal comparison lemma, Lemma ., for
 ≤ k < l,
Cov I
k
(Xi ≤ uki ) , I
i=
l
(Xi ≤ uli )
i=k+
= P(X ≤ uk , . . . , Xk ≤ ukk , Xk+ ≤ ul,k+ , . . . , Xl ≤ ull )
– P(X ≤ uk , . . . , Xk ≤ ukk )P(Xk+ ≤ ul,k+ , . . . , Xl ≤ ull )
k l
i= j=k+
uki + ulj
|rij | exp –
( + |rij |)


+
.
γ
l
ln l(ln Dl )+ε
Hence, () holds.
Proof of Lemma . Notice, for  ≤ k < l,
l
Sl
Sl
(Xi ≤ uli ), ≤ y – I
(Xi ≤ uli ), ≤ y
EI
σl
σl
i=
i=k+
l
l
Sl
Sl
(Xi ≤ uli ), ≤ y – P
(Xi ≤ uli ), ≤ y
=P
σl
σl
i=
i=k+
l
l
Sl
Sl
(Xi ≤ uli ), ≤ y – P
(Xi ≤ uli ) P
≤y
≤ P
σl
σl
i=
i=
l
l
Sl
Sl
(Xi ≤ uli ), ≤ y – P
(Xi ≤ uli ) P
≤y
+ P
σl
σl
i=k+
i=k+
l
l
Sl
≤y P
(Xi ≤ uli ) – P
(Xi ≤ uli )
+P
σl
i=
l
i=k+
:= H + H + H .
()
Using l – | ≤i<j≤l rij | ≤ σl ≤ l + | ≤i<j≤l rij | and (), there exist constants ci > ,
i = , , such that
c l ≤ σl ≤ c l.
()
Hence, using (), for  ≤ i ≤ l ≤ n,
l
Sl
ln/ l
 |rij | /
≤
Cov Xi ,
σl
σl j=
l ln Dl
()
l
k
k / (ln l)/
 Sk Sl
,
|rij | /
.
≤
Cov
σk σl
σk σl i= j=
l
ln Dl
()
and
Wu Journal of Inequalities and Applications (2015) 2015:109
Page 13 of 15
Noting the fact that ln l and ln Dl are slowly varying functions at infinity, () and ()
imply that there exists  < μ <  such that, for sufficiently large l,
Sl
max Cov Xi ,
≤μ
≤i≤l
σl
and
Sk Sl
≤ μ.
max Cov
,
≤k<l/ ln l
σk σl
Combining (), (), and the normal comparison lemma, Lemma ., for i = , ,
Hi l
i=
≤
l
i=
≤

uli + y
Sl
exp –
Cov Xi ,
σl
( + μ)
vl
Sl
exp –
Cov Xi ,
σl
( + μ)

(ln l)/+/(+μ)
≤ γ
/(+μ)–/
l
ln Dl l 
()
for  < γ < /( + μ) – /.
By the proof of (), we have H k/l. This, combined with () and (), implies that
() holds for γ = min(, γ ) > .
Now we prove (). Again applying the normal comparison lemma, Lemma ., for  ≤
k < l/ ln l,
l
Sk
Sl
Cov I
(Xi ≤ uki ),
≤ y ,I
(Xi ≤ uli ), ≤ y
σk
σl
i=
i=k+
Sk
Sl
= P X ≤ uk , . . . , Xk ≤ ukk ,
≤ y, Xk+ ≤ ul,k+ , . . . , Xl ≤ ull , ≤ y
σk
σl
Sk
≤y
– P X ≤ uk , . . . , Xk ≤ ukk ,
σk
Sl
× P Xk+ ≤ ul,k+ , . . . , Xl ≤ ull , ≤ y
σl
k l
uki + ulj
|rij | exp –
( + |rij |)
i=
k
j=k+
+
k
i=
+
l
j=k+
uki + y
Sl
exp –
Cov Xi ,
σl
( + | Cov(Xi , Sl /σl )|)
ulj + y
Sk
exp –
Cov Xj ,
σk
( + | Cov(Xj , Sk /σk )|)
y
Sk Sl
exp –
,
+ Cov
σk σl
 + | Cov(Sk /σk , Sl /σl )|
:= H + H + H + H .
Wu Journal of Inequalities and Applications (2015) 2015:109
Page 14 of 15
By (), () to (),
k l
H ≤
i= j=
k
 
v + vl
|rij | exp – k
( + δ)
(ln l)/ (ln k)/(+δ) (ln l)/(+δ)
ln Dl
k /(+δ)
l/(+δ)
(ln l)/+/(+δ)

≤ γ
l/(+δ)– ln Dl
l
≤
for  < γ < /( + δ) – ,
H k
i=
vk
Sl
exp –
Cov Xi ,
σl
( + μ)
(ln l)/+/(+μ)

(ln l)/ (ln k)/(+μ)
≤
≤ ,
l/ ln Dl k /(+μ)
l/(+μ)–/ ln Dl lγ
l
vl
Sk
H ≤
exp –
Cov Xj ,
σk
( + μ)
k
j=k+
l
k
vl
 |rij | exp –
≤
σk i= j=
( + μ)

k (ln l)/ (ln l)/(+μ)
≤ γ
/
/(+μ)
k
ln Dl
l
l
and
Sk Sl
k / (ln l)/
,
.
/
H ≤ Cov
σk σl
l
ln Dl
Hence () follows for γ = min(γ , γ ) >  and thus () and () hold for γ = min(γ ,
γ , ) > .
Proof of Lemma . On applying (), it follows from the normal comparison lemma,
Lemma ., that
n
n
uni + unj
(Xi ≤ uni ) –
(uni ) |rij | exp –
P
( + |rij |)
i=
i=
≤i<j≤n
Hence, by
n
i= ( – (uni )) → τ ,


+
→ .
nγ (ln Dn )+ε
we get
n
n
n
lim P
(Xi ≤ uni ) = lim
(uni ) = lim exp
ln (uni )
n→∞
i=
n→∞
i=
n→∞
= lim exp –
n→∞
n
i=
i=
 – (uni )
= e–τ .
Wu Journal of Inequalities and Applications (2015) 2015:109
Page 15 of 15
That is, () holds. Using the proof of H and (), () follows from
n
n
Sn
Sn
(Xi ≤ uni ),
≤ y = lim P
(Xi ≤ uni ) lim P
≤y
lim P
n→∞
n→∞
n→∞
σn
σn
i=
i=
= 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
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