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We give here an almost sure central limit theorem for product of sums of strongly mixing positive random variables.. Introduction and Results In recent decades, there has been a lot of w

Volume 2011, Article ID 576301, 9 pages

doi:10.1155/2011/576301

Research Article

Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables

Daxiang Ye and Qunying Wu

College of Science, Guilin University of Technology, Guilin 541004, China

Correspondence should be addressed to Daxiang Ye,3040801111@163.com

Received 19 September 2010; Revised 1 January 2011; Accepted 26 January 2011

Academic Editor: Ondˇrej Doˇsl ´y

Copyrightq 2011 D Ye and Q Wu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We give here an almost sure central limit theorem for product of sums of strongly mixing positive random variables

1 Introduction and Results

In recent decades, there has been a lot of work on the almost sure central limit theorem

ASCLT, we can refer to Brosamler 1, Schatte 2, Lacey and Philipp 3, and Peligrad and Shao4

Khurelbaatar and Rempala5 gave an ASCLT for product of partial sums of i.i.d random variables as follows

Theorem 1.1 Let {X n, n ≥ 1} be a sequence of i.i.d positive random variables with EX1  μ > 0

and Var X1  σ2 Denote γ  σ/μ the coefficient of variation Then for any real x

lim

n → ∞

1

ln n

n



k1

1

k I

⎛

i1 Si k!μ k

1/γ√

k

≤ x

⎞

where Sn  n

k1 Xk, I∗ is the indicator function, F· is the distribution function of the random variable eN, and N is a standard normal variable.

Recently, Jin6 had proved that 1.1 holds under appropriate conditions for strongly mixing positive random variables and gave an ASCLT for product of partial sums of strongly mixing as follows

Theorem 1.2 Let {X n, n ≥ 1} be a sequence of identically distributed positive strongly mixing random variable with EX1  μ > 0 and VarX1  σ2, dk  1/k, D n  n

k1 dk Denote by

γ  σ/μ the coefficient of variation, σ2

n Var n

k1 S k − kμ/kσ and B2

n  VarS n  Assume

E |X1|2δ< ∞ for some δ > 0, lim

n → ∞

B2

n

n  σ2

0 > 0,

α n  On −r

for some r > 1  2

δ , n∈Ninf

σ2

n

n > 0.

1.2

Then for any real x

lim

n → ∞

1

Dn

n



k1 dkI

⎛

i1 Si k!μ k

1/γσ k

≤ x

⎞

The sequence{d k, k ≥ 1} in 1.3 is called weight Under the conditions ofTheorem 1.2,

it is easy to see that1.3 holds for every sequence d∗

kwith 0≤ d∗

k ≤ d k and D n∗ k≤n d k∗ → ∞

7 Clearly, the larger the weight sequence d k is, the stronger is the result 1.3

In the following sections, let d k  e ln k α

/k, 0 ≤ α < 1/2, Dn n

k1 dk, “ ” denote the inequality “≤” up to some universal constant

We first give an ASCLT for strongly mixing positive random variables

Theorem 1.3 Let {X n, n ≥ 1} be a sequence of identically distributed positive strongly mixing random variable with EX1  μ > 0 and VarX1  σ2, dk and Dn as mentioned above Denote

by γ  σ/μ the coefficient of variation, σ2

n Var n

k1 S k − kμ/kσ and B2

n  VarS n  Assume

that

E |X1|2δ< ∞ for some δ > 0, 1.4

α n  On −r

for some r > 1 2

lim

n → ∞

B2

n

n  σ2

inf

n∈N

σ2

n

Then for any real x

lim

n → ∞

1

Dn

n



k1 dkI

⎛

i1 Si k!μ k

1/γσ k

≤ x

⎞

In order to proveTheorem 1.3we first establish ASCLT for certain triangular arrays of

random variables In the sequel we shall use the following notation Let b k,n n

ik 1/i and

s2k,n  k

i1 b2i,n for k ≤ n with b k,n  0 if k > n Y k  X k n  n

k1 Ykand

Sn,n n

k1 bk,nYk

In this setting we establish an ASCLT for the triangular arrayb k,nYk.

Theorem 1.4 Under the conditions of Theorem 1.3 , for any real x

lim

n → ∞

1

Dn

n



k1 dkI



Sk,k

σk ≤ x



where Φx is the standard normal distribution function.

2 The Proofs

2.1 Lemmas

To prove theorems, we need the following lemmas

Lemma 2.1 see 8 Let {X n, n ≥ 1} be a sequence of strongly mixing random variables with zero mean, and let {a k,n, 1 ≤ k ≤ n, n ≥ 1} be a triangular array of real numbers Assume that

sup

n

n



k1

a2k,n < ∞, max

1≤k≤n|a k,n | −→ 0 as n −→ ∞. 2.1

If for a certain δ > 0, {|Xk|2δ} is uniformly integrable, inf kVarXk  > 0,

∞



n1

n 2/δ α n < ∞, Var

 n



n1 ak,nXk



then

n



k1

Lemma 2.2 see 9 Let d k eln k α

/k, 0 ≤ α < 1/2, Dn  n

k1 dk ; then

Dn ∼ Cln n1−αexp

ln n α

where C  1/α as 0 < α < 1/2, C  1 as α  0.

Lemma 2.3 see 8 Let {X n, n ≥ 1} be a strongly mixing sequence of random variables such that

supn E|Xn|2δ < ∞ for a certain δ > 0 and every n ≥ 1 Then there is a numerical constant cδ depending only on δ such that for every n > 1 one has

sup

j

nj



ij1

Cov

Xi, Xj  ≤ cδn

i1

i 2/δ α i

δ/2δ

sup

k

X k2

where X k p  E|X k|p1/p , p > 1.

Lemma 2.4 see 9 Let {ξ k, k ≥ 1} be a sequence of random variables, uniformly bounded below and with finite variances, and let {d k, k ≥ 1} be a sequence of positive number Let for n ≥ 1, Dn 

n

k1 dk and Tn  1/D n n

k1 dkξk Assume that

Dn−→ ∞ Dn1

as n → ∞ If for some ε > 0, C and all n

ET n2≤ Cln−1−ε Dn

then

Lemma 2.5 see 10 Let {X n, n ≥ 1} be a strongly mixing sequence of random variables with zero mean and sup n E|Xn|2δ< ∞ for a certain δ > 0 Assume that 1.5 and 1.6 hold Then

lim sup

n → ∞

|S n|



2σ02n ln ln n

2.2 Proof of Theorem 1.4

From the definition of strongly mixing we know that{Y k, k ≥ 1} remain to be a sequence of

identically distributed strongly mixing random variable with zero mean and unit variance

Let a k,n  b k,n/σn; note that

n



k1

b2k,n  b 1,n 2n

k2

k−1



i1

1

k  b 1,n 2n

k2

k − 1

and via1.7 we have

sup

n

n



k1

a2k,n sup

n

n



k1

b2k,n

σ2

n

sup

n

2n − b 1,n

n < ∞,

max 1≤k≤n|a k,n|  max

1≤k≤n

bk,n

σn ln n√

2.11

From the definition of Y kand1.4 we have that {|Y k|2δ} is uniformly integrable; note that

inf

k VarYk   EY2

1  1 > 0, Var

n



k1 ak,nYk



 Var n k1 bk,nYk

σ2

n

and applying1.5

∞



n1

n 2/δ α n ∞

n1

Consequently usingLemma 2.1, we can obtain

Sn,n σn

d

which is equivalent to

Ef



Sn,n σn



for any bounded Lipschitz-continuous function f; applying Toeplitz Lemma

1

Dn

n



k1 dkEf



Sk,k σk



We notice that1.9 is equivalent to

lim

n → ∞

1

Dn

n



k1 dkf



Sk,k σk



for all bounded Lipschitz continuous f; it therefore remains to prove that

Tn 1

Dn

n



k1 dk



f



Sk,k σk



− Ef



Sk,k σk



a.s.

Let ξ k  fS k,k/σk  − EfS k,k /σk,

E

n

k1 dkξk

2

≤ E



1≤k≤l≤n

dkdlξkξl



1≤k≤l≤n

dkdl |Eξ kξl|

1≤k≤l≤n

l≤2k

dkdl |Eξ kξl|  

1≤k≤l≤n

l>2k dkdl |Eξ kξl|

 T 1,n  T 2,n

2.19

FromLemma 2.2, we obtain for some constant C1

eln n α ∼ C1Dn ln D n1−1/α. 2.20

Using2.20 and property of f, we have

T 1,n eln n αn

k1 dk

2k



lk

1

l D neln n α D2

We estimate now T 2,n For l > 2k,

Sl,l − S 2k,2k  b 1,l Y1 b 2,l Y2 · · ·  b l,lYl  − b 1,2k Y1 b 2,2k Y2 · · ·  b 2k,2k Y 2k

 b 2k1,l 2k  b 2k1,l Y 2k1  · · ·  b l,lYl . 2.22

Notice that

|Eξ kξl| 

Covf



Sk,k σk



, f



Sl,l σl





≤



Cov



f



Sk,k σk



, f



Sl,l σl



− f



Sl,l − S 2k,2k − b 2k1,l 2k

σl











Cov



f



Sk,k σk



, f



Sl,l − S 2k,2k − b 2k1,l 2k

σl





,

2.23

and the properties of strongly mixing sequence imply





Cov



f



Sk,k σk



, f



Sl,l − S 2k,2k − b 2k1,l 2k

σl





ApplyingLemma 2.3and2.10,

VarS2k,2k 2k

i1

b2i,2k EY i2 22k−1

j1

2k



ij1 bi,2kbj,2kCov

Yi, Yj

≤2k

i1

b2i,2k 22k−1

j1

b j,2k2

2k



ij1

Cov

Yi, Yj  k,

Var

2k



 E

2k

i1 Yi

2

2k

i1

EY i2 22k−1

i1

2k



ji1

Cov

Yi, Yj

k.

2.25

Consequently, via the properties of f, the Jensen inequality, and 1.7,





Cov



f



Sk,k σk



, f



Sl,l σl



− f



Sl,l − S 2k,2k − b 2k1,l 2k

σl









S 2k,2k  b 2k1,l 2k



ES2

2k,2k

σl 



E

b 2k1,l 2k

2

σl



 VarS2k,2k

σl  b 2k1,l

 Var

2k





k l

β

,

2.26

where 0 < β < 1/2 Hence for l > 2k we have

|Eξ kξl | αk 



k l

β

Consequently, we conclude from the above inequalities that

T 2,n  1≤k≤l≤n

l>2k dkdl



α k 



k l

β

1≤k≤l≤n

l>2k

dkdlα k  

1≤k≤l≤n

l>2k dkdl



k l

β

 T 2,n,1  T 2,n,2

2.28

Applying1.5 andLemma 2.2we can obtain for any η > 0

T 2,n,1≤n

k1

n



l1

dkdlα k ln D n−1−ηn

k1 dk n



l1

dl  D2

n ln D n−1−η 2.29

Notice that

T 2,n,2 

1≤k≤l≤n

l>2k

l/k≥ln D n2/β

dkdl



k l

β

1≤k≤l≤n

l>2k

l/k<ln D n2/β

dkdl



k l

β

 T 2,n,2,1  T 2,n,2,2 ,

2.30

T 2,n,2,1≤ 

1≤k≤l≤n

l>2k

dkdl ln D n−2≤ ln D n−2n

k1 dk n



l1

dl  D2

n ln D n−2. 2.31

Let n0 max{l : k ≤ l ≤ n, l/k < ln D n2/β}, then

T 2,n,2,2≤n

k1

n0



l2k

dkdl≤ eln n αn

k1 dk

n0



l2k

1

l eln n αn

k1

dk ln n0− ln 2k

eln n α

Dn ln ln D n D2

nln1−1/αDn ln ln D n.

2.32

By2.21, 2.29, 2.31, and 2.32, for some ε > 0 such that

ET n2 1

D2n E

n

k1 dkξk

2

applyingLemma 2.4, we have

2.3 Proof of Theorem 1.3

Let C k  S k/μk; we have

1

γσn

n



k1

C k− 1  1

γσn

n



k1



Sk

μk − 1



 1

σn

n



k1

bk,nYk Sn,n

We see that1.9 is equivalent to

lim

n → ∞

1

Dn

n



k1 dkI

 1

γσk

k



i1

C i − 1 ≤ x



Note that in order to prove1.8 it is sufficient to show that

lim

n → ∞

1

Dn

n



k1 dkI

 1

γσk

k



i1

ln C i ≤ x



FromLemma 2.5, for sufficiently large k, we have

|C k − 1|  Olnln k

k

1/2

Since ln1  x  x  Ox2 for |x| < 1/2, thus







n



k1

lnCk −n

k1

C k− 1





n



k1

C k− 12 n

k1

lnln k

Hence for any ε > 0 and for sufficiently large n, we have

I



1

γσn

n



k1

C k − 1 ≤ x − ε



≤ I

 1

γσn

n



k1

ln C k ≤ x



≤ I

 1

γσn

n



k1

C k − 1 ≤ x  ε



2.40 and thus2.36 implies 2.37

Acknowledgment

This work is supported by the National Natural Science Foundation of China11061012, Innovation Project of Guangxi Graduate Education200910596020M29

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Theorem 1.3 Let {X n, n ≥ 1} be a sequence of identically distributed positive strongly mixing< /small> random variable