Hindawi Publishing Corporation
Journal of Inequalities and Applications
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,
Received 19 September 2010; Revised 1 January 2011; Accepted 26 January 2011
Academic Editor: Ond
ˇ
rej Do
ˇ
sl
´
y
Copyright q 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 Shao 4.
Khurelbaatar and Rempala 5 gave an ASCLT for product of partial sums of i.i.d.
random variables as follows.
Theorem 1.1. Let {X
n
√
k
≤ x
⎞
⎠
F
x
a.s.,
1.1
where S
n
n
k1
X
k
, I∗ is the indicator function, F· is the distribution function of the random
variable e
N
, and N is a standard normal variable.
Recently, Jin 6 had p roved 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.
2 Journal of Inequalities and Applications
Theorem 1.2. Let {X
n
,n ≥ 1} be a sequence of identically distributed positive strongly mixing
n
. Assume
E
|
X
1
|
2δ
< ∞ for some δ>0, lim
n →∞
B
2
n
n
σ
2
0
> 0,
α
n
O
n
−r
for some r>1
2
δ
1/γσ
k
≤ x
⎞
⎠
F
x
a.s. 1.3
The sequence {d
k
,k ≥ 1} in 1.3 is called weight. Under the conditions of Theorem 1.2,
it is easy to see that 1.3 holds for every sequence d
∗
k
with 0 ≤ d
∗
k
≤ d
k
and D
∗
n
k≤n
d
∗
, d
k
and D
n
as mentioned above. Denote
by γ σ/μ the coefficient of variation, σ
2
n
Var
n
k1
S
k
− kμ/kσ and B
2
n
VarS
n
. Assume
that
E
|
X
1
|
2δ
< ∞ for some δ>0,
1.4
α
Then for any real x
lim
n →∞
1
D
n
n
k1
d
k
I
⎛
⎝
k
i1
S
i
k!μ
k
1/γσ
k
≤ x
⎞
⎠
F
n
n
k1
Y
k
and
S
n,n
n
k1
b
k,n
Y
k
.
Journal of Inequalities and Applications 3
In this setting we establish an ASCLT for the triangular array b
k,n
Y
k
.
Theorem 1.4. Under the conditions of Theorem 1.3, for any real x
lim
n →∞
1
D
sup
n
n
k1
a
2
k,n
< ∞, max
1≤k≤n
|
a
k,n
|
−→ 0 as n −→ ∞.
2.1
If for a certain δ>0, {|X
k
|
2δ
} is uniformly integrable, inf
k
VarX
k
> 0,
∞
n1
n
2/δ
2.3
Lemma 2.2 see 9. Let d
k
e
ln k
α
/k, 0 ≤ α<1/2,D
n
n
k1
d
k
;then
D
n
∼ C
ln n
1−α
exp
ln n
α
,
δ
n
i1
i
2/δ
α
i
δ/2δ
sup
k
X
k
2
2δ
,
2.5
where X
k
p
E|X
ξ
k
. Assume that
D
n
−→ ∞
D
n1
D
n
−→ 1,
2.6
as n →∞.Ifforsomeε>0, C and all n
ET
2
n
≤ C
ln
−1−ε
D
n
, 2.7
then
T
n
a.s.
−−−−→ 0 as n −→ ∞.
2.8
b
k,n
/σ
n
;notethat
n
k1
b
2
k,n
b
1,n
2
n
k2
k−1
i1
1
k
b
1,n
2
n
k2
k − 1
k
max
1≤k≤n
|
a
k,n
|
max
1≤k≤n
b
k,n
σ
n
ln n
√
n
−→ 0,n−→ ∞.
2.11
From the definition of Y
k
and 1.4 we have that {|Y
k
|
2δ
} is uniformly integrable; note
that
inf
k
Var
1,
2.12
Journal of Inequalities and Applications 5
and applying 1.5
∞
n1
n
2/δ
α
n
∞
n1
n
−r2/δ
< ∞.
2.13
Consequently using Lemma 2.1, we can obtain
S
n,n
σ
n
d
−−−→N
0, 1
σ
k
−→ Ef
N
as n −→ ∞.
2.16
We notice that 1.9 is equivalent to
lim
n →∞
1
D
n
n
k1
d
k
f
S
k,k
σ
k
Φ
x
a.s.
−−−−→
0,n−→ ∞.
2.18
Let ξ
k
fS
k,k
/σ
k
− EfS
k,k
/σ
k
,
E
n
k1
d
k
ξ
k
2
≤ E
2
d
k
d
l
|
E
ξ
k
ξ
l
|
1≤k≤l≤n
l>2k
d
k
d
l
|
E
ξ
k
ξ
l
|
k1
d
k
2k
lk
1
l
D
n
e
ln n
α
D
2
n
ln D
n
1−1/α
.
2.21
We estimate now T
2,n
. For l>2k,
S
l,l
− S
b
2k1,l
S
2k
b
2k1,l
Y
2k1
··· b
l,l
Y
l
.
2.22
Notice that
|
Eξ
k
ξ
l
|
S
k,k
σ
k
,f
S
l,l
σ
l
− f
S
l,l
− S
2k,2k
− b
2k1,l
S
2k
σ
l
,
2.23
and the properties of strongly mixing sequence imply
Cov
f
S
k,k
σ
k
,f
S
l,l
− S
2k,2k
− b
i,2k
EY
2
i
2
2k−1
j1
2k
ij1
b
i,2k
b
j,2k
Cov
Y
i
,Y
j
≤
2k
i1
b
2
i,2k
2
i1
Y
i
2
2k
i1
EY
2
i
2
2k−1
i1
2k
ji1
Cov
Y
i
,Y
j
k.
2.25
Journal of Inequalities and Applications 7
S
2k
σ
l
E
S
2k,2k
b
2k1,l
S
2k
σ
l
≤
Var
S
2k
σ
l
k
l
β
,
2.26
where 0 <β<1/2. Hence for l>2k we have
|
Eξ
k
ξ
l
|
α
k
k
1≤k≤l≤n
l>2k
d
k
d
l
α
k
1≤k≤l≤n
l>2k
d
k
d
l
k
l
β
T
2,n,1
T
2,n,2
.
2.28
Applying 1.5 and Lemma 2.2 we can obtain for any η>0
l1
d
l
D
2
n
ln D
n
−1−η
.
2.29
Notice that
T
2,n,2
1≤k≤l≤n
l>2k
l/k≥
ln D
n
2/β
d
k
d
l
T
2,n,2,1
≤
1≤k≤l≤n
l>2k
d
k
d
l
ln D
n
−2
≤
ln D
n
−2
n
k1
d
k
n
l1
d
k
d
l
≤ e
ln n
α
n
k1
d
k
n
0
l2k
1
l
e
ln n
α
n
k1
d
k
ln n
0
− ln 2k
k1
d
k
ξ
k
2
ln D
n
−1−ε
,
2.33
applying Lemma 2.4, we have
T
n
a.s.
−−−−→ 0.
2.34
2.3. Proof of Theorem 1.3
Let C
k
S
k
/μk; we have
1
γσ
k,n
Y
k
S
n,n
σ
n
.
2.35
We see that 1.9 is equivalent to
lim
n →∞
1
D
n
n
k1
d
k
I
1
γσ
k
k
i1
i1
ln C
i
≤ x
Φ
x
, a.s. ∀x.
2.37
From Lemma 2.5,forsufficiently large k, we have
|
C
k
− 1
|
O
ln
ln k
k
1/2
. 2.38
Since ln1 xx Ox
n
k1
C
k
− 1
2
n
k1
ln
ln k
k
ln n ln
ln n
a.s.
2.39
Journal of Inequalities and Applications 9
Hence for any ε>0andforsufficiently large n, we have
I
1
γσ
k1
C
k
− 1
≤ x ε
2.40
and thus 2.36 implies 2.37.
Acknowledgment
This work is supported by the National Natural Science Foundation of China 11061012,
Innovation Project of Guangxi Graduate Education 200910596020M29.
References
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249–256, 1988.
3 M. T. Lacey and W. Philipp, “A note on the almost sure central limit theorem,” Statistics & Probability
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4 M. Peligrad and Q. M. Shao, “A note on the almost sure central limit theorem for weakly dependent
random variables,” Statistics & Probability Letters, vol. 22, no. 2, pp. 131–136, 1995.
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partial sums,” Applied Mathematics Letters, vol. 19, pp. 191–196, 2004.
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random variables,” Journal of Zhejiang University, vol. 34, no. 1, pp. 24–27, 2007.
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´
aki, “A universal result in almost sure central limit theory,” Stochastic Processes and
Their Applications, vol. 94, no. 1, pp. 105–134, 2001.
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