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, 3040801111@163.com
Received 19 September 2010; Revised 1 January 2011; Accepted 26 January 2011
Academic Editor: Ondřej Došlý
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 {Xn , n ≥ 1} be a sequence of i.i.d. positive random variables with EX1 μ > 0
and VarX1 σ 2 . Denote γ σ/μ the coefficient of variation. Then for any real x
⎛ ⎞
1/γ √k
k
n
S
1 1 ⎝
i
i1
I
lim
≤ x⎠ Fx
n → ∞ ln n
k
k!μk
k1
a.s.,
1.1
where Sn nk1 Xk , I∗ is the indicator function, F· is the distribution function of the random
variable eN , and N is a standard normal variable.
Recently, Jin 6 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.
2
Journal of Inequalities and Applications
Theorem 1.2. Let {Xn , n ≥ 1} be a sequence of identically distributed positive strongly mixing
random variable with EX1 μ > 0 and VarX1 σ 2 , dk 1/k, Dn nk1 dk . Denote by
γ σ/μ the coefficient of variation, σn2 Var nk1 Sk − kμ/kσ and Bn2 VarSn . Assume
E|X1 |2δ < ∞ for some δ > 0,
−r
αn O n
Bn2
σ02 > 0,
n→∞ n
lim
2
for some r > 1 ,
δ
σ2
inf n > 0.
n∈N n
1.2
Then for any real x
⎛ ⎞
1/γσk
k
n
S
1 i1 i
lim
dk I ⎝
≤ x⎠ Fx a.s.
k
n → ∞ Dn
k!μ
k1
1.3
The sequence {dk , 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 dk∗ with 0 ≤ dk∗ ≤ dk and Dn∗ k≤n dk∗ → ∞
7. Clearly, the larger the weight sequence dk is, the stronger is the result 1.3.
α
In the following sections, let dk eln k /k, 0 ≤ α < 1/2, Dn nk1 dk , “ ” denote the
inequality “≤” up to some universal constant.
We first give an ASCLT for strongly mixing positive random variables.
Theorem 1.3. Let {Xn , n ≥ 1} be a sequence of identically distributed positive strongly mixing
random variable with EX1 μ > 0 and VarX1 σ 2 , dk and Dn as mentioned above. Denote
by γ σ/μ the coefficient of variation, σn2 Var nk1 Sk − kμ/kσ and Bn2 VarSn . Assume
that
1.4
E|X1 |2δ < ∞ for some δ > 0,
αn O n−r
for some r > 1 2
,
δ
1.5
Bn2
σ02 > 0,
n→∞ n
1.6
σn2
> 0.
n∈N n
1.7
⎛ ⎞
1/γσk
k
n
S
1 i
i1
lim
dk I ⎝
≤ x⎠ Fx a.s.
n → ∞ Dn
k!μk
k1
1.8
lim
inf
Then for any real x
In order to prove Theorem 1.3 we first establish ASCLT for certain triangular arrays of
random variables. In the sequel we shall use the following notation. Let bk,n nik 1/i and
2
s2k,n ki1 bi,n
for k ≤ n with bk,n 0 if k > n. Yk Xk − μ/σ, k ≤ 1, Sn nk1 Yk and
n
Sn,n k1 bk,n Yk .
Journal of Inequalities and Applications
3
In this setting we establish an ASCLT for the triangular array bk,n Yk .
Theorem 1.4. Under the conditions of Theorem 1.3, for any real x
n
Sk,k
1 dk I
≤ x Φx
n → ∞ Dn
σk
k1
1.9
a.s.,
lim
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 {Xn , n ≥ 1} be a sequence of strongly mixing random variables with zero
mean, and let {ak,n , 1 ≤ k ≤ n, n ≥ 1} be a triangular array of real numbers. Assume that
sup
n
a2k,n < ∞,
n k1
max |ak,n | −→ 0 as n −→ ∞.
2.1
1≤k≤n
If for a certain δ > 0, {|Xk |2δ } is uniformly integrable, infk VarXk > 0,
∞
2/δ
n
αn < ∞,
n
Var
ak,n Xk
n1
1,
2.2
n1
then
n
d
ak,n Xk −−−→ N0, 1.
2.3
k1
α
Lemma 2.2 see 9. Let dk eln k /k, 0 ≤ α < 1/2, Dn n
k1
dk ; then
Dn ∼ Cln n1−α exp ln nα ,
2.4
where C 1/α as 0 < α < 1/2, C 1 as α 0.
Lemma 2.3 see 8. Let {Xn , 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
δ/2δ
nj
n
2/δ
Cov Xi , Xj ≤ cδ
i αi
sup Xk
sup
j
where Xk
p
ij1
E|Xk |p 1/p , p > 1.
i1
k
2
2δ ,
2.5
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Journal of Inequalities and Applications
Lemma 2.4 see 9. Let {ξk , k ≥ 1} be a sequence of random variables, uniformly bounded below
and with finite variances, and let {dk , k ≥ 1} be a sequence of positive number. Let for n ≥ 1, Dn n
n
k1 dk and Tn 1/Dn k1 dk ξk . Assume that
Dn1
−→ 1,
Dn
Dn −→ ∞
2.6
as n → ∞. If for some ε > 0, C and all n
ETn2 ≤ C ln−1−ε Dn ,
2.7
then
a.s.
Tn −−−−→ 0
2.8
as n −→ ∞.
Lemma 2.5 see 10. Let {Xn , n ≥ 1} be a strongly mixing sequence of random variables with zero
mean and supn E|Xn |2δ < ∞ for a certain δ > 0. Assume that 1.5 and 1.6 hold. Then
lim sup n→∞
|Sn |
1
2σ02 n ln ln n
a.s.
2.9
2.2. Proof of Theorem 1.4
From the definition of strongly mixing we know that {Yk , k ≥ 1} remain to be a sequence of
identically distributed strongly mixing random variable with zero mean and unit variance.
Let ak,n bk,n /σn ; note that
k−1
n
n n
1
k−1
2
b1,n 2
2n − b1,n ,
bk,n
b1,n 2
k
k
k1
k2 i1
k2
n ≥ 1,
2.10
and via 1.7 we have
n
n b2
k,n
2
sup ak,n sup
2
n k1
n k1 σn
bk,n
max |ak,n | max
1≤k≤n
1≤k≤n σn
sup
n
2n − b1,n
< ∞,
n
ln n
√ −→ 0,
n
2.11
n −→ ∞.
From the definition of Yk and 1.4 we have that {|Yk |2δ } is uniformly integrable; note
that
inf VarYk k
EY12
1 > 0,
Var
n
k1
ak,n Yk
Var
n
k1 bk,n Yk
σn2
1,
2.12
Journal of Inequalities and Applications
5
and applying 1.5
∞
n2/δ αn
∞
n1
n1
n−r2/δ < ∞.
2.13
as n −→ ∞,
2.14
Consequently using Lemma 2.1, we can obtain
Sn,n d
−−−→ N0, 1
σn
which is equivalent to
Sn,n
Ef
σn
−→ EfN
2.15
as n −→ ∞
for any bounded Lipschitz-continuous function f; applying Toeplitz Lemma
n
Sk,k
1 dk Ef
−→ EfN
Dn k1
σk
as n −→ ∞.
2.16
We notice that 1.9 is equivalent to
n
Sk,k
1 lim
dk f
Φx a.s.
n → ∞ Dn
σk
k1
2.17
for all bounded Lipschitz continuous f; it therefore remains to prove that
n
Sk,k
Sk,k
1 a.s.
−−−−→ 0,
Tn dk f
− Ef
Dn k1
σk
σk
n −→ ∞.
2.18
Let ξk fSk,k /σk − EfSk,k /σk ,
2
n
dk ξk
≤E 2
dk dl ξk ξl
E
k1
1≤k≤l≤n
dk dl |Eξk ξl | 1≤k≤l≤n
l≤2k
dk dl |Eξk ξl |
1≤k≤l≤n
dk dl |Eξk ξl |
2.19
1≤k≤l≤n
l>2k
T1,n T2,n .
From Lemma 2.2, we obtain for some constant C1
α
eln n ∼ C1 Dn ln Dn 1−1/α .
2.20
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Journal of Inequalities and Applications
Using 2.20 and property of f, we have
α
eln n
T1,n
n
dk
k1
2k
1
lk
α
Dn eln n
l
Dn2 ln Dn 1−1/α .
2.21
We estimate now T2,n . For l > 2k,
Sl,l − S2k,2k b1,l Y1 b2,l Y2 · · · bl,l Yl − b1,2k Y1 b2,2k Y2 · · · b2k,2k Y2k b2k1,l S2k b2k1,l Y2k1 · · · bl,l Yl .
2.22
Notice that
Sk,k
Sl,l
,f
|Eξk ξl | Cov f
σk
σl
Sk,k
Sl,l
Sl,l − S2k,2k − b2k1,l S2k
≤ Cov f
,f
−f
σk
σl
σl
Sk,k
Sl,l − S2k,2k − b2k1,l S2k
Cov f
,f
,
σk
σl
2.23
and the properties of strongly mixing sequence imply
Sk,k
Sl,l − S2k,2k − b2k1,l S2k
,f
Cov f
σk
σl
2.24
αk.
Applying Lemma 2.3 and 2.10,
VarS2k,2k 2k
i1
≤
2k
i1
Var S2k
2k−1
2k
2
bi,2k
EYi2 2
2k−1
2
bi,2k
2
2k
E
Yi
i1
j1
2
bi,2k bj,2k Cov Yi , Yj
j1 ij1
2
bj,2k
2k Cov Yi , Yj 2k
2k
2k−1
EYi2 2
Cov Yi , Yj
i1
2.25
k,
ij1
i1 ji1
k.
Journal of Inequalities and Applications
7
Consequently, via the properties of f, the Jensen inequality, and 1.7,
Sk,k
Sl,l − S2k,2k − b2k1,l S2k
Sl,l
,f
−f
Cov f
σk
σl
σl
ES2k,2k b2k1,l S2k σl
≤
VarS2k,2k b2k1,l
σl
ES22k,2k
σl
2
E b2k1,l S2k
σl
Var S2k
2.26
β
k
,
l
σl
where 0 < β < 1/2. Hence for l > 2k we have
|Eξk ξl |
β
k
.
l
αk 2.27
Consequently, we conclude from the above inequalities that
T2,n
dk dl
1≤k≤l≤n
l>2k
β k
αk l
β
k
dk dl αk dk dl
T2,n,1 T2,n,2 .
l
1≤k≤l≤n
1≤k≤l≤n
l>2k
2.28
l>2k
Applying 1.5 and Lemma 2.2 we can obtain for any η > 0
T2,n,1 ≤
n
n dk dl αk
ln Dn −1−η
k1 l1
n
dk
k1
n
dl Dn2 ln Dn −1−η .
2.29
l1
Notice that
T2,n,2 dk dl
1≤k≤l≤n
l>2k
l/k≥ln Dn 2/β
T2,n,2,1 ≤
β
k
l
dk dl
1≤k≤l≤n
l>2k
l/k<ln Dn 2/β
dk dl ln Dn −2 ≤ ln Dn −2
1≤k≤l≤n
l>2k
β
k
T2,n,2,1 T2,n,2,2 ,
l
n
n
dk dl Dn2 ln Dn −2 .
k1
l1
2.30
2.31
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Journal of Inequalities and Applications
Let n0 max{l : k ≤ l ≤ n, l/k < ln Dn 2/β }, then
T2,n,2,2 ≤
n0
n dk dl ≤ eln n
k1 l2k
α
n0
n
1
dk
l
k1
l2k
α
eln n Dn ln ln Dn
eln n
α
n
dk ln n0 − ln 2k
k1
2.32
Dn2 ln1−1/α Dn ln ln Dn .
By 2.21, 2.29, 2.31, and 2.32, for some ε > 0 such that
ETn2
2
n
1
2E
dk ξk
Dn
k1
ln Dn −1−ε ,
2.33
applying Lemma 2.4, we have
a.s.
Tn −−−−→ 0.
2.34
n
n n
Sn,n
1 1
Sk
1 −1 bk,n Yk .
Ck − 1 γσn k1
γσn k1 μk
σn k1
σn
2.35
2.3. Proof of Theorem 1.3
Let Ck Sk /μk; we have
We see that 1.9 is equivalent to
n
1 lim
dk I
n → ∞ Dn
k1
k
1 Ci − 1 ≤ x
γσk i1
Φx,
2.36
a.s. ∀x.
Note that in order to prove 1.8 it is sufficient to show that
n
1 lim
dk I
n → ∞ Dn
k1
k
1 ln Ci ≤ x
γσk i1
Φx,
2.37
a.s. ∀x.
From Lemma 2.5, for sufficiently large k, we have
|Ck − 1| O
lnln k
k
1/2 2.38
.
Since ln1 x x Ox2 for |x| < 1/2, thus
n
n
lnCk − Ck − 1
k1
k1
n
Ck − 12
n
lnln k
k1
k1
k
ln n lnln n
a.s.
2.39
Journal of Inequalities and Applications
9
Hence for any ε > 0 and for sufficiently large n, we have
I
n
1 Ck − 1 ≤ x − ε
γσn k1
≤I
n
1 ln Ck ≤ x
γσn k1
≤I
n
1 Ck − 1 ≤ x ε
γσn k1
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
1 G. A. Brosamler, “An almost everywhere central limit theorem,” Mathematical Proceedings of the
Cambridge Philosophical Society, vol. 104, no. 3, pp. 561–574, 1988.
2 P. Schatte, “On strong versions of the central limit theorem,” Mathematische Nachrichten, vol. 137, pp.
249–256, 1988.
3 M. T. Lacey and W. Philipp, “A note on the almost sure central limit theorem,” Statistics & Probability
Letters, vol. 9, no. 3, pp. 201–205, 1990.
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.
5 G. Khurelbaatar and G. Rempala, “A note on the almost sure central limit theorem for the product of
partial sums,” Applied Mathematics Letters, vol. 19, pp. 191–196, 2004.
6 J. S. Jin, “An almost sure central limit theorem for the product of partial sums of strongly missing
random variables,” Journal of Zhejiang University, vol. 34, no. 1, pp. 24–27, 2007.
7 I. Berkes and E. Csáki, “A universal result in almost sure central limit theory,” Stochastic Processes and
Their Applications, vol. 94, no. 1, pp. 105–134, 2001.
8 M. Peligrad and S. Utev, “Central limit theorem for linear processes,” The Annals of Probability, vol. 25,
no. 1, pp. 443–456, 1997.
9 F. Jonsson, Almost Sure Central Limit Theory, Uppsala University: Department of Mathematics, 2007.
10 L. Chuan-Rong and L. Zheng-Yan, Limit Theory for Mixing Dependent Random Variabiles, Science Press,
Beijing, China, 1997.