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Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables
Journal of Inequalities and Applications volume 2011, Article number: 576301 (2011)
Abstract
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 be a sequence of i.i.d. positive random variables with
and
. Denote
the coefficient of variation. Then for any real
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ1_HTML.gif)
where ,
is the indicator function,
is the distribution function of the random variable
, and
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.
Theorem 1.2.
Let be a sequence of identically distributed positive strongly mixing random variable with
and
,
,
. Denote by
the coefficient of variation,
and
. Assume
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ2_HTML.gif)
Then for any real
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ3_HTML.gif)
The sequence 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
with
and
[7]. Clearly, the larger the weight sequence
is, the stronger is the result (1.3).
In the following sections, let ,
, "
" denote the inequality "
" up to some universal constant.
We first give an ASCLT for strongly mixing positive random variables.
Theorem 1.3.
Let be a sequence of identically distributed positive strongly mixing random variable with
and
,
and
as mentioned above. Denote by
the coefficient of variation,
and
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ4_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ5_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ7_HTML.gif)
Then for any real
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ8_HTML.gif)
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 and
for
with
if
.
,
,
and
.
In this setting we establish an ASCLT for the triangular array .
Theorem 1.4.
Under the conditions of Theorem 1.3, for any real
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ9_HTML.gif)
where 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 be a sequence of strongly mixing random variables with zero mean, and let
be a triangular array of real numbers. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ10_HTML.gif)
If for a certain is uniformly integrable,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ11_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ12_HTML.gif)
Lemma 2.2 (see [9]).
Let ; then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ13_HTML.gif)
where as
as
.
Lemma 2.3 (see [8]).
Let be a strongly mixing sequence of random variables such that
for a certain
and every
. Then there is a numerical constant
depending only on
such that for every
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ14_HTML.gif)
where .
Lemma 2.4 (see [9]).
Let be a sequence of random variables, uniformly bounded below and with finite variances, and let
be a sequence of positive number. Let for
and
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ15_HTML.gif)
as . If for some
,
and all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ16_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ17_HTML.gif)
Lemma 2.5 (see [10]).
Let be a strongly mixing sequence of random variables with zero mean and
for a certain
. Assume that (1.5) and (1.6) hold. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ18_HTML.gif)
2.2. Proof of Theorem 1.4
From the definition of strongly mixing we know that remain to be a sequence of identically distributed strongly mixing random variable with zero mean and unit variance. Let
; note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ19_HTML.gif)
and via (1.7) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ20_HTML.gif)
From the definition of and (1.4) we have that
is uniformly integrable; note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ21_HTML.gif)
and applying (1.5)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ22_HTML.gif)
Consequently using Lemma 2.1, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ23_HTML.gif)
which is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ24_HTML.gif)
for any bounded Lipschitz-continuous function ; applying Toeplitz Lemma
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ25_HTML.gif)
We notice that (1.9) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ26_HTML.gif)
for all bounded Lipschitz continuous ; it therefore remains to prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ27_HTML.gif)
Let ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ28_HTML.gif)
From Lemma 2.2, we obtain for some constant
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ29_HTML.gif)
Using (2.20) and property of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ30_HTML.gif)
We estimate now . For
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ31_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ32_HTML.gif)
and the properties of strongly mixing sequence imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ33_HTML.gif)
Applying Lemma 2.3 and (2.10),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ34_HTML.gif)
Consequently, via the properties of , the Jensen inequality, and (1.7),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ35_HTML.gif)
where . Hence for
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ36_HTML.gif)
Consequently, we conclude from the above inequalities that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ37_HTML.gif)
Applying (1.5) and Lemma 2.2 we can obtain for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ38_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ39_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ40_HTML.gif)
Let , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ41_HTML.gif)
By (2.21), (2.29), (2.31), and (2.32), for some such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ42_HTML.gif)
applying Lemma 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ43_HTML.gif)
2.3. Proof of Theorem 1.3
Let ; we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ44_HTML.gif)
We see that (1.9) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ45_HTML.gif)
Note that in order to prove (1.8) it is sufficient to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ46_HTML.gif)
From Lemma 2.5, for sufficiently large , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ47_HTML.gif)
Since for
, thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ48_HTML.gif)
Hence for any and for sufficiently large
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F576301/MediaObjects/13660_2010_Article_2351_Equ49_HTML.gif)
and thus (2.36) implies (2.37).
References
Brosamler GA: An almost everywhere central limit theorem. Mathematical Proceedings of the Cambridge Philosophical Society 1988,104(3):561–574. 10.1017/S0305004100065750
Schatte P: On strong versions of the central limit theorem. Mathematische Nachrichten 1988, 137: 249–256. 10.1002/mana.19881370117
Lacey MT, Philipp W: A note on the almost sure central limit theorem. Statistics & Probability Letters 1990,9(3):201–205. 10.1016/0167-7152(90)90056-D
Peligrad M, Shao QM: A note on the almost sure central limit theorem for weakly dependent random variables. Statistics & Probability Letters 1995,22(2):131–136. 10.1016/0167-7152(94)00059-H
Khurelbaatar G, Rempala G: A note on the almost sure central limit theorem for the product of partial sums. Applied Mathematics Letters 2004, 19: 191–196.
Jin JS: An almost sure central limit theorem for the product of partial sums of strongly missing random variables. Journal of Zhejiang University 2007,34(1):24–27.
Berkes I, Csáki E: A universal result in almost sure central limit theory. Stochastic Processes and Their Applications 2001,94(1):105–134. 10.1016/S0304-4149(01)00078-3
Peligrad M, Utev S: Central limit theorem for linear processes. The Annals of Probability 1997,25(1):443–456.
Jonsson F: Almost Sure Central Limit Theory. Uppsala University: Department of Mathematics; 2007.
Chuan-Rong L, Zheng-Yan L: Limit Theory for Mixing Dependent Random Variabiles. Science Press, Beijing, China; 1997.
Acknowledgment
This work is supported by the National Natural Science Foundation of China (11061012), Innovation Project of Guangxi Graduate Education (200910596020M29).
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Ye, D., Wu, Q. Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables. J Inequal Appl 2011, 576301 (2011). https://doi.org/10.1155/2011/576301
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DOI: https://doi.org/10.1155/2011/576301