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Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence
Journal of Inequalities and Applications volume 2010, Article number: 130915 (2010)
Abstract
Let be a standardized non-stationary Gaussian sequence, and let denote
,
. Under some additional condition, let the constants
satisfy
as
for some
and
, for some
, then, we have
almost surely for any
, where
is the indicator function of the event
and
stands for the standard normal distribution function.
1. Introduction
When is a sequence of independent and identically distributed (i.i.d.) random variables and
for
. If
, the so-called almost sure central limit theorem (ASCLT) has the simplest form as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ1_HTML.gif)
almost surely for all , where
is the indicator function of the event
and
stands for the standard normal distribution function. This result was first proved independently by Brosamler [1] and Schatte [2] under a stronger moment condition; since then, this type of almost sure version was extended to different directions. For example, Fahrner and Stadtmüller [3] and Cheng et al. [4] extended this almost sure convergence for partial sums to the case of maxima of i.i.d. random variables. Under some natural conditions, they proved as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ2_HTML.gif)
for all , where
and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ3_HTML.gif)
for any continuity point of
.
In a related work, Csáki and Gonchigdanzan [5] investigated the validity of (1.2) for maxima of stationary Gaussian sequences under some mild condition whereas Chen and Lin [6] extended it to non-stationary Gaussian sequences. Recently, Dudziński [7] obtained two-dimensional version for a standardized stationary Gaussian sequence. In this paper, inspired by the above results, we further study ASCLT in the joint version for a non-stationary Gaussian sequence.
2. Main Result
Throughout this paper, let be a non-stationary standardized normal sequence, and
. Here
and
stand for
and
, respectively.
is the standard normal distribution function, and
is its density function;
will denote a positive constant although its value may change from one appearance to the next. Now, we state our main result as follows.
Theorem 2.1.
Let be a sequence of non-stationary standardized Gaussian variables with covariance matrix
such that
for
, where
for all
and
. If the constants
satisfy
as
for some
and
, for some
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ4_HTML.gif)
almost surely for any .
Remark 2.2.
The condition is inspired by (a1) in Dudziński [8], which is much more weaker.
3. Proof
First, we introduce the following lemmas which will be used to prove our main result.
Lemma 3.1.
Under the assumptions of Theorem 2.1, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ5_HTML.gif)
Proof.
This lemma comes from Chen and Lin [6].
The following lemma is Theorem 2.1 and Corollary in Li and Shao [9].
Lemma 3.2.
-
(1)
Let
and
be sequences of standard Gaussian variables with covariance matrices
and
, respectively. Put
. Then one has
(3.2)
for any real numbers ,
.
-
(2)
Let
be standard Gaussian variables with
. Then
(3.3)
for any real numbers ,
.
Lemma 3.3.
Let be a sequence of standard Gaussian variables and satisfy the conditions of Theorem 2.1, then for
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ8_HTML.gif)
for any .
Proof.
By the conditions of Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ9_HTML.gif)
then, for , by
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ10_HTML.gif)
Then, there exist numbers ,
, such that, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ11_HTML.gif)
We can write that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ12_HTML.gif)
where is a random variable, which has the same distribution as
, but it is independent of
For
apply Lemma 3.2 (1) with
Then
for
and
for
. Thus, we have (for
)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ13_HTML.gif)
Since (3.5), (3.7) hold, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ14_HTML.gif)
Now define by
. By the well-known fact
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ15_HTML.gif)
it is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ16_HTML.gif)
Thus, according to the assumption , we have
for some
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ17_HTML.gif)
Now, we are in a position to estimate . Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ18_HTML.gif)
For , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ19_HTML.gif)
By Lemma 3.2 (2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ20_HTML.gif)
Thus by Lemma 3.1 we obtain the desired result.
Lemma 3.4.
Let be a sequence of standard Gaussian variables satisfying the conditions of Theorem 2.1, then for
, any
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ21_HTML.gif)
Proof.
Apply Lemma 3.2 (1) with (,
,
,
,
,
), (
), where
has the same distribution as
, but it is independent of
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ22_HTML.gif)
Thus, combined with (3.5), (3.7), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ23_HTML.gif)
Using Lemma 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ24_HTML.gif)
By the similar technique that was applied to prove (3.10), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ25_HTML.gif)
For , by
, and (3.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ26_HTML.gif)
As to , by (3.5) and (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ27_HTML.gif)
Thus the proof of this lemma is completed.
Proof of Theorem 2.1.
First, by assumptions and Theorem in Leadbetter et al. [10], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ28_HTML.gif)
Let denote a random variable which has the same distribution as
, but it is independent of
then by (3.10), we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ29_HTML.gif)
Thus, by the standard normal property of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ30_HTML.gif)
Hence, to complete the proof, it is sufficient to show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ31_HTML.gif)
In order to show this, by Lemma 3.1 in Csáki and Gonchigdanzan [5], we only need to prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ32_HTML.gif)
for and any
. Let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ33_HTML.gif)
Since , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ34_HTML.gif)
Now, we turn to estimate . Observe that for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ35_HTML.gif)
By Lemma 3.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ36_HTML.gif)
Using Lemma 3.4, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ37_HTML.gif)
Hence for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ38_HTML.gif)
Consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F130915/MediaObjects/13660_2010_Article_2059_Equ39_HTML.gif)
Thus, we complete the proof of (3.28) by (3.30) and (3.35). Further, our main result is proved.
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Acknowledgments
The author thanks the referees for pointing out some errors in a previous version, as well as for several comments that have led to improvements in this paper. The authors would like to thank Professor Zuoxiang Peng of Southwest University in China for his help. The paper has been supported by the young excellent talent foundation of Huaiyin Normal University.
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Zang, Qp. Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence. J Inequal Appl 2010, 130915 (2010). https://doi.org/10.1155/2010/130915
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DOI: https://doi.org/10.1155/2010/130915