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Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence

Journal of Inequalities and Applications20102010:130915

Received: 4 May 2010

Accepted: 12 August 2010

Published: 16 August 2010


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.


Central Limit TheoremIndicator FunctionCovariance MatriceMoment ConditionStandard Normal Distribution

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:
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:
for all , where and satisfy

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

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


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. (1)
    Let and be sequences of standard Gaussian variables with covariance matrices and , respectively. Put . Then one has
for any real numbers , .
  1. (2)
    Let be standard Gaussian variables with . Then

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

for any .


By the conditions of Theorem 2.1, we have
then, for , by , it follows that
Then, there exist numbers , , such that, for any , we have
We can write that
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 )
Since (3.5), (3.7) hold, we obtain
Now define by . By the well-known fact
it is easy to see that
Thus, according to the assumption , we have for some . Hence
Now, we are in a position to estimate . Observe that
For , it follows that
By Lemma 3.2 (2), we have

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


Apply Lemma 3.2 (1) with ( , , , , , ), ( ), where has the same distribution as , but it is independent of . Then,
Thus, combined with (3.5), (3.7), it follows that
Using Lemma 3.1, we have
By the similar technique that was applied to prove (3.10), we obtain
For , by , and (3.12), we have
As to , by (3.5) and (3.6), we have

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
Let denote a random variable which has the same distribution as , but it is independent of then by (3.10), we derive
Thus, by the standard normal property of , we have
Hence, to complete the proof, it is sufficient to show
In order to show this, by Lemma 3.1 in Csáki and Gonchigdanzan [5], we only need to prove
for and any . Let . Then
Since , it follows that
Now, we turn to estimate . Observe that for
By Lemma 3.3, we have
Using Lemma 3.4, it follows that
Hence for , we have

Thus, we complete the proof of (3.28) by (3.30) and (3.35). Further, our main result is proved.



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.

Authors’ Affiliations

School of Mathematical Science, Huaiyin Normal University, Huaian, China


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© Qing-pei Zang. 2010

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