Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence
© Qing-pei Zang. 2010
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.
In a related work, Csáki and Gonchigdanzan  investigated the validity of (1.2) for maxima of stationary Gaussian sequences under some mild condition whereas Chen and Lin  extended it to non-stationary Gaussian sequences. Recently, Dudziński  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.
The condition is inspired by (a1) in Dudziński , which is much more weaker.
First, we introduce the following lemmas which will be used to prove our main result.
This lemma comes from Chen and Lin .
The following lemma is Theorem 2.1 and Corollary in Li and Shao .
Thus by Lemma 3.1 we obtain the desired result.
Thus the proof of this lemma is completed.
Proof of Theorem 2.1.
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.
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