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Almost Sure Central Limit Theorem for a Nonstationary Gaussian Sequence
Journal of Inequalities and Applications volumeÂ 2010, ArticleÂ number:Â 130915 (2010)
Abstract
Let be a standardized nonstationary 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 socalled 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 StadtmÃ¼ller [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, CsÃ¡ki 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 nonstationary Gaussian sequences. Recently, DudziÅ„ski [7] obtained twodimensional 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 nonstationary Gaussian sequence.
2. Main Result
Throughout this paper, let be a nonstationary 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 nonstationary 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 DudziÅ„ski [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
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
for any .
Proof.
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 wellknown 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
Proof.
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 CsÃ¡ki 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
Consequently
Thus, we complete the proof of (3.28) by (3.30) and (3.35). Further, our main result is proved.
References
Brosamler GA: An almost everywhere central limit theorem. Mathematical Proceedings of the Cambridge Philosophical Society 1988, 104(3):561â€“574. 10.1017/S0305004100065750
Schatte P: On strong versions of the central limit theorem. Mathematische Nachrichten 1988, 137: 249â€“256. 10.1002/mana.19881370117
Fahrner I, StadtmÃ¼ller U: On almost sure maxlimit theorems. Statistics & Probability Letters 1998, 37(3):229â€“236. 10.1016/S01677152(97)001211
Cheng S, Peng L, Qi Y: Almost sure convergence in extreme value theory. Mathematische Nachrichten 1998, 190: 43â€“50. 10.1002/mana.19981900104
CsÃ¡ki E, Gonchigdanzan K: Almost sure limit theorems for the maximum of stationary Gaussian sequences. Statistics & Probability Letters 2002, 58(2):195â€“203. 10.1016/S01677152(02)001281
Chen S, Lin Z: Almost sure maxlimits for nonstationary Gaussian sequence. Statistics & Probability Letters 2006, 76(11):1175â€“1184. 10.1016/j.spl.2005.12.018
DudziÅ„ski M: The almost sure central limit theorems in the joint version for the maxima and sums of certain stationary Gaussian sequences. Statistics & Probability Letters 2008, 78(4):347â€“357. 10.1016/j.spl.2007.07.007
DudziÅ„ski M: An almost sure limit theorem for the maxima and sums of stationary Gaussian sequences. Probability and Mathematical Statistics 2003, 23(1):139â€“152.
Li WV, Shao Q: A normal comparison inequality and its applications. Probability Theory and Related Fields 2002, 122(4):494â€“508. 10.1007/s004400100176
Leadbetter MR, Lindgren G, RootzÃ©n H: Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics. Springer, New York, NY, USA; 1983:xii+336.
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