# A Note on Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Sums

- Qing-pei Zang
^{1}Email author, - Zhi-xiang Wang
^{1}and - Ke-ang Fu
^{2}

**2010**:234964

https://doi.org/10.1155/2010/234964

© Qing-pei Zang et al. 2010

**Received: **30 March 2010

**Accepted: **25 May 2010

**Published: **20 June 2010

## Abstract

Let be a sequence of independent and identically distributed (i.i.d.) random variables and denote , . In this paper, we investigate the almost sure central limit theorem in the joint version for the maxima and sums. If for some numerical sequences , we have for a nondegenerate distribution , and is a bounded Lipschitz 1 function, then almost surely, where stands for the standard normal distribution function, ,and , .

## Keywords

## 1. Introduction and Main Results

For Gaussian sequences, Csáki and Gonchigdanzan [5] investigated the validity of (1.2) maxima of stationary Gaussian sequences under some mild condition. Furthermore, Chen and Lin [6] extended it to nonstationary Gaussian sequences. As for some other dependent random variables, Peligrad and Shao [7] and Dudziński [8] derived some corresponding results about ASCLT. The almost sure central limit theorem in the joint version for log average in the case of independent and identically distributed random variables is obtained by Peng et al. [9]; a joint version of almost sure limit theorem for log average of maxima and partial sums in the case of stationary Gaussian random variables is derived by Dudziński [10].

All the above results are related to the almost sure logarithmic version; in this paper; inspired by the results of Berkes and Csáki [11], we further study ASCLT in the joint version for the maxima and partial sums with another weighted sequence ( ). Now, we state our main result as follows.

Theorem 1.1.

almost surely, where stands for the standard normal distribution function, and , .

Remark 1.2.

Since a set of bounded Lipschitz 1 functions is tight in a set of bounded continuous functions, Theorem 1.1 is true for all bounded continuous functions .

Remark 1.3.

almost surely.

Example 1.4.

Then, we can derive a corresponding result in Theorem 1.1.

## 2. Proof of Our Main Result

In this section, denote and , for , unless it is specially mentioned. Here and stand for and , respectively. is the standard normal distribution function.

Proof of Theorem 1.1.

almost surely. Since , the convergence of the subsequence implies that the whole sequence converges almost surely. Hence the proof of (1.5) is completed for . Via the same arguments, we can obtain (1.5) for .

## Declarations

### Acknowledgments

The authors thank two anonymous referees for their useful comments. They would like to thank professor Zhengyan Lin of Zhejiang University in China for his help. The work has been supported by the Young Excellent Talent Foundation of Huaiyin Normal University.

## Authors’ Affiliations

## References

- Brosamler GA: An almost everywhere central limit theorem.
*Mathematical Proceedings of the Cambridge Philosophical Society*1988, 104(3):561–574. 10.1017/S0305004100065750MathSciNetView ArticleMATHGoogle Scholar - Schatte P: On strong versions of the central limit theorem.
*Mathematische Nachrichten*1988, 137: 249–256. 10.1002/mana.19881370117MathSciNetView ArticleMATHGoogle Scholar - Fahrner I, Stadtmüller U: On almost sure max-limit theorems.
*Statistics & Probability Letters*1998, 37(3):229–236. 10.1016/S0167-7152(97)00121-1MathSciNetView ArticleMATHGoogle Scholar - Cheng S, Peng L, Qi Y: Almost sure convergence in extreme value theory.
*Mathematische Nachrichten*1998, 190: 43–50. 10.1002/mana.19981900104MathSciNetView ArticleMATHGoogle Scholar - Csáki E, Gonchigdanzan K: Almost sure limit theorems for the maximum of stationary Gaussian sequences.
*Statistics & Probability Letters*2002, 58(2):195–203. 10.1016/S0167-7152(02)00128-1MathSciNetView ArticleMATHGoogle Scholar - Chen S, Lin Z: Almost sure max-limits for nonstationary Gaussian sequence.
*Statistics & Probability Letters*2006, 76(11):1175–1184. 10.1016/j.spl.2005.12.018MathSciNetView ArticleMATHGoogle Scholar - 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-HMathSciNetView ArticleMATHGoogle Scholar - Dudziński M: A note on the almost sure central limit theorem for some dependent random variables.
*Statistics & Probability Letters*2003, 61(1):31–40. 10.1016/S0167-7152(02)00291-2MathSciNetView ArticleMATHGoogle Scholar - Peng Z, Wang L, Nadarajah S: Almost sure central limit theorem for partial sums and maxima.
*Mathematische Nachrichten*2009, 282(4):632–636. 10.1002/mana.200610760MathSciNetView ArticleMATHGoogle Scholar - Dudziński 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):347–357. 10.1016/j.spl.2007.07.007MathSciNetView ArticleMATHGoogle Scholar - 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)00078-3MathSciNetView ArticleMATHGoogle Scholar - 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-DMathSciNetView ArticleMATHGoogle Scholar - Leadbetter MR, Lindgren G, Rootzén H:
*Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics*. Springer, New York, NY, USA; 1983:xii+336.View ArticleMATHGoogle Scholar - Hsing T: A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables.
*The Annals of Probability*1995, 23(2):938–947. 10.1214/aop/1176988296MathSciNetView ArticleMATHGoogle Scholar

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