A note on the almost sure central limit theorems for the maxima and sums
© Zang; licensee Springer 2012
Received: 10 April 2012
Accepted: 4 September 2012
Published: 8 October 2012
In this note, we derive, under some natural assumptions, a general pattern of the almost sure central limit theorem in the joint version for the maxima and sums.
1 Introduction and main results
for some . These three distributions are often called the Gumbel, the Frechet and the Weibull distributions, respectively.
For Gaussian sequences, Csáki and Gonchigdanzan  investigated, under some mild conditions, the validity of (1.2) for maxima of stationary Gaussian sequences. Furthermore, Chen and Lin  extended it to non-stationary Gaussian sequences. As for some other dependent random variables, Peligrad and Shao  and Dudziński  derived some corresponding results about ASCLT. In addition, the almost sure central limit theorem in the joint version for log-average of maxima and partial sums of independent and identically distributed random variables was obtained by Peng et al. , whereas the joint version of the almost sure limit theorem for log-average of maxima and partial sums of stationary Gaussian random variables was derived by Dudziński .
In statistical context, we are very concerned with the ASCLT in the joint version for the maxima and partial sums. The goal of this note is to investigate the general pattern of the ASCLT for the maxima and partial sums of i.i.d. random variables by the method provided by Hörmann . He showed the following result.
with some (possibly degenerate) distribution function H.
for some positive constants C, β.
Now, we may state our main result as follows.
Suppose, moreover, that the positive weights , , satisfy the following conditions:
() is nonincreasing for some ;
() for some , where .
Remark 1.2 Since a set of bounded Lipschitz functions is tight in a set of bounded continuous functions, Theorem 1.1 is true for all bounded continuous functions .
which at least includes a ‘halfway’ from logarithmic to ordinary averaging. Moreover, Hörmann  shows that this sequence obeys the a.s. central limit theorem for all . Due to the similar conditions on the sequence of weights, our result also holds for this sequence provided .
2 Proof of our main result
The following notations will be used throughout this section: , , , and for . Furthermore, and stand for and , respectively, and is the standard normal distribution function. The proof of our main result is based on the following lemmas.
Thus, using (2.1)-(2.3), we get the desired result. □
We will also prove the following auxiliary result.
The relations in (2.4), (2.5) imply the claim in Lemma 2. □
The following lemma will also be used.
imply the desired result. □
We will also prove the following lemma.
Proof This lemma can be obtained from Lemmas 1 and 3 by making slight changes in the proof of Lemma 4 of Hörmann . □
The following result will be needed in the proof of our main result.
Proof This result follows from Lemma 5 in Hörmann . □
This follows from Lemmas 4 and 5 by applying similar arguments to those used in Hörmann .
Since , the convergence of the subsequence implies that the whole sequence converges almost surely. This completes the proof of Theorem 1.1. □
The author would like to thank the Editor and the two referees for careful reading of the paper and for their comments which have led to the improvement of this work. This work was supported by the Natural Science Research Project of Ordinary Universities in Jiangsu Province, P.R. China. Grant No. 12KJB110003.
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