A note on the almost sure central limit theorem for the product of some partial sums
© Chen et al.; licensee Springer 2014
Received: 4 January 2014
Accepted: 4 June 2014
Published: 23 June 2014
Let be a sequence of i.i.d., positive, square integrable random variables with , . Denote by and by the coefficient of variation. Our goal is to show the unbounded, measurable functions g, which satisfy the almost sure central limit theorem, i.e.,
The almost sure central limit theorem (ASCLT) has been first introduced independently by Schatte  and Brosamler . Since then, many studies have been done to prove the ASCLT in different situations, for example, in the case of function-typed almost sure central limit theorem (FASCLT) (see Berkes et al. , Ibragimov and Lifshits ). The purpose of this paper is to investigate the FASCLT for the product of some partial sums.
Recently Gonchigdanzan and Rempala  obtained the almost sure limit theorem related to (1) as follows.
where is the distribution function of , is a standard normal random variable. Some extensions on the above result can be found in Ye and Wu and the reference therein.
A similar result on the product of partial sums was provided by Miao , which stated the following.
The purpose of this paper is to investigate the validity of (4) for some class of unbounded measurable functions g.
Throughout this article, is a sequence of i.i.d. positive, square integrable random variables with and . We denote by and by the coefficient of variation. Furthermore, is the standard normal random variable, Φ is the standard normal distribution function, ϕ is its density function and stands for .
2 Main result
We state our main result as follows.
where is the distribution function of the random variable .
Let . By a simple calculation, we can get the following result.
Remark 2 Lu et al.  proved the function-typed almost sure central limit theorem for a type of random function, which can include U-statistics, Von-Mises statistics, linear processes and some other types of statistics, but their results cannot imply Theorem 1.
3 Auxiliary results
In this section, we state and prove several auxiliary results, which will be useful in the proof of Theorem 1.
Our proof mainly relies on decomposition (7). Properties (8) and (9) will be extensively used in the following parts of this section.
for every and x.
Proof It is Lemma 1.3 of Petrov . □
with some absolute constant A, provided .
Proof It can be obtained from Berkes et al. . □
Lemma 3 Assume that (6) is true for all indicator functions of intervals and for a fixed a.e. continuous function . Then (6) is also true for all a.e. continuous functions f such that , , and, moreover, the exceptional set of probability 0 can be chosen universally for all such f.
Proof See Berkes et al. . □
Lemma 4 Under the conditions of Theorem 1, we have .
where in the last step we use the assumption and a version of the Kolmogorov-Erdös-Feller-Petrovski test (see Feller , Theorem 2). This completes the proof of Lemma 4. □
Clearly, , .
which completes the proof. □
and Lemma 7 is proved. □
4 Proof of the main result
We only prove the property in (6), since, in view of Remark 1, it is sufficient for the proof of Theorem 1.
by the positivity of each term of . Noting that as , we get (6) by (22) and (23). □
The authors wish to thank the editor and the referees for their very valuable comments by which the quality of the paper has been improved. The authors would also like to thank Professor Zuoxiang Peng for several discussions and suggestions. Research supported by the National Science Foundation of China (No. 11326175), the Natural Science Foundation of Zhejiang Province of China (No. LQ14A010012) and the Research Start-up Foundation of Jiaxing University (No. 70512021).
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