On the almost sure central limit theorem for self-normalized products of partial sums of ϕ-mixing random variables
© Hwang; licensee Springer 2013
Received: 21 December 2012
Accepted: 12 March 2013
Published: 4 April 2013
Let be a sequence of strictly stationary ϕ-mixing positive random variables which are in the domain of attraction of the normal law with , possibly infinite variance and mixing coefficient rates satisfying . Under suitable conditions, we here give an almost sure central limit theorem for self-normalized products of partial sums, i.e.,
where F is the distribution function of the random variable and is a standard normal random variable.
1 Introduction and main results
Here and in the sequel, denotes an indicator function and is the distribution function of the standard normal random variable. It is known (see Berkes ) that the class of sequences satisfying the ASCLT is larger than the class of sequences satisfying the central limit theorem. In recent years, the ASCLT for products of partial sums has received more and more attention. We refer to Gonchigdanzan and Rempala  on the ASCLT for the products of partial sums, Gonchigdanzan  on the ASCLT for the products of partial sums with stable distribution. Li and Wang  and Zhang et al.  showed ASCLT for products of sums and products of sums of partial sums under association. Huang and Pang , Zhang and Yang  obtained the ASCLT results of self-normalized versions. Zhang and Yang  proved the following ASCLT for self-normalized products of sums of i.i.d. random variables.
where is the distribution function of the random variable and is a standard normal random variable.
where is a set of all -measurable random variables with second moments. It is well known that , and hence a ϕ-mixing sequence is ρ-mixing.
and for .
In this paper we study the almost sure central limit theorem, containing the general weight sequences, for weakly dependent random variables. Let be a sequence of strictly stationary ϕ-mixing positive random variables which are in the domain of attraction of the normal law with , possibly infinite variance and mixing coefficients satisfying . We here give an almost sure central limit theorem for self-normalized products of partial sums under a fairly general condition.
Throughout this paper, the following notations are frequently used. For any two positive sequences, means that for a certain numerical constant C not depending on n, we have for all n, and means as . denotes the largest integer smaller or equal to x, and C denotes a generic positive constant, whose value can differ in different places.
Our main theorem is as follows.
then with .
We have the following corollaries.
Corollary 1.1 Let be a strictly stationary ϕ-mixing sequence of positive random variables such that , and , then (1.4) holds.
Corollary 1.2 Let be a strictly stationary ϕ-mixing sequence of positive random variables such that , and . Set and , then (1.4) holds.
Remark 1.2 Let and . If is a sequence of i.i.d. positive random variables such that and belongs to the domain of attraction of the normal law, then Theorem 1.1 is just Theorem A.
Remark 1.3 By the terminology of summation procedures (see [, p.35]), Theorem 1.1 remains valid if we replace the weight sequence by any such that and .
In this section, we introduce some lemmas which are used to prove our theorem.
Lemma 2.1 (Csörgő et al. )
X is in the domain of attraction of the normal law,
- (d)for .
which yields (2.5). Hence (2.3) holds true. □
Proof The relations (2.11)-(2.13) follow by the same method as in the proof of Lemma 2.2, and the details are omitted here. □
To prove that under the hypotheses of Theorem 1.1, the relations (2.2) and (2.8)-(2.10) hold true, we show them by using the following four lemmas.
Hence, combining (2.15) with (2.17) and (2.20) yields (2.14). □
which means , and hence (2.21) is proved. □
Proof This follows by the same method as the proof of Lemma 2.4, and the details are omitted. □
Therefore, combining (2.23) with (2.24) and (2.27), we obtain (2.22), which is our claim. □
3 Proof of Theorem 1.1
which implies that (3.4). The proof is completed.
- Brosamler GA: An almost everywhere central limit theorem. Math. Proc. Camb. Philos. Soc. 1988, 104: 561–574. 10.1017/S0305004100065750MathSciNetView ArticleGoogle Scholar
- Schatte P: On strong versions of the central limit theorem. Math. Nachr. 1988, 137: 249–256. 10.1002/mana.19881370117MathSciNetView ArticleGoogle Scholar
- Berkes I: Results and problems related to the pointwise central limit theorem. In Asymptotic Results in Probability and Statistics (A Volume in Honor of Miklós Csörgő). Edited by: Szyszkowicz B. Elsevier, Amsterdam; 1998:59–60.View ArticleGoogle Scholar
- Gonchigdanzan K, Rempala GA: A note on the almost sure limit theorem for the product of partial sums. Appl. Math. Lett. 2006, 19: 191–196. 10.1016/j.aml.2005.06.002MathSciNetView ArticleGoogle Scholar
- Gonchigdanzan K: An almost sure limit theorem for the product of partial sums with stable distribution. Stat. Probab. Lett. 2008, 78: 3170–3175. 10.1016/j.spl.2008.06.003MathSciNetView ArticleGoogle Scholar
- Li YX, Wang JF: An almost sure central limit theorem for products of sums under association. Stat. Probab. Lett. 2008, 78(4):367–375. 10.1016/j.spl.2007.07.009View ArticleGoogle Scholar
- Zhang Y, Yang XY, Dong ZS: An almost sure central limit theorem for products of sums of partial sums under association. J. Math. Anal. Appl. 2009, 355(2):708–716. 10.1016/j.jmaa.2009.01.071MathSciNetView ArticleGoogle Scholar
- Huang SH, Pang TX: An almost sure central limit theorem for self-normalized partial sums. Comput. Math. Appl. 2010, 60: 2639–2644. 10.1016/j.camwa.2010.08.093MathSciNetView ArticleGoogle Scholar
- Zhang Y, Yang XY: An almost sure central limit theorem for self-normalized products of sums of i.i.d. random variables. J. Math. Anal. Appl. 2011, 376: 29–41. 10.1016/j.jmaa.2010.10.021MathSciNetView ArticleGoogle Scholar
- Balan, RM, Kulik, R: Self-normalized weak invariance principle for mixing sequences. Tech. Rep. Ser. 417, Lab. Reas. Probab. Stat., Univ. Ottawa-Carleton Univ. (2005)Google Scholar
- Balan RM, Kulik R: Weak invariance principle for mixing sequences in the domain of attraction of normal law. Studia Sci. Math. Hung. 2009, 46(3):329–343.MathSciNetGoogle Scholar
- de la Pena H, Lai TL, Shao QM: Self-Normalized Processes. Springer, Berlin; 2010.Google Scholar
- Chandrasekharan K, Minakshisundaram S: Typical Means. Oxford University Press, Oxford; 1952.Google Scholar
- Csörgő M, Szyszkowicz B, Wang Q: Donsker’s theorem for self-normalized partial sums processes. Ann. Probab. 2003, 31(3):1228–1240. 10.1214/aop/1055425777MathSciNetView ArticleGoogle Scholar
- Liu W, Lin ZY: Asymptotics for self-normalized random products of sums for mixing sequences. Stoch. Anal. Appl. 2007, 25: 739–762. 10.1080/07362990701419938MathSciNetView ArticleGoogle Scholar
- Seneta E: Regularly Varying Functions. Springer, Berlin; 1976.View ArticleGoogle Scholar
- Lin Z, Lu C: Limit Theory for Mixing Dependent Random Variables. Kluwer Academic, Boston; 1996.Google Scholar
- Xue LG: Convergence rates of the strong law of large numbers for a mixing sequence. J. Syst. Sci. Math. Sci. 1994, 14: 213–221. (in Chinese)Google Scholar
- Shao QM: Almost sure invariance principle for mixing sequences of random variables. Stoch. Process. Appl. 1993, 48: 319–334. 10.1016/0304-4149(93)90051-5View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.