On strong law of large numbers and growth rate for a class of random variables
© Shen et al.; licensee Springer. 2013
Received: 9 August 2013
Accepted: 4 November 2013
Published: 25 November 2013
In this paper, we study the strong law of large numbers for a class of random variables satisfying the maximal moment inequality with exponent 2. Our results embrace the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for this class of random variables. In addition, strong growth rate for weighted sums of this class of random variables is presented.
Keywordsstrong law of large numbers weighted sums with exponent 2
holds. Note that the SLLN of the form (1.1) embraces the Kolmogorov SLLN (, ) and the Marcinkiewicz SLLN (, , ). When , fundamental results for the SLLN were obtained.
Under an independent assumption, many SLLNs for the weighted sums are obtained. One can refer to Adler and Rosalsky , Chow and Teicher , Fernholz and Teicher , Jamison et al.  and Teicher .
Under a pairwise independent assumption, Rosalsky  obtained some SLLNs for weighted sums of pairwise independent and identically distributed random variables. Sung  obtained sufficient conditions for (1.1) if is a sequence of pairwise independent random variables satisfying . Sung  presented the following result: a.s., where is a sequence of positive constants with and is a sequence of pairwise independent and identically distributed random variables.
Recently Sung  gave the following definition.
Definition 1.1 (Sung )
We can see that a wide class of mean zero random variables satisfies (1.2). Inspired by Sung [7, 15], we establish SLLN of the form (1.1) for a class of random variables satisfying the maximal moment inequality with exponent 2.
The rest of the paper is organized as follows. In Section 2, some preliminary definition and lemmas are presented. In Section 3, main results and their proofs are provided.
Throughout the paper, let be the indicator function of the set A. C denotes a positive constant not depending on n, which may be different in various places. Let and be sequences of positive numbers, represents that there exists a constant such that for all n.
The following lemmas and definition will be needed in this paper.
Lemma 2.1 (Sung )
for any sequence satisfying .
Lemma 2.2 (Sung )
Let be a sequence of random variables satisfying the maximal moment inequality with exponent 2. If , then converges almost surely.
for all and .
where C is a positive constant.
Lemma 2.5 (Hu )
3 Main results
By Kronecker’s lemma, we can obtain (3.6) immediately.
follows from (3.9)-(3.11). Hence the result is proved. □
which implies (3.19) from Kronecker’s lemma. We complete the proof of theorem. □
The proof is completed. □
Remark 3.4 It is easy to see that a wide class of mean zero random variables satisfies the maximal moment inequality with exponent 2. Examples include independent random variables, negatively associated random variables (see Matula ), negatively superadditive dependent random variables (see Shen et al. ), φ-mixing random variables and AANA random variables (see Wang et al. [21, 22]), and -mixing random variables (see Utev et al. ). So Theorems 3.1-3.3 hold for this wide class of random variables.
The authors are most grateful to the editor and the anonymous referee for their careful reading and insightful comments. This work is supported by the National Natural Science Foundation of China (11171001, 11201001), Natural Science Foundation of Anhui Province (1208085QA03), Humanities and Social Sciences Project from Ministry of Education of China (12YJC91007), Key Program of Research and Development Foundation of Hefei University (13KY05ZD) and Doctoral Research Start-up Funds Projects of Anhui University.
- Adler A, Rosalsky A: On the strong law of large numbers for normed weighted sums of i.i.d. random variables. Stoch. Anal. Appl. 1987, 5: 467–483. 10.1080/07362998708809131MathSciNetView ArticleGoogle Scholar
- Chow YS, Teicher H: Almost certain summability of independent, identically distributed random variables. Ann. Math. Stat. 1971, 42: 401–404. 10.1214/aoms/1177693533MathSciNetView ArticleGoogle Scholar
- Fernholz LT, Teicher H: Stability of random variables and iterated logarithm laws of martingales and quadratic forms. Ann. Probab. 1980, 8: 765–774. 10.1214/aop/1176994664MathSciNetView ArticleGoogle Scholar
- Jamison B, Orey S, Pruitt W: Convergence of weighted averages of independent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 1965, 4: 40–44. 10.1007/BF00535481MathSciNetView ArticleGoogle Scholar
- Teicher H: Almost certain convergence in double arrays. Z. Wahrscheinlichkeitstheor. Verw. Geb. 1985, 69: 331–345. 10.1007/BF00532738MathSciNetView ArticleGoogle Scholar
- Rosalsky A: Strong stability of normed weighted sums of pairwise i.i.d. random variables. Bull. Inst. Math. Acad. Sin. 1987, 15: 203–219.MathSciNetGoogle Scholar
- Sung SH: Strong law of large numbers for weighted sums of pairwise independent random variables. Bull. Inst. Math. Acad. Sin. 1999, 27(1):23–28.Google Scholar
- Sung SH: On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions. Stat. Probab. Lett. 2013, 83: 1963–1968. 10.1016/j.spl.2013.05.009View ArticleGoogle Scholar
- Wu QY: A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J. Inequal. Appl. 2010., 2010: Article ID 383805Google Scholar
- Wu QY, Jiang YY:Some strong limit theorems for weighted product sums of -mixing sequences of random variables. J. Inequal. Appl. 2009., 2009: Article ID 174768Google Scholar
- Hu SH, Wang XJ, Yang WZ, Zhao T: The Hàjek-Rènyi-type inequality for associated random variables. Stat. Probab. Lett. 2009, 79: 884–888. 10.1016/j.spl.2008.11.014MathSciNetView ArticleGoogle Scholar
- Shen Y, Wang XJ, Yang WZ, Hu SH: Almost sure convergence theorem and strong stability for weighted sums of NSD random variables. Acta Math. Sin. Engl. Ser. 2013, 29: 743–756. 10.1007/s10114-012-1723-6MathSciNetView ArticleGoogle Scholar
- Zhou XC, Tan CC, Lin JG:On the strong laws for weighted sums of -mixing random variables. J. Inequal. Appl. 2011., 2011: Article ID 157816Google Scholar
- Zhou XC: Complete moment convergence of moving average processes under φ -mixing assumptions. Stat. Probab. Lett. 2010, 80: 285–292. 10.1016/j.spl.2009.10.018View ArticleGoogle Scholar
- Sung SH: On the strong law of large numbers for weighted sums of random variables. Comput. Math. Appl. 2011, 62: 4277–4287. 10.1016/j.camwa.2011.10.018MathSciNetView ArticleGoogle Scholar
- Hu SH: Some new results for the strong law of large numbers. Acta Math. Sin. (Chin. Ser.) 2003, 46(6):1123–1134. (Chinese)Google Scholar
- Fazekas I, Klesov O: A general approach to the strong law of large numbers. Theory Probab. Appl. 2002, 45(3):436–449.MathSciNetView ArticleGoogle Scholar
- Rao MM: Measure Theory and Integration. Wiley, New York; 1987.Google Scholar
- Hu SH, Hu M: A general approach rate to the strong law of large numbers. Stat. Probab. Lett. 2006, 76: 843–851. 10.1016/j.spl.2005.10.016View ArticleGoogle Scholar
- Matula P: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab. Lett. 1992, 12: 209–213.MathSciNetView ArticleGoogle Scholar
- Wang XJ, Hu SH, Shen Y, Yang WZ: Moment inequality for φ -mixing sequences and its applications. J. Inequal. Appl. 2011., 2009: Article ID 379743Google Scholar
- Wang XJ, Hu SH, Yang WZ: Convergence properties for asymptotically almost negatively associated sequence. Discrete Dyn. Nat. Soc. 2010., 2010: Article ID 218380Google Scholar
- Utev S, Peligrad M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J. Theor. Probab. 2003, 16: 101–115. 10.1023/A:1022278404634MathSciNetView ArticleGoogle Scholar
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