On Chung‐Teicher Type Strong Law of Large Numbers for ρ^∗‐Mixing Random Variables

Abstract

In this paper the classical strong laws of large number of Kolmogorov, Chung, and Teicher for independent random variables were generalized on the case of -mixing sequence. The main result was applied to obtain a Marcinkiewicz SLLN.

On a General Approach to the Strong Laws of Large Numbers*

“…, X n be fixed. If the first Kolmogorov type maximal inequality (19) for the probability is satisfied, then the first Hájek-Rényi type maximal inequality (20) for the probability is satisfied with C = 4K.…”

“…, r > 0, and K > 0 be fixed. Assume that for each m 1 the first Kolmogorov type maximal inequality (19) for the probability is satisfied . Let b 1 , b 2 …”

“…A version of Theorem 12 was proved by Tómács [41] when instead of the moment inequality (2) the probability inequality (19) was assumed.…”

On Complete Convergence for Weighted Sums ofρ*-Mixing Random Variables

“…Many authors have studied this concept providing interesting results and applications. See, for example, Bradley [13] for the central limit theorem, Bryc and Smoleń ski [14], Peligrad and Gut [15], and Utev and Peligrad [16] for moment inequalities, Gan [17], Kuczmaszewska [18], Wu and Jiang [19] and Wang et al [20,21] for almost sure convergence, Peligrad and Gut [15], Cai [22], Kuczmaszewska [23], Zhu [24], An and Yuan [25], Wang et al [26], and Sung [27] for complete convergence, Peligrad [28] for invariance principle, Wu and Jiang [29] for strong limit theorems for weighted product sums of * -mixing sequences of random variables, Wu and Jiang [30] for Chover-type laws of the -iterated logarithm, Wu [31] for strong consistency of estimator in linear model, Wang et al [32] for complete consistency of the estimator of nonparametric regression models, Wu et al [33] and Guo and Zhu [34] for complete moment convergence, and so forth. When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.…”

Strong laws of large numbers for ρ˜-mixing random variables

“…for almost sure results, Utev and Peligrad [10] for maximal inequalities, and Kuczmaszewska [11] for Chung-Teicher type SLLN. However, by far, general strong laws of large numbers in which the coefficient of sum and the weight are both general functions have not been obtained yet.…”