On complete convergence for arrays of rowwise dependent random variables
Introduction
Let be a sequence of random variables defined on probability space . A sequence is said to converge completely to a constant C if Hsu and Robbins (1947), who introduce this concept, proved that the sequence of arithmetic means of independent identically distributed random variables converges completely to the expected value of the summands, provided the variance is finite. The converse theorem was proved by Erdös (1949).
The extensions of Hsu–Robbins–Erdös's result, due to Katz (1963), Baum and Katz (1965), Chow (1973), form a complete convergence theorem with a Marcinkiewicz–Zygmunt type normalization (see Gut, 1983). Theorem 1.1 Let be a sequence of independent identically distributed random variables and let , . The following statements are equivalent: If , the above are also equivalent to
Many authors generalized and extended this result without assumption of identical distribution in several directions. They studied the cases of independent, stochastically dominated random variables, triangular arrays of rowwise independent, stochastically dominated in the Cesaro sense random variables and sequences of independent random variables taking value in a Banach space (Pruitt, 1966, Rohatgi, 1971, Hu et al., 1989, Gut, 1992, Wang et al., 1993; Kuczmaszewska and Szynal, 1988, Kuczmaszewska and Szynal, 1991, Kuczmaszewska and Szynal, 1994; Sung, 1997, Bozorgnia et al., 1993, Hu et al., 1999; Hu et al., 1998, Hu and Volodin, 2000, Ahmed et al., 2002, Kuczmaszewska, 2004, Sung et al., 2005; Tómács, 2005).
In this paper, we consider a complete convergence in the strong law of large numbers for arrays of dependent random variables. We study the complete convergence for -mixing and -mixing sequences of random variables. The obtained results extend some previous known. Some results for complete convergence for -mixing and -mixing random variables one can find Zhengyan and Chuanrong (1996) (Shao, Kong and Zang, Section 8.4), Shao (1995), Shixin (2004). Definition 1.1 A sequence of random variables is said to be a -mixing sequence if the maximal correlation coefficientas , where is the -field generated by random variables , . Definition 1.2 A sequence of random variable is said to be a -mixing sequence if there exists such thatwhere S, T are the finite subsets of positive integers such that dist and is the -field generated by the random variable .
The -mixing conception is similar to the -mixing, but they are quite different from each other.
In our further consideration we need the following definition and lemmas. Definition 1.3 A real valued function , positive and measurable on for some , is said to be slowly varying if Lemma 1.1 Shao, 1995 Let be a -mixing sequence of random variables such that , . Then for any , there exists a constant depending only on q and such that: Lemma 1.2 Shanchao, 1998; Peligrad and Gut, 1999 Let be a -mixing sequence with . Let , , , , . Then there exists a positive constant C such that:
We also assume that in our consideration constant is not the same constant in each case.
Section snippets
The main results
In this paper we consider arrays of random variables.
Let denotes the maximal correlation coefficient defined in (1.1) for the nth row of an array i.e. for the sequence , .
Similarly, we will use the notation for denoting the coefficient defined in (1.2) for the sequence , .
Moreover, let be a sequence of nonnegative, even, continuous and nondecreasing on functions with and such that:
Acknowledgment
The author is very grateful to the referee for the suggestions which allowed her to improve this paper.
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