Strong Law of Large Numbers for α-Mixing Sequences

摘要

For independent identically distributed random variables, the Marcinkiewicz strong law of large numbers is that sppose EX_n=0, Then n~(-1)S_n→0, n→∞, a.s. if and only if E|x_1|~p＜∞. Let {X_n, n≥1} be an identically distributed α-mixing sequence of random variables, in this paper, the Marcinkiewicz's strong law of large numbers for {X_n, n≥1} is discussed, by using Wang Xiaoming's Borell-Cantelli Lemma (Wang X. M.1997), we have the following Equiverlences, {arbitrary}＞0. By use of Herrndorf's maximal inequality (Herrndorf N. 1983), necessary conditions for Marcinkiewicz's strong law of large numbers are obtained, which require low mixing speed. As a consequence, by use of Shao Qiman's result(1995), we obtain the Marcinkiewicz's strong law of large numbers for ρ-mixing sequence of random variables.