Abstract
In this paper, we study the almost sure convergence for ρ ˜ -mixing sequences of random variables. As a result, the authors improve the corresponding results of Yang [Yang, Shanchao, 1998. Some moment inequalities for partial sums of random variables and their applications. Chinese Sci. Bull. 43 (17), 1823–1827], Gan [Gan, Shixin, 2004. Almost sure convergence for ρ ˜ -mixing random variable sequences. Statist. Probab. Lett. 67, 289–298], and Wu [Wu, Qunying, 2001. Some convergence properties for ρ ˜ -mixing sequences. J. Engng. Math. 18 (3), 58–64 (in Chinese)]. We extend the classical Khintchine–Kolmogorov convergence theorem, the Marcinkiewicz strong law of large numbers, and the three series theorem for independent sequences of random variables to ρ ˜ -mixing sequences of random variables without necessarily adding any extra conditions.
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