complete convergence in mean; double array of random variables with values in Banach space; martingale difference double array; strong law of large numbers; $p$-uniformly smooth space
The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order $p$). In this paper, we give some new results of complete convergence in mean of order $p$ and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces.
 Chow, Y. S.: On the rate of moment convergence of sample sums and extremes
. Bull. Inst. Math., Acad. Sin. 16 (1988), 177-201. MR 1089491
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