A general Hsu-Robbins-Erdős Type estimate of tail probabilities of sums of independent identically distributed random variables

Abstract. Let X1, X2, . . . be a sequence of independent and identically distributed random variables, and put Sn = X1 + · · ·+ Xn. Under some conditions on the positive sequence τn and the positive increasing sequence an, we give necessary and sufficient conditions for the convergence of ∑ ∞ n=1 τnP (|Sn| ≥ εan) for all ε > 0, generalizing Baum and Katz’s (1965) generalization of the Hsu-Robbins-Erdős (1947, 1949) law of large numbers, also allowing us to characterize the convergence of the above series in the case where τn = n −1 and an = (n logn) 1/2 for n ≥ 2, thereby answering a question of Spătaru. Moreover, some results for nonidentically distributed independent random variables are obtained by a recent comparison inequality. Our basic method is to use a central limit theorem estimate of Nagaev (1965) combined with the Hoffman-Jørgensen inequality (1974).