for all & > 0, generalizing Baum and Katz's~(1965) generalization of the Hsu–Robbins–Erds (1947, 1949) law of large numbers,
also allowing us to characterize the convergence of the above series in the case where τn = n-1and
for n ≤ 2, thereby answering a question of Spătaru. Moreover, some results for non-identically 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-Jrgensen inequality~(1974).