A general Hsu-Robbins-Erdos type estimate of tail probabilities of sums   of independent identically distributed random variables
  Let $X_1,X_2,...$ be a sequence of independent and identically distributed random variables, and put $S_n=X_1+...+X_n$. Under some conditions on the positive sequence $\tau_n$ and the positive increasing sequence $a_n$, we give necessary and sufficient conditions for the convergence of $\sum_{n=1}^\infty \tau_n P(|S_n|\ge \epsilon a_n)$ for all $\epsilon>0$, generalizing Baum and Katz's (1965) generalization of the Hsu-Robbins-Erdos (1947, 1949) law of large numbers, also allowing us to characterize the convergence of the above series in the case where $\tau_n=n^{-1}$ and $a_n=(n\log n)^{1/2}$ for $n\ge 2$, thereby answering a question of Spataru. 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-Jorgensen inequality (1974). 
Alexander R. Pruss