  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).
