  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|>t a_n) for all t>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=1/n and a_n=(n log n)^{1/2} for n>1, 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).
