Refinement of convergence rates for tail probabilities

Let X-1, X-2,... be, i.i.d. random variables, and put S-n = X-1 + ... + X-n. We find necessary and sufficient moment conditions for integral(infinity)(epsilon) f(x(q))dx < infinity, epsilon > delta, where delta >= 0 and q > 0, and f (x) = Sigma(n) a(n)P(vertical bar S-n vertical bar > xb(n)) with a(n) > 0 and b(n) is either n(1/p), 0 < p < 2, root n log n or root n log log n. The series f (x) we deal with are classical series studied by Hsu and Robbins, Erdos, Spitzer, Baum and Katz, Davis, Lai, Gut, etc.