Precise asymptotics in the Baum-Katz and Davis laws of large numbers

Let X, X-1, X-2,... be a sequence of i.i.d. random variables such that EX = 0, let Z be a random variable possessing a stable distribution G with exponent alpha, 1 < alpha less than or equal to 2, assume the distribution of X is attracted to G, and set S-n = X-1 + ... + X-n. We prove that [GRAPHICS] for 1 less than or equal to p < r < alpha, under the additional assumption that there is normal attraction to G, and that [GRAPHICS] for 1 less than or equal to p < alpha. We close with a related result for the limiting case alpha = r = p = 2. (C) 2000 Academic Press.