Asymptotic behavior of tail and local probabilities for sums of subexponential random variables

Let {X-k, k greater than or equal to 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (-infinity, infinity), and let tau be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {X-k, k greater than or equal to 1}. This paper investigates asymptotic behavior of the tail probabilities P((.) > x) and the local probabilities P(x < (.) less than or equal to x + h) of the quantities X-(n) = max(0less than or equal tokless than or equal ton) X-k, S-n = Sigma(k=0) X-k and S-(n) max(0less than or equal tokless than or equal ton) S-k for n greater than or equal to 1, and their randomized versions X-(tau), S-tau and S-(tau), where X-0 = 0 by convention and It > 0 is arbitrarily fixed.