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

Let {Xk, k ¿ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (-¿,¿), and let ¿ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {Xk, k ¿ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x) and the local probabilities P(x < · ¿ x + h) of the quantities X(n) = max0¿k¿nXk, Sn = ¿k=0nXk and S(n) = max0¿k¿nSk for n ¿ 1, and their randomized versions X(¿), S¿ and S(¿), where X0 = 0 by convention and h > 0 is arbitrarily fixed.