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

Abstract

Let {Xk, k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (-∞, ∞), and let τ 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≤n Xk, Sn = ∑n k=0 Xk and S(n) = max0≤k≤n Sk for n ≥ 1, and their randomized versions X(τ), Sτ and S(τ), where X0 = 0 by convention and h > 0 is arbitrarily fixed.