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

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Abstract

Let { X k , 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 { X k , k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x ) and the local probabilities P( x < · ≤ x + h ) of the quantities and for n ≥ 1, and their randomized versions X ( τ ) , S τ and S ( τ ) , where X 0 = 0 by convention and h > 0 is arbitrarily fixed.

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