J. Appl. Prob. 41, 108–116 (2004) Printed in Israel  Applied Probability Trust 2004 ASYMPTOTIC BEHAVIOR OF TAIL AND LOCAL PROBABILITIES FOR SUMS OF SUBEXPONENTIAL RANDOM VARIABLES KAI W. NG, ∗ University of Hong Kong QIHE TANG, ∗∗ University of Amsterdam 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 X (n) = max 0≤k≤n X k , S n = ∑ n k=0 X k and S (n) = max 0≤k≤n S k 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. Keywords: Asymptotics; local probability; partial sum; subexponentiality; tail probability 2000 Mathematics Subject Classification: Primary 60G50 Secondary 62E20 1. Introduction Throughout this paper, {X k ,k ≥ 1} denotes a sequence of independent and identically distributed (i.i.d.) random variables with common distribution function F(x) = 1 - F(x) = P(X 1 ≤ x) which satisfies F(x) > 0 for any x ∈ R. To simplify notation, we may say that the X k for k ≥ 1 are independent copies of a generic random variable X. The mean of X, if it exists, is denoted by μ F . With the convention X 0 = 0, we write, for n ≥ 1, X (n) = max 0≤k≤n X k , S n = n  k=0 X k , S (n) = max 0≤k≤n S k . Since in most cases accurate distributions for the sum S n and the maximum S (n) are not available, deriving asymptotic relationships for their tail probabilities becomes important. In this, an important role is played by a well-known class of heavy-tailed distribution functions, namely the subexponential class S. By definition, a distribution function F concentrated on (0, ∞) belongs to S if and only if, for each n ≥ 1, the tail probabilities of the sum and the maximum of the first n random variables of {X k ,k ≥ 1} are asymptotically of the same order, i.e. P(S n > x) ∼ P(X (n) > x) as x →∞. (1.1) Received 9 August 2002; revision received 8 September 2003. ∗ Postal address: Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong. Email address: kaing@hku.hk ∗∗ Postal address: Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. Email address: q.tang@uva.nl 108