Blanchet J, Glynn P. Uniform renewal theory with applications to expansions of random geometric sums. Advances in Applied Probability. 2007;39(4):1070-1097. doi:10.1239/aap/1198177240
Abstract
Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.
Authors
J Blanchet, P Glynn
Publication date
2007/12
Journal
Advances in Applied Probability
Volume
39
Issue
4
Pages
1070-1097
Publisher
Cambridge University Press
