Uniform Renewal Theory with Applications to Expansions of Random Geometric Sums

J. Blanchet and P. W. Glynn

Advances in Applied Probability , 39, 1070-1097 (2007)

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/E X1). 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.