Uniform renewal theory with applications to expansions of random geometric sums

Consider a sequence X = (X-n : n >= 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable S-M = X-1 + ... +X-M is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of S-M as p SE arrow 0. If E X-1 > 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(pS(M) > x) approximate to exp(-x/EX1). Conversely, if E X-1 = 0 then the expansion is given in powers of root 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.