Uniform renewal theory with applications to expansions of random geometric sums

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.