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 ↘ 0. If EX 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) ≈ exp(-x/EX 1). Conversely, if EX 1 = 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.
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