## 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 = (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/E X_{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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