Uniform Renewal Theory with Applications to Expansions of Random Geometric Sums
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چکیده
Consider a sequence X = (Xn : n ≥ 1) of iid random variables (r.v.’s) and a geometrically distributed r.v. M with parameter p and independent of X. The r.v. SM = X1 + ... + XM is called a geometric sum. In this paper, we obtain asymptotic expansions for the distribution of SM as p& 0. If EX > 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/EX). On the other hand, if EX = 0 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 uniform renewal theory results that are also developed in this paper.
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