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Uniform renewal theory with applications to expansions of random geometric sums

  • J. Blanchet (a1) and P. Glynn (a2)
Abstract

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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Copyright
Corresponding author
Postal address: Statistics Department, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA. Email address: blanchet@fas.harvard.edu
∗∗ Postal address: Management Science and Engineering, Stanford University, 380 Panama Way, Stanford, CA 94305, USA.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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