Uniform Renewal Theory with Applications to Expansions of Random Geometric Sums
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 $S_{M}=X_{1}+\cdots +X_{M}$ is called a geometric sum. In this paper we obtain asymptotic expansi...
| Veröffentlicht in: | Advances in Applied Probability. - Applied Probability Trust. - 39(2007), 4, Seite 1070-1097 |
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| Format: | Online-Aufsatz |
| Sprache: | English |
| Veröffentlicht: |
2007
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| Zugriff auf das übergeordnete Werk: | Advances in Applied Probability |
| Schlagworte: | Renewal theory random geometric sum corrected diffusion approximation Mathematics |
| Zusammenfassung: | 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 $S_{M}=X_{1}+\cdots +X_{M}$ is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of $S_{M}$ as p ↘ 0. If E X₁ > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that $\text{P}(pS_{M}>x)$ ≈ exp(-x/EX₁). Conversely, if E X₁ = 0 then the expansion is given in powers of $\sqrt{p}$. 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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| ISSN: | 00018678 |