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|a (JST)20443559
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|b ger
|c DE-627
|e rakwb
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|a eng
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|a 60K05
|2 MSC
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|2 MSC
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|2 MSC
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|a Blanchet, J.
|e verfasserin
|4 aut
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|a Uniform Renewal Theory with Applications to Expansions of Random Geometric Sums
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|c 2007
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|a Text
|b txt
|2 rdacontent
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|a Computermedien
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|a 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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|a Copyright 2007 Applied Probability Trust
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| 650 |
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|a Renewal theory
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| 650 |
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|a random geometric sum
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| 650 |
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|a corrected diffusion approximation
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| 650 |
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4 |
|a Mathematics
|x Mathematical procedures
|x Approximation
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| 650 |
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4 |
|a Mathematics
|x Mathematical expressions
|x Mathematical theorems
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| 650 |
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4 |
|a Mathematics
|x Applied mathematics
|x Statistics
|x Random walk
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| 650 |
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4 |
|a Mathematics
|x Mathematical expressions
|x Mathematical functions
|x Implicit functions
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| 650 |
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4 |
|a Mathematics
|x Pure mathematics
|x Probability theory
|x Random variables
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| 650 |
|
4 |
|a Mathematics
|x Applied mathematics
|x Statistics
|x Applied statistics
|x Descriptive statistics
|x Statistical distributions
|x Distribution functions
|x Probability distributions
|x Mathematical moments
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| 650 |
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4 |
|a Mathematics
|x Mathematical objects
|x Mathematical series
|x Power series
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| 650 |
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|a Mathematics
|x Pure mathematics
|x Calculus
|x Differential calculus
|x Mathematical integration
|x Integration techniques
|x Integration by parts
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| 650 |
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4 |
|a Mathematics
|x Mathematical expressions
|x Mathematical functions
|x Mathematical transformations
|x Integral transformations
|x Fourier transformations
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| 650 |
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|a Mathematics
|x Mathematical objects
|x Mathematical series
|x Series convergence
|x Absolute convergence
|x General Applied Probability
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| 655 |
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4 |
|a research-article
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| 700 |
1 |
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|a Glynn, P.
|e verfasserin
|4 aut
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| 773 |
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|i Enthalten in
|t Advances in Applied Probability
|d Applied Probability Trust
|g 39(2007), 4, Seite 1070-1097
|w (DE-627)269247009
|w (DE-600)1474602-5
|x 00018678
|7 nnns
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|g volume:39
|g year:2007
|g number:4
|g pages:1070-1097
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|u https://www.jstor.org/stable/20443559
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|d 39
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|h 1070-1097
|