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July, 1990 Error Bounds for Exponential Approximations of Geometric Convolutions
Mark Brown
Ann. Probab. 18(3): 1388-1402 (July, 1990). DOI: 10.1214/aop/1176990750


Define $Y_0$ to be a geometric convolution of $X$ if $Y_0$ is the sum of $N_0$ i.i.d. random variables distributed as $X$, where $N_0$ is geometrically distributed and independent of $X$. It is known that if $X$ is nonnegative with finite second moment, then as $p \rightarrow 0, Y_0/EY_0$ converges in distribution to an exponential distribution with mean 1. We derive an upper bound for $d(Y_0)$, the sup norm distance between $Y_0$ and an exponential with mean $EY_0$. This upper bound is $d(Y_0) \leq \operatorname{cp}$ for $0 < p \leq \frac{1}{2}$, where $c = EX^2/(EX)^2$. It is asymptotically $(p \rightarrow 0)$ tight. Also derived is a bound for $d(Y_0 + Z)$, where $Z$ is independent of $Y_0$.


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Mark Brown. "Error Bounds for Exponential Approximations of Geometric Convolutions." Ann. Probab. 18 (3) 1388 - 1402, July, 1990.


Published: July, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0709.60016
MathSciNet: MR1062073
Digital Object Identifier: 10.1214/aop/1176990750

Primary: 60F05
Secondary: 60E15

Keywords: Exponential approximations , Geometric convolutions , NWU distributions , queuing theory , regenerative processes , reliability theory , renewal theory

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 3 • July, 1990
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