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October, 1988 Bounds on the Coarseness of Random Sums
James Allen Fill
Ann. Probab. 16(4): 1644-1664 (October, 1988). DOI: 10.1214/aop/1176991589


Let $X^0_1, \ldots, X^0_N$ be integer-valued random variables and let $a_1, \ldots, a_N$ be (fixed) nonzero vectors. We introduce the notion of coarseness of a discrete distribution and obtain upper bounds on the coarseness of the distribution of $S = \Sigma X^0_ia_i$ by comparison with the case $a_i \equiv a$. The bounds derived are seen to be tight and to apply for example when $S$ is formed (a) from independent summands or (b) by using any of a large family of sampling schemes. We show how such bounds can easily and efficiently substitute for use of Berry-Esseen theorems and other analytical methods in applications.


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James Allen Fill. "Bounds on the Coarseness of Random Sums." Ann. Probab. 16 (4) 1644 - 1664, October, 1988.


Published: October, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0655.60015
MathSciNet: MR958208
Digital Object Identifier: 10.1214/aop/1176991589

Primary: 60E15
Secondary: 60G50

Keywords: Berry-Esseen theorems , coarseness , log concavity , Random sums , sampling , Stochastic monotonicity , top $K$ probability , top $K$ set

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 4 • October, 1988
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