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Total variation asymptotics for sums of independent integer random variables


Barbour, A D; Cekanavicius, V (2002). Total variation asymptotics for sums of independent integer random variables. The Annals of Probability, 30(2):509-545.

Abstract

Let $W_n := \sum_{j=1}^n Z_j$ be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability $\mathbb{P}[W_n \in A]$ of an arbitrary subset $A \in \mathbb{Z}$. Our approximation improves upon the classical expansions by including an explicit, uniform error estimate, involving only easily computable properties of the distributions of the $Z_j:$ an appropriate number of moments and the total variation distance $d_{\mathrm{TV}}(\mathscr{L}(Z_j), \mathscr{L}(Z_j + 1))$. The proofs are based on Stein’s method for signed compound Poisson approximation.

Let $W_n := \sum_{j=1}^n Z_j$ be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability $\mathbb{P}[W_n \in A]$ of an arbitrary subset $A \in \mathbb{Z}$. Our approximation improves upon the classical expansions by including an explicit, uniform error estimate, involving only easily computable properties of the distributions of the $Z_j:$ an appropriate number of moments and the total variation distance $d_{\mathrm{TV}}(\mathscr{L}(Z_j), \mathscr{L}(Z_j + 1))$. The proofs are based on Stein’s method for signed compound Poisson approximation.

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33 citations in Web of Science®
36 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:compound Poisson; Stein's method; total variation distance; Kolmogorov's problem
Language:English
Date:2002
Deposited On:07 Apr 2010 10:12
Last Modified:05 Apr 2016 13:25
Publisher:Institute of Mathematical Statistics
ISSN:0091-1798
Publisher DOI:https://doi.org/10.1214/aop/1023481001
Permanent URL: https://doi.org/10.5167/uzh-21917

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