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April 2002 Total variation asymptotics for sums of independent integer random variables
A. D. Barbour, V. Ćekanavićius
Ann. Probab. 30(2): 509-545 (April 2002). DOI: 10.1214/aop/1023481001


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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A. D. Barbour. V. Ćekanavićius. "Total variation asymptotics for sums of independent integer random variables." Ann. Probab. 30 (2) 509 - 545, April 2002.


Published: April 2002
First available in Project Euclid: 7 June 2002

zbMATH: 1018.60049
MathSciNet: MR1905850
Digital Object Identifier: 10.1214/aop/1023481001

Primary: 60F05 , 60G50 , 62E20

Keywords: Compound Poisson , Kolmogorov's problem , Stein's method , total variation distance

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.30 • No. 2 • April 2002
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