The Annals of Probability

Total variation asymptotics for sums of independent integer random variables

A. D. Barbour and V. Ćekanavićius

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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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Ann. Probab., Volume 30, Number 2 (2002), 509-545.

First available in Project Euclid: 7 June 2002

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Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory

compound Poisson Stein's method total variation distance Kolmogorov's problem


Barbour, A. D.; Ćekanavićius, V. Total variation asymptotics for sums of independent integer random variables. Ann. Probab. 30 (2002), no. 2, 509--545. doi:10.1214/aop/1023481001.

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