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.

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.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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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 |

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