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December, 1973 Another Note on the Borel-Cantelli Lemma and the Strong Law, with the Poisson Approximation as a By-product
David Freedman
Ann. Probab. 1(6): 910-925 (December, 1973). DOI: 10.1214/aop/1176996800

Abstract

Here is another way to prove Levy's conditional form of the Borel-Cantelli lemmas, and his strong law. Consider a sequence of dependent variables, each bounded between 0 and 1. Then the sum $S$ of the variables tends to be close to the sum $T$ of the conditional expectations. Indeed, the chance that $S$ is above one level and $T$ is below another is exponentially small. So is the chance that $S$ is below one level and $T$ is above another. The inequalities also show that for a sequence of dependent events, such that each has uniformly small conditional probability given the past, and the sum of the conditional probabilities is nearly constant at $a$, the number of events which occur is nearly Poisson with parameter $a$.

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David Freedman. "Another Note on the Borel-Cantelli Lemma and the Strong Law, with the Poisson Approximation as a By-product." Ann. Probab. 1 (6) 910 - 925, December, 1973. https://doi.org/10.1214/aop/1176996800

Information

Published: December, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0301.60025
MathSciNet: MR370711
Digital Object Identifier: 10.1214/aop/1176996800

Subjects:
Primary: 60F05
Secondary: 60F10 , 60F15 , 60G40 , 60G45

Keywords: Borel-Cantelli lemmas , Poisson approximation , strong law

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 6 • December, 1973
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