## The Annals of Probability

### Another Note on the Borel-Cantelli Lemma and the Strong Law, with the Poisson Approximation as a By-product

David Freedman

#### 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 $a$.

#### Article information

Source
Ann. Probab., Volume 1, Number 6 (1973), 910-925.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996800

Digital Object Identifier
doi:10.1214/aop/1176996800

Mathematical Reviews number (MathSciNet)
MR370711

Zentralblatt MATH identifier
0301.60025

JSTOR