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

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 $a$.