## The Annals of Probability

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

### 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 $a$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60F10: Large deviations 60F15: Strong theorems 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G45

**Keywords**

Borel-Cantelli lemmas strong law Poisson approximation

#### Citation

Freedman, David. Another Note on the Borel-Cantelli Lemma and the Strong Law, with the Poisson Approximation as a By-product. Ann. Probab. 1 (1973), no. 6, 910--925. doi:10.1214/aop/1176996800. https://projecteuclid.org/euclid.aop/1176996800