## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 3 (1975), 534-545.

### Poisson Approximation for Dependent Trials

#### Abstract

Let $X_1, \cdots, X_n$ be an arbitrary sequence of dependent Bernoulli random variables with $P(X_i = 1) = 1 - P(X_i = 0) = p_i.$ This paper establishes a general method of obtaining and bounding the error in approximating the distribution of $\sum^n_{i=1} X_i$ by the Poisson distribution. A few approximation theorems are proved under the mixing condition of Ibragimov (1959), (1962). One of them yields, as a special case and with some improvement, an approximation theorem of Le Cam (1960) for the Poisson binomial distribution. The possibility of an asymptotic expansion is also discussed and a refinement in the independent case obtained. The method is similar to that of Charles Stein (1970) in his paper on the normal approximation for dependent random variables.

#### Article information

**Source**

Ann. Probab., Volume 3, Number 3 (1975), 534-545.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996359

**Digital Object Identifier**

doi:10.1214/aop/1176996359

**Mathematical Reviews number (MathSciNet)**

MR428387

**Zentralblatt MATH identifier**

0335.60016

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 62E20: Asymptotic distribution theory 60G99: None of the above, but in this section

**Keywords**

Poisson approximation rates of convergence dependent trials

#### Citation

Chen, Louis H. Y. Poisson Approximation for Dependent Trials. Ann. Probab. 3 (1975), no. 3, 534--545. doi:10.1214/aop/1176996359. https://projecteuclid.org/euclid.aop/1176996359