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June, 1975 Poisson Approximation for Dependent Trials
Louis H. Y. Chen
Ann. Probab. 3(3): 534-545 (June, 1975). DOI: 10.1214/aop/1176996359


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.


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Louis H. Y. Chen. "Poisson Approximation for Dependent Trials." Ann. Probab. 3 (3) 534 - 545, June, 1975.


Published: June, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0335.60016
MathSciNet: MR428387
Digital Object Identifier: 10.1214/aop/1176996359

Primary: 60F05
Secondary: 60G99 , 62E20

Keywords: dependent trials , Poisson approximation , rates of convergence

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 3 • June, 1975
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