The Annals of Probability

Poisson Approximation for Dependent Trials

Louis H. Y. Chen

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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

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

First available: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 60G99: None of the above, but in this section

Poisson approximation rates of convergence dependent trials


Chen, Louis H. Y. Poisson Approximation for Dependent Trials. The Annals of Probability 3 (1975), no. 3, 534--545. doi:10.1214/aop/1176996359.

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