Le Cam's theorem

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In probability theory, Le Cam's theorem, named after Lucien le Cam (1924 — 2000), is as follows.

Suppose:

• X₁, ..., X_n are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.

• Pr(X_i = 1) = p_i for i = 1, 2, 3, ...

• 

• 

Then

In other words, the sum has approximately a Poisson distribution.

By setting p_i = 2λ_n²/n, we see that this generalizes the usual Poisson limit theorem.

[edit] References

• Le Cam, L. "An Approximation Theorem for the Poisson Binomial Distribution," Pacific Journal of Mathematics, volume 10, pages 1181 - 1197 (1960).

• Le Cam, L. "On the Distribution of Sums of Independent Random Variables," Bernouli, Bayes, Laplace: Proceedings of an International Research Seminar (Jerzy Neyman and Lucien le Cam, editors), Springer-Verlag, New York, pages 179 - 202 (1963).

[edit] External links

• Le Cam's Inequality on Mathworld