Le Cam's theorem

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

Suppose:

* "X"₁, ..., "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, ...

* $lambda_n = p_1 + cdots + p_n.,$

* $S_n = X_1 + cdots + X_n.,$

Then

: $sum_{k=0}^infty left| Pr(S_n=k) - {lambda_n^k e^{-lambda_n} over k!}
ight| < 2 sum_{i=1}^n p_i^2.$

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

By setting "p"_"i" = 2λ_"n"²/"n", we see that this generalizes the usual Poisson limit theorem.

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).

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