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:

  • Pr(Xi = 1) = pi 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!} \right| < 2 \sum_{i=1}^n p_i^2.

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

By setting pi = 2λn2/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 link

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