- Le Cam's theorem
In
probability theory , Le Cam's theorem, named afterLucien le Cam (1924 – 2000), is as follows.Suppose:
* "X"1, ..., "X""n" are independent
random variable s, each with aBernoulli 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).External links
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