Le Cam's theorem

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In probability theory, Le Cam's theorem, named after Lucien Le Cam, states the following. [1] [2] [3]

Suppose:

Then

In other words, the sum has approximately a Poisson distribution and the above inequality bounds the approximation error in terms of the total variation distance.

By setting pi = λn/n, we see that this generalizes the usual Poisson limit theorem.

When is large a better bound is possible: , [4] where represents the operator.

It is also possible to weaken the independence requirement. [4]

References

  1. Le Cam, L. (1960). "An Approximation Theorem for the Poisson Binomial Distribution". Pacific Journal of Mathematics. 10 (4): 1181–1197. doi: 10.2140/pjm.1960.10.1181 . MR   0142174. Zbl   0118.33601 . Retrieved 2009-05-13.
  2. Le Cam, L. (1963). "On the Distribution of Sums of Independent Random Variables". In Jerzy Neyman; Lucien le Cam (eds.). Bernoulli, Bayes, Laplace: Proceedings of an International Research Seminar. New York: Springer-Verlag. pp. 179–202. MR   0199871.
  3. Steele, J. M. (1994). "Le Cam's Inequality and Poisson Approximations". The American Mathematical Monthly. 101 (1): 48–54. doi:10.2307/2325124. JSTOR   2325124.
  4. 1 2 den Hollander, Frank. Probability Theory: the Coupling Method.