Le Cam's theorem explained

In probability theory, Le Cam's theorem, named after Lucien Le Cam, states the following.

Suppose:

X _{1,} X _{2,} X _{3,} \ldots 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 , \ldots. λ _{n} = p _{1} + … + p _{n.} S _{n} = X _{1} + … + X _{n.} (i.e. S _{n} follows a Poisson binomial distribution)

Then

\sum ^{inf ty} _{k =0} \left| \Pr(S _{n=k)} - { λ ^{k} _{n} e ^{-λ _{n}} \over k!} \right| < 2 \left( \sum ^{n} _{i =1} p ^{2} _{i} \right).

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 p_i = λ_n/n, we see that this generalizes the usual Poisson limit theorem.

When

λ _{n} is large a better bound is possible: \sum ^{inf ty} _{k =0} \left| \Pr(S _{n=k)} - { λ ^{k} _{n} e ^{-λ _{n}} \over k!} \right| < 2 \left( 1 \wedge (1)/(λ) _{n\right)} \left( \sum ^{n} _{i =1} p ^{2} _{i} \right) , where \wedge represents the min operator.

It is also possible to weaken the independence requirement.

References

^{[1] [2] [3] [4]}

Notes and References

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