Le Cam's theorem

In probability theory, Le Cam's theorem, named after Lucien le Cam (1924 – 2000), states the following.^[1][2][3]

Suppose:

• X₁, ..., 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, ...
• $\lambda _{n}=p_{1}+\cdots +p_{n}.\,$
• $S_{n}=X_{1}+\cdots +X_{n}.\,$ (i.e. $S_{n}$ follows a Poisson binomial distribution)

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 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 $\lambda _{n}$ is large a better bound is possible: ${\displaystyle \sum _{k=0}^{\infty }\left|\Pr(S_{n}=k)-{\lambda _{n}^{k}e^{-\lambda _{n}} \over k!}\right|<2(1\wedge {\frac {1}{\lambda }}_{n})\sum _{i=1}^{n}p_{i}^{2}.}$ ^[4]

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. 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. den Hollander, Frank. Probability Theory: the Coupling Method.

External links

• Weisstein, Eric W. "Le Cam's Inequality". MathWorld.