# Le Cam's theorem

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

Suppose:

• ${\displaystyle X_{1},X_{2},X_{3},\ldots }$ are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.
• ${\displaystyle \Pr(X_{i}=1)=p_{i},{\text{ for }}i=1,2,3,\ldots .}$
• ${\displaystyle \lambda _{n}=p_{1}+\cdots +p_{n}.}$
• ${\displaystyle S_{n}=X_{1}+\cdots +X_{n}.}$ (i.e. ${\displaystyle S_{n}}$ follows a Poisson binomial distribution)

Then

${\displaystyle \sum _{k=0}^{\infty }\left|\Pr(S_{n}=k)-{\lambda _{n}^{k}e^{-\lambda _{n}} \over k!}\right|<2\left(\sum _{i=1}^{n}p_{i}^{2}\right).}$

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

When ${\displaystyle \lambda _{n}}$ is large a better bound is possible: ${\displaystyle \sum _{k=0}^{\infty }\left|\Pr(S_{n}=k)-{\lambda _{n}^{k}e^{-\lambda _{n}} \over k!}\right|<2\left(1\wedge {\frac {1}{\lambda }}_{n}\right)\left(\sum _{i=1}^{n}p_{i}^{2}\right)}$,[4] where ${\displaystyle \wedge }$ represents the ${\displaystyle \min }$ 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. ^ a b den Hollander, Frank. Probability Theory: the Coupling Method.