Le Cam's theorem

In probability theory, Le Cam's theorem, named after Lucien le Cam (1924 – 2000), states the following.

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.

External links

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