Le Cam's theorem

In probability theory, Le Cam's theorem, named after Lucien le Cam (1924 &mdash; 2000), is as follows.

Suppose:


 * X1, ..., Xn are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.


 * Pr(Xi = 1) = pi for i = 1, 2, 3, ...


 * $$\lambda_n = p_1 + \cdots + p_n.\,$$


 * $$S_n = X_1 + \cdots + X_n.\,$$

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.

By setting pi = 2λn²/n, we see that this generalizes the usual Poisson limit theorem.