Le Cam's theorem

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

Suppose:

• are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.
• (i.e. follows a Poisson binomial distribution)

Then

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 is large a better bound is possible: , where represents the operator.

It is also possible to weaken the independence requirement.

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