Le Cam's theorem

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

Suppose:

• $X_1,X_2,X_3,\dots$ are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.
• $\mathrm{Pr}(X_i=1)=p_i,\text{ for }i=1,2,3,\dots .$
• ${\lambda }_n=p_1+\cdots +p_n.$
• $S_n=X_1+\cdots +X_n.$ (i.e. $S_n$ follows a Poisson binomial distribution)

Then

$\sum _{k=0}^{\mathrm{\infty }}|\mathrm{Pr}(S_n=k)-\frac{{\lambda }_n^k{e}^{-{\lambda }_n}}{k!}|<2\left(\sum _{i=1}^n{p}_i^2\right).$

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 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: $\sum _{k=0}^{\mathrm{\infty }}|\mathrm{Pr}(S_n=k)-\frac{{\lambda }_n^k{e}^{-{\lambda }_n}}{k!}|<2(1\wedge {\frac{1}{\lambda }}_n)\left(\sum _{i=1}^n{p}_i^2\right)$, where $\wedge$ represents the $\min$ operator.

It is also possible to weaken the independence requirement.

External links

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