Le Cam's Theorem

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

Suppose:

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

Then

:$\sum_^\infty \left, \Pr(S_n=k) - \ < 2 \left( \sum_^n p_i^2 \right).$

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: $\sum_^\infty \left, \Pr(S_n=k) - \ < 2 \left(1 \wedge \frac 1 \lambda_n\right) \left( \sum_^n p_i^2 \right)$, where $\wedge$ represents the $\min$ operator.

It is also possible to weaken the independence requirement.

References

External links

* {{MathWorld, urlname=LeCamsInequality, title=Le Cam's Inequality

Theorems in probability theory
Probabilistic inequalities
Statistical inequalities
Theorems in statistics