In probability theory, Le Cam's theorem, named after Lucien le Cam (1924 – 2000), states the following.
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, ... λ n = p 1 + ⋯ + p n . S n = X 1 + ⋯ + X n . (i.e. S n follows a Poisson binomial distribution)Then
∑ k = 0 ∞ | Pr ( S n = k ) − λ n k e − λ n k ! | < 2 ∑ 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 pi = λn/n, we see that this generalizes the usual Poisson limit theorem.
When λ n is large a better bound is possible: ∑ k = 0 ∞ | Pr ( S n = k ) − λ n k e − λ n k ! | < 2 ( 1 ∧ 1 λ n ) ∑ i = 1 n p i 2 .
It is also possible to weaken the independence requirement.
