Rahul Sharma (Editor)

Le Cam's theorem

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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.

    References

    Le Cam's theorem Wikipedia
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