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:

  • Pr(Xi = 1) = pi for i = 1, 2, 3, ...
  • \lambda_n = p_1 + \cdots + p_n.\,

Then

\sum_{k=0}^\infty \left| \Pr(S_n=k) - {\lambda_n^k e^{-\lambda_n} \over k!} \right| < 2 \sum_{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.

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