Le Cam's Theorem bounds the total variation distance between the distribution of a sum of independent indicator random variables, each with its own small probability of success, and a Poisson distribution with mean equal to the sum of those probabilities, showing the two distributions are close whenever the individual probabilities are small. Named for Lucien Le Cam, it makes precise and quantifies the classical heuristic that a sum of many rare, roughly independent events is well approximated by a Poisson distribution.
Facts
StatementThe sum of independent Bernoulli variables has approximately a Poisson distribution, with the approximation error bounded in total variation distance. 1 Sources
1. Le Cam's theorem (Wikipedia)
Statement
the sum has approximately a Poisson distribution and the above inequality bounds the approximation error
References
Le Cam, L. (1960). 'An Approximation Theorem for the Poisson Binomial Distribution'.
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