Le Cam's theorem

Le Cam's theorem bounds the total variation distance between a sum of independent Bernoulli random variables and a Poisson random variable with the same mean, by a quantity controlled by the Bernoulli success probabilities.

Poisson-binomial distribution: The distribution of a sum of independent Bernoulli random variables with possibly different success probabilities. It is the exact distribution Le Cam's theorem approximates.
Independent random variables: Random variables whose joint probabilities factor into the product of their marginal probabilities. Independence is a central hypothesis of the theorem.
Poisson limit theorem: A limit theorem in which suitably scaled binomial distributions converge to a Poisson distribution. It gives an asymptotic approximation, while Le Cam's theorem provides a finite-sample error bound.
Lucien Le Cam: A French-American statistician known for foundational contributions to asymptotic statistics and probability. His work established the theorem's quantitative Poisson approximation bound.
Total variation distance: A metric measuring the largest probability difference between two probability distributions over the same space. This is the distance Le Cam's theorem bounds.
Expected value: The probability-weighted average of a random variable's possible values. The Poisson approximation uses the sum's expected value as its parameter.
Poisson point process: A random collection of points whose counts in disjoint regions are independent Poisson variables. Poisson approximations for event counts underpin broader models of random points.
Poisson (1937): Lucien Le Cam's 1960 paper on approximating sums of independent integer-valued random variables by Poisson laws. This paper is the source of the theorem and its original bound.
Poisson distribution: A discrete probability distribution on nonnegative integers, determined by its mean parameter. It supplies the approximating distribution, with mean matching the Bernoulli sum.
Rare event: An event with small probability relative to the setting in which it is considered. Small Bernoulli success probabilities make the Poisson approximation especially informative.

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