In probability theory, Le Cam's theorem, named after Lucien Le Cam, states the following. [1] [2] [3]
Suppose:
are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.

(i.e.
follows a Poisson binomial distribution)
Then

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
is large a better bound is possible:
, [4] where
represents the
operator.
It is also possible to weaken the independence requirement. [4]
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