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| In [[probability theory]], '''Le Cam's theorem''', named after [[Lucien le Cam]] (1924 – 2000), is as follows.
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| Suppose:
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| * ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[statistical independence|independent]] [[random variable]]s, each with a [[Bernoulli distribution]] (i.e., equal to either 0 or 1), not necessarily identically distributed.
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| * Pr(''X''<sub>''i''</sub> = 1) = ''p''<sub>''i''</sub> for ''i'' = 1, 2, 3, ...
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| * <math>\lambda_n = p_1 + \cdots + p_n.\,</math>
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| * <math>S_n = X_1 + \cdots + X_n.\,</math> (i.e. <math>S_n</math> follows a [[Poisson binomial distribution]])
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| Then
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| :<math>\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. </math>
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| In other words, the sum has approximately a [[Poisson distribution]].
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| By setting ''p''<sub>''i''</sub> = λ<sub>''n''</sub>/''n'', we see that this generalizes the usual [[Poisson limit theorem]].
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| ==References==
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| * {{cite journal
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| |last=Le Cam |first=L. |authorlink=Lucien le Cam
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| |title=An Approximation Theorem for the Poisson Binomial Distribution
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| |journal=Pacific Journal of Mathematics
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| |volume=10 |issue=4 |pages=1181–1197 |year=1960
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| |url=http://projecteuclid.org/euclid.pjm/1103038058 |accessdate=2009-05-13
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| |mr=0142174 | zbl = 0118.33601
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| }}
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| * {{cite conference
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| |last=Le Cam |first=L. |authorlink=Lucien le Cam
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| |title=On the Distribution of Sums of Independent Random Variables
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| |booktitle=Bernoulli, Bayes, Laplace: Proceedings of an International Research Seminar
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| |editor1=[[Jerzy Neyman]] |editor2=Lucien le Cam
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| |publisher=Springer-Verlag |location=New York
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| |pages=179–202 |year=1963
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| |mr=0199871
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| }}
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| * {{cite jstor|2325124}}
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| ==External links==
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| * {{MathWorld|urlname=LeCamsInequality|title=Le Cam's Inequality}}
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| [[Category:Probability theorems]]
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| [[Category:Probabilistic inequalities]]
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| [[Category:Statistical inequalities]]
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| [[Category:Statistical theorems]]
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