Le Cam's theorem

Le Cam's theorem In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following.[1][2][3] Suppose: {displaystyle X_{1},X_{2},X_{3},ldots } are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed. {displaystyle Pr(X_{i}=1)=p_{i},{text{ for }}i=1,2,3,ldots .} {displaystyle lambda _{n}=p_{1}+cdots +p_{n}.} {displaystyle S_{n}=X_{1}+cdots +X_{n}.} (i.e. {displaystyle S_{n}} follows a Poisson binomial distribution) Then {displaystyle sum _{k=0}^{infty }left|Pr(S_{n}=k)-{lambda _{n}^{k}e^{-lambda _{n}} over k!}right|<2left(sum _{i=1}^{n}p_{i}^{2}right).} 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 {displaystyle lambda _{n}} is large a better bound is possible: {displaystyle sum _{k=0}^{infty }left|Pr(S_{n}=k)-{lambda _{n}^{k}e^{-lambda _{n}} over k!}right|<2left(1wedge {frac {1}{lambda }}_{n}right)left(sum _{i=1}^{n}p_{i}^{2}right).} [4] It is also possible to weaken the independence requirement.[4] References ^ Le Cam, L. (1960). "An Approximation Theorem for the Poisson Binomial Distribution". Pacific Journal of Mathematics. 10 (4): 1181–1197. doi:10.2140/pjm.1960.10.1181. MR 0142174. Zbl 0118.33601. Retrieved 2009-05-13. ^ Le Cam, L. (1963). "On the Distribution of Sums of Independent Random Variables". In Jerzy Neyman; Lucien le Cam (eds.). Bernoulli, Bayes, Laplace: Proceedings of an International Research Seminar. New York: Springer-Verlag. pp. 179–202. MR 0199871. ^ Steele, J. M. (1994). "Le Cam's Inequality and Poisson Approximations". The American Mathematical Monthly. 101 (1): 48–54. doi:10.2307/2325124. JSTOR 2325124. ^ Jump up to: a b den Hollander, Frank. Probability Theory: the Coupling Method. External links Weisstein, Eric W. "Le Cam's Inequality". MathWorld. Categories: Probability theoremsProbabilistic inequalitiesStatistical inequalitiesTheorems in statistics