Misplaced Pages

Le Cam's theorem

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

In probability theory, Le Cam's theorem, named after Lucien Le Cam, states the following.

Suppose:

  • X 1 , X 2 , X 3 , … {\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.
  • Pr ( X i = 1 ) = p i ,  for  i = 1 , 2 , 3 , … . {\displaystyle \Pr(X_{i}=1)=p_{i},{\text{ for }}i=1,2,3,\ldots .}
  • λ n = p 1 + ⋯ + p n . {\displaystyle \lambda _{n}=p_{1}+\cdots +p_{n}.}
  • S n = X 1 + ⋯ + X n . {\displaystyle S_{n}=X_{1}+\cdots +X_{n}.} (i.e. S n {\displaystyle S_{n}} follows a Poisson binomial distribution)

Then

∑ k = 0 ∞ | Pr ( S n = k ) − λ n k e − λ n k ! | < 2 ( ∑ i = 1 n p i 2 ) . {\displaystyle \sum _{k=0}^{\infty }\left|\Pr(S_{n}=k)-{\lambda _{n}^{k}e^{-\lambda _{n}} \over k!}\right|<2\left(\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 λ n {\displaystyle \lambda _{n}} is large a better bound is possible: ∑ k = 0 ∞ | Pr ( S n = k ) − λ n k e − λ n k ! | < 2 ( 1 ∧ 1 λ n ) ( ∑ i = 1 n p i 2 ) {\displaystyle \sum _{k=0}^{\infty }\left|\Pr(S_{n}=k)-{\lambda _{n}^{k}e^{-\lambda _{n}} \over k!}\right|<2\left(1\wedge {\frac {1}{\lambda }}_{n}\right)\left(\sum _{i=1}^{n}p_{i}^{2}\right)} , where ∧ {\displaystyle \wedge } represents the min {\displaystyle \min } operator.

It is also possible to weaken the independence requirement.

References

  1. 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.
  2. 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.
  3. 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.
  4. ^ den Hollander, Frank. Probability Theory: the Coupling Method.

External links

Categories:
Le Cam's theorem Add topic