le Cam's theorem

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

Suppose:

$X_1, X_2, X_3, \ldots$ are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.
$\Pr(X_i = 1) = p_i, \text{ for } i = 1, 2, 3, \ldots.$
$\lambda_n = p_1 + \cdots + p_n.$
$S_n = X_1 + \cdots + X_n.$ (i.e. $S_n$ follows a Poisson binomial distribution)

Then

: $\sum_{k=0}^\infty \left| \Pr(S_n=k) - {\lambda_n^k e^{-\lambda_n} \over k!} \right| < 2 \left( \sum_{i=1}^n p_i^2 \right).$

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 p_i = λ_n/n, we see that this generalizes the usual Poisson limit theorem.

When $\lambda_n$ is large a better bound is possible: $\sum_{k=0}^\infty \left| \Pr(S_n=k) - {\lambda_n^k e^{-\lambda_n} \over k!} \right| < 2 \left(1 \wedge \frac 1 \lambda_n\right) \left( \sum_{i=1}^n p_i^2 \right)$ , where $\wedge$ represents the $\min$ operator.

It is also possible to weaken the independence requirement.

References

{{cite book | last1 = den Hollander | first1 = Frank | title = Probability Theory: the Coupling Method}}

{{cite journal

|last=Le Cam |first=L. |author-link=Lucien le Cam

|title=An Approximation Theorem for the Poisson Binomial Distribution

|journal=Pacific Journal of Mathematics

|volume=10 |issue=4 |pages=1181–1197 |year=1960

|url=http://projecteuclid.org/euclid.pjm/1103038058 |access-date=2009-05-13

|mr=0142174 | zbl = 0118.33601 |doi=10.2140/pjm.1960.10.1181

|doi-access=free }}

{{cite conference

|last=Le Cam |first=L. |author-link=Lucien le Cam

|title=On the Distribution of Sums of Independent Random Variables

|book-title=Bernoulli, Bayes, Laplace: Proceedings of an International Research Seminar

|editor1=Jerzy Neyman

|editor-link=Jerzy Neyman

|editor2=Lucien le Cam

|publisher=Springer-Verlag |location=New York

|pages=179–202 |year=1963

|mr=0199871

}}

{{Cite journal | last1 = Steele | first1 = J. M.| title = Le Cam's Inequality and Poisson Approximations | jstor = 2325124 | journal = The American Mathematical Monthly | volume = 101 | issue = 1 | pages = 48–54 | year = 1994 | doi = 10.2307/2325124| url = https://repository.upenn.edu/oid_papers/271}}

External links

{{MathWorld|urlname=LeCamsInequality|title=Le Cam's Inequality}}

Category:Theorems in probability theory

Category:Probabilistic inequalities

Category:Statistical inequalities

Category:Theorems in statistics