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:
- [math]\displaystyle{ X_1, X_2, X_3, \ldots }[/math] are independent random variables, each with a Bernoulli distribution (i.e., equal to either 0 or 1), not necessarily identically distributed.
- [math]\displaystyle{ \Pr(X_i = 1) = p_i, \text{ for } i = 1, 2, 3, \ldots. }[/math]
- [math]\displaystyle{ \lambda_n = p_1 + \cdots + p_n. }[/math]
- [math]\displaystyle{ S_n = X_1 + \cdots + X_n. }[/math] (i.e. [math]\displaystyle{ S_n }[/math] follows a Poisson binomial distribution)
Then
- [math]\displaystyle{ \sum_{k=0}^\infty \left| \Pr(S_n=k) - {\lambda_n^k e^{-\lambda_n} \over k!} \right| \lt 2 \left( \sum_{i=1}^n p_i^2 \right). }[/math]
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 [math]\displaystyle{ \lambda_n }[/math] is large a better bound is possible: [math]\displaystyle{ \sum_{k=0}^\infty \left| \Pr(S_n=k) - {\lambda_n^k e^{-\lambda_n} \over k!} \right| \lt 2 \left(1 \wedge \frac 1 \lambda_n\right) \left( \sum_{i=1}^n p_i^2 \right). }[/math],[4] where [math]\displaystyle{ \wedge }[/math] represents the [math]\displaystyle{ \min }[/math] operator.
It is also possible to weaken the independence requirement.[4]
References
- ↑ "An Approximation Theorem for the Poisson Binomial Distribution". Pacific Journal of Mathematics 10 (4): 1181–1197. 1960. doi:10.2140/pjm.1960.10.1181. http://projecteuclid.org/euclid.pjm/1103038058. Retrieved 2009-05-13.
- ↑ "On the Distribution of Sums of Independent Random Variables". New York: Springer-Verlag. 1963. pp. 179–202.
- ↑ Steele, J. M. (1994). "Le Cam's Inequality and Poisson Approximations". The American Mathematical Monthly 101 (1): 48–54. doi:10.2307/2325124. https://repository.upenn.edu/oid_papers/271.
- ↑ 4.0 4.1 den Hollander, Frank. Probability Theory: the Coupling Method.
External links
![]() | Original source: https://en.wikipedia.org/wiki/Le Cam's theorem.
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