Le Cam's theorem - Wikipedia
In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following.[1][2][3] Suppose: Then 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 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 ) . ,[4] where ∧ represents the min operator. It is also possible to weaken the independence requirement.[4]
Le Cam's theorem - Wikipedia Jump to content From Wikipedia, the free encyclopedia Probability theorem In probability theory , Le Cam's theorem , named after Lucien Le Cam , states the following. [ 1 ] [ 2 ] [ 3 ] 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}+\cd
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