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Le Cam’s inequality and Poisson approximations. (English) Zbl 0802.60019
For the sum \(S_ n\) of \(n\) independent, non-identically distributed Bernoulli random variables \(X_ i\) with \(P (X_ i=1)= p_ i\), Le Cam established the remarkable inequality \[ \sum_{k=0}^ \infty | P (S_ n=k)- e^{-\lambda} \lambda^ k /k!| <2 \sum_{i=1}^ n p^ 2_ i, \] where \(\lambda= p_ 1+ p_ 2+ \dots +p_ n\). One purpose is to provide a proof of Le Cam’s inequality using some basic facts from matrix analysis. This proof is simple, but simplicity is not its raison d’√™tre. It also serves as a concrete introduction to the semigroup method for approximation of probability distributions. This method was used by L. Le Cam [Pac. J. Math. 10, 1181-1197 (1960; Zbl 0118.336)], and it has been used again most recently by P. Deheuvels and D. Pfeifer [Ann. Probab. 14, 663-676 (1986; Zbl 0597.60019)] to provide impressively precise results.
The semigroup method is elegant and powerful, but it faces tough competition, especially from the coupling method and the Chen-Stein method. The literature of these methods is reviewed, and it is shown how they also lead to proofs of Le Cam’s inequality.

MSC:
60E15 Inequalities; stochastic orderings
60E05 Probability distributions: general theory
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