The aim of this paper is twofold: first, to show that Poisson approximation problems for independent summands can in a natural way be treated in a suitable operator semigroup framework, allowing at the same time for an asymptotically precise evaluation of the leading term with respect to the total variation distance; second, to determine asymptotically those Poisson distributions which minimize this distance for given Bernoulli summands. Besides semigroup methods, coupling techniques as well as direct computations are used.
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Deheuvels et al. (1986) studied this question.
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