Abstract
The problem of evaluating the accuracy of Poisson approximation to the distribution of a sum of independent integer-valued random variables has attracted a lot of attention in the past six decades. From a practical point of view, it has important applications in insurance, reliability theory, extreme value theory, etc.; from a theoretical point of view, the topic provides insights into Kolmogorov’s problem. The task of establishing an estimate with the best possible constant at the leading term remained open for decades. The paper presents a solution to that problem. A first-order asymptotic expansion is established as well. We generalise and sharpen the corresponding inequalities of Prokhorov, LeCam, Barbour, Hall, Deheuvels, Pfeifer, and Roos. A new result is established for the intensively studied topic of Poisson approximation to the binomial distribution.
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