On Poisson Approximation

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. Among authors who contributed to the topic are Prokhorov, Kolmogorov, LeCam, Shorgin, Barbour, Hall, Deheuvels, Pfeifer, Roos, and many others. From a practical point of view, the problem 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 concerning the accuracy of approximation of the distribution of a sum of independent random variables by infinitely divisible laws. The task of establishing an estimate of the accuracy of Poisson approximation with a correct (the best possible) constant at the leading term remained open for decades. We present a solution to that problem in the case where the accuracy of approximation is evaluated in terms of the point metric. We generalise and sharpen the corresponding inequalities established by preceding authors. A new result is established for the intensively studied topic of compound Poisson approximation to the distribution of a sum of integer-valued r.v.s.