A Charlier–Parseval approach to poisson approximation and its applications

Abstract

A new approach to Poisson approximation is proposed. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. A selected survey on diverse Poisson approximation results is also given.MSC 2000 Subject Classifications: Primary 62E17; secondary 60C05 60F05.The early history of Poisson approximation. Poisson distribution appeared naturally as the limit of the sum of a large number of independent trials each with very small probability of success. Such a limit form, being the most primitive version of Poisson approximation, dates back to at least de Moivre's work [32] in the early eighteenth century and Poisson's book [61] in the nineteenth century. Haight [38] writes: ". . . although Poisson (or de Moivre) discovered the mathematical expression (1.1-1) [which is e −λ λ k /k!], Bortkiewicz discovered the probability distribution (1.1-1)." And according to Good [37], "perhaps the Poisson distribution should have been named after von Bortkiewicz (1898) because he was the first to write extensively about rare events whereas Poisson added little to what de Moivre had said on the matter and was probably aware of de Moivre's work;" see also Seneta's account in [74] on Abbe's work. In addition to Bortkiewicz's book [17], another important contribution to the early history of Poisson approximation was made by Charlier [21] for his type B expansion, which will play a crucial role in our development of arguments.The next half a century or so after Bortkiewicz and Charlier then witnessed an increase of interests in the properties and applications of the Poisson distribution and Charlier's expansion. In particular, Jordan [47] proved the orthogonality of the Charlier polynomials with respect to the Poisson measure, and considered a formal expansion pair, expressing the Taylor coefficients of a given function in terms of series of Charlier polynomials and vice versa. A sufficient condition justifying the validity of such an expansion pair was later on provided by Uspensky [83]; he also derived very precise estimates for the coefficients in the case of binomial distribution. His complex-analytic approach was later on extended by Shorgin [80] to the more general Poisson-binomial distribution (each trial with a different probability; see next paragraph). Schmidt [73] then gives a sufficient and necessary condition for justifying the Charlier-Jordan expansion; see also Boas [13] and the references therein. Prohorov [65] was the first to study, using elementary arguments, the total variation distance between binomial and Poisson distributions, thus upgrading the classical limit theorem to an approximation theorem.

Poisson Approximations for Sum of Bernoulli Random Variables and its Application to Ewens Sampling Formula

“…For the Charlier polynomials, see, for example, Barbour et al (1992;p. 175), Takeuchi (1975), and Zacharovas and Hwang (2010). We put α (m, n, λ)…”

Approximation Methods in Probability Theory

“…Research on the accuracy of approximations is in no way restricted to the four metrics considered in this book, see, for example, [28,167].…”

Standard and robust intensity parameter estimation for stationary determinantal point processes

“…We let Π(θ) denote a Poisson random variable with parameter θ. The following proposition is based on results obtained by Zacharovas and Hwang (2010).…”