The domain of partial attraction of a Poisson law

Abstract

Groshev gave a characterization of the union of domains of partial attraction of all Poisson laws in 1941. His classical condition is expressed by the underlying distribution function and disguises the role of the mean λ of the attracting distribution. In the present paper we start out from results of the recent “probabilistic approach” and derive characterizations for any fixed λ>0 in terms of the underlying quantile function. The approach identifies the portion of the sample that contributes the limiting Poisson behavior of the sum, delineates the effect of extreme values, and leads to necessary and sufficient conditions all involving λ. It turns out that the limiting Poisson distributions arise in two qualitatively different ways depending upon whether λ>1 or λ<1. A concrete construction, illustrating all the results, also shows that in the boundary case when λ=1 both possibilities may occur.