Asymptotic gaussianity of some estimators for reduced factorial moment measures and product densities of stationary poisson cluster processes

Abstract

The main purpose of this paper is to present cer~trai limit tileoreins including functional limit theorems for empirical factorial moment measures and kernel-type product density estimators when the underlying point process is a regular infinitely divisible one.The requied moment conditions are minimal, they are necessary to ensure finite variances of the estimators under consideration. In the special case of a stationary poisson process the obtained results are used to construct a goodness-of-fit test for the function λK(t), 0≤t≤T, denoting the mean number of points within a sphere with radius t around a typical point of the process.