Asymptotic Methods in Statistics of Random Point Processes

Abstract

First we put together basic definitions and fundamental facts and results from the theory of (un)marked point processes defined on Euclidean spaces ${\mathbb{R}}^{d}$. We introduce the notion random marked point process together with the concept of Palm distributions in a rigorous way followed by the definitions of factorial moment and cumulant measures and characteristics related with them. In the second part we define a variety of estimators of second-order characteristics and other so-called summary statistics of stationary point processes based on observations on a “convex averaging sequence” of windows $\{W_{n},\,n \in \mathbb{N}\}$. Although all these (mostly edge-corrected) estimators make sense for fixed bounded windows our main issue is to study their behaviour when W _n grows unboundedly as n → ∞. The first problem of large-domain statistics is to find conditions ensuring strong or at least mean-square consistency as n → ∞ under ergodicity or other mild mixing conditions put on the underlying point process. The third part contains weak convergence results obtained by exhausting strong mixing conditions or even m-dependence of spatial random fields generated by Poisson-based point processes. To illustrate the usefulness of asymptotic methods we give two Kolmogorov–Smirnov-type tests based on K-functions to check complete spatial randomness of a given point pattern in ${\mathbb{R}}^{d}$.