Asymptotic Methods in Statistics of Random Point Processes
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 Pal
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Asymptotic Methods in Statistics of Random Point Processes Lothar Heinrich
Abstract First we put together basic definitions and fundamental facts and results from the theory of (un)marked point processes defined on Euclidean spaces Rd . 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 fWn ; n 2 Ng. Although all these (mostly edgecorrected) estimators make sense for fixed bounded windows our main issue is to study their behaviour when Wn grows unboundedly as n ! 1. The first problem of large-domain statistics is to find conditions ensuring strong or at least mean-square consistency as n ! 1 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 Rd .
4.1 Marked Point Processes: An Introduction First we present a rigorous definition of the marked point process on Euclidean spaces with marks in some Polish space and formulate an existence theorem for marked point processes based on their finite-dimensional distributions. Further, all essential notions and tools of point process theory such as factorial moment
L. Heinrich () Augsburg University, Augsburg, Germany e-mail: [email protected] E. Spodarev (ed.), Stochastic Geometry, Spatial Statistics and Random Fields, Lecture Notes in Mathematics 2068, DOI 10.1007/978-3-642-33305-7 4, © Springer-Verlag Berlin Heidelberg 2013
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and cumulant measures with their densities and reduced versions as well as the machinery of Palm distributions in the marked and unmarked case are considered in detail.
4.1.1 Marked Point Processes: Definitions and Basic Facts Point processes are mathematical models for irregular point patterns formed by randomly scattered points in some locally compact Hausdorff space. Throughout this chapter, this space will be the Euclidean space Rd of dimension d 2 N. In many applications to each point Xi of the pattern a further random element Mi , called mark, can be assigned which carries additional information and may take values in a rather general mark space M equipped with an appropriate -algebra M. For example, for d D 1, the Xi ’s could be arrival times of customers and the Mi ’s their sojourn times in a queueing system and, for d D 2, one can interpret the Xi ’s as locations of trees in a forest with the associated random vectors Mi
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