On the second-order characteristics of marked point processes

Abstract

New definitions for the second-order characteristics of marked point processes are presented. Non-reduced moment measures are involved in obtaining consistency with ana\-logous definitions in the random field context. A certain σ-algebra replaces the current assumptions on stationarity and isotropy so that the characteristics are still well defined even if such assumptions do not hold. The new definitions and the present ones coincide for positive arguments if the marked point process is simple, stationary and isotropic, and if the second-order product density exists. The different second-order characteristics given in the literature are discussed. A renaming of one of the characteristics is suggested since distinct characteristics have been known under identical names. Furthermore, it is shown that arbitrary measurable functions with compact support can appear as second-order characteristics of marked point processes.