Point processes: Distributions, partial orders and compensators.

Abstract

Jacod ([23]) established that the compensator of a simple point process on R+, when taken with respect to the point process' internal history, exists as an essentially unique predictable increasing process which determines the point process' distribution. In the present thesis, we endeavour to infer other distributional properties of a point process from its compensator. Specifically, regarding point processes on R +, we show that the compensator, under appropriate assumptions, (i) determines a sequence of "locally Cox" point processes of discrete support which approximate the original point process' distribution, (ii) determines the stochastic order of two point process distributions with respect to three known partial orders on a certain space of point process realizations, and (iii) determines the association of a point process under any one of the same three partial orders. For the purposes of points (ii) and (iii), we develop a tool called a "representation map", which enables one to infer important distributional properties of random elements of a partially ordered Polish space by "representing" these elements as random sequences of R&d1;infinity+ . Regarding point processes on the quadrant R2+:=&sqbl0;0, infinity&parr0;x&sqbl0;0,infinity&parr0; , we define the compensator as a family of compensators on R+ induced by the planar point process, and show that, under the assumption of strict simplicity and mild regularity conditions, this family exists, is essentially unique, and characterizes the planar point process' distribution---thus generalizing Jacod's result. As a subsidiary result, we develop a regenerative form for the compensator of the non-simple, marked point process on R +, generalizing Jacod's formula ([23]: Proposition 3.1) for the compensator of the simple, marked point process.