A Compensator Characterization of Planar Point Processes

TL;DR

This paper studies the *-compensator of planar point processes, proving it determines the process law under certain conditions and providing explicit formulas, extending martingale techniques from one-dimensional to two-dimensional models.

Contribution

It establishes that the *-compensator uniquely determines the law of planar point processes satisfying (F4) and derives explicit regenerative formulas for these processes.

Findings

The *-compensator determines the law under (F4) condition.

Explicit regenerative formulas for planar point processes are provided.

Conditional avoidance probabilities are key in characterizing the process law.

Abstract

Martingale techniques play a fundamental role in the analysis of point processes on the real line. In particular, the compensator of a point process uniquely determines and is determined by its distribution, and an explicit formula involving conditional interarrival distributions is well-known. In two dimensions there are many possible definitions of a point process compensator and we focus here on the one that has been the most useful in practice: the so-called *-compensator. Although existence of the *-compensator is well understood, in general it does not determine the law of the point process and it must be calculated on a case-by-case basis. However, it will be proven that when the point process satisfies a certain property of conditional independence (usually denoted by (F4)), the *- compensator determines the law of the point process and an explicit regenerative formula can be given. The basic building block of the planar model is the single line process (a point process with incomparable jump points). Its law can be characterized by a class of avoidance probabilities that are the two-dimensional counterpart of the survival function in one dimension. Conditional avoidance probabilities then play the same role in the construction of the *-compensator as conditional survival probabilities do for compensators in one dimension.