  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.
