Compensators and Cox convergence

In a previous paper (4), the author showed that the martingale approach to point processes may be used to derive Poisson distributional limit theorems. Here this approach is extended to Cox convergence, general increasing processes are considered, and a regularity condition needed in (4) is removed (that of calculability). In Section 3 we give examples to show the need for the various hypotheses used in the main theorem of Section 2. This theorem is applied to various (dependent) thinnings and compound-ings of point processes in Section 4.