General thinning characterizations of distributions and point processes

TL;DR

This paper explores the inverse operation of thinning in point processes, establishing links to integration-by-parts formulas and extending classical results to general thinnings and finite point processes.

Contribution

It generalizes thinning and condensing operations, linking them to integration-by-parts formulas and extending classical characterizations to broader classes of point processes.

Findings

Extended the link between thinning and integration-by-parts formulas.

Generalized the representation of splitting kernels using reduced Palm kernels.

Provided a discrete stick-breaking characterization of distributions.

Abstract

For general thinning procedures, its inverse operation, the condensing, is studied and a link to integration-by-parts formulas is established. This extends the recent results on that link for independent thinnings of point processes to general thinnings of finite point processes. In particular, the classical integration-by-parts formulas appear as the example of independent thinnings. Moreover, the representation of the splitting kernel of finite point processes in terms of its reduced Palm kernels is extended to general thinnings. This link is studied in the context of discrete random variables and yields analogue characterizations of their distributions. Results on independent thinnings are complemented by a discrete stick breaking characterization of distributions.