The Polya branching process and limit theorems for conditioned random fields

TL;DR

This paper generalizes Polya type point processes through branching mechanisms, establishing conditions for integration by parts, computing Palm kernels, and characterizing processes with similar local features as mixtures of Polya branchings.

Contribution

It introduces a new class of generalized Polya point processes via branching, providing their characterization and properties, including Palm kernels and integration by parts formulas.

Findings

Derived conditions for integration by parts in generalized processes

Computed Palm kernels as superpositions of point processes

Characterized processes with similar local characteristics as mixtures of Polya branchings

Abstract

The first aim is to construct generalizations of Polya type point process by applying a branching mechanism to these point processes. Conditions are given under which these point processes satisfy an integration by parts formula. Furthermore we compute their Palm kernels, which turn out to be superpositions of different point processes. Secondly we identify all point processes whose local characteristics agree with these of a fixed branching of a Polya type point process as mixtures of branchings of Polya type point process and show that in this case also they are characterized by an integration by parts formula.