On the Palm distribution of superposition of point processes

TL;DR

This paper characterizes the Palm distribution of a superposed point process as a mixture of the individual processes' Palm distributions, providing a simple representation useful for analyzing combined point processes.

Contribution

It introduces a novel mixture representation for the Palm distribution of superposed point processes, extending to multiple processes and higher-order distributions.

Findings

Palm distribution of superposition is a mixture of individual distributions.

Extension to multiple superpositions and higher-order distributions.

Simplifies analysis of combined point processes.

Abstract

Palm distributions are critical in the study of point processes. In the present paper we focus on a point process $\Phi$ defined as the superposition, i.e., sum, of two independent point processes, say $\Phi = \Phi_1 + \Phi_2$, and we characterize its Palm distribution. In particular, we show that the Palm distribution of $\Phi$ admits a simple mixture representation depending only on the Palm distribution of $\Phi_j$, as $j=1, 2$, and the associated moment measures. Extensions to the superposition of multiple point processes, and higher order Palm distributions, are treated analogously.