# Palm distributions of superposed point processes for statistical inference

## TL;DR

This paper characterizes the Palm distributions of superposed point processes, providing new tools for statistical inference such as likelihood-based methods and applications to shot noise Cox processes.

## Contribution

It introduces a simple mixture representation for Palm distributions of superposed processes and derives explicit formulas for higher-order distributions and Janossy densities.

## Key findings

- Derived explicit Palm distribution formulas for superposed processes
- Provided tractable Janossy density expressions for likelihood inference
- Extended results to multiple superpositions and higher-order distributions

## Abstract

Palm distributions play a central role in the study of point processes and their associated summary statistics. In this paper, we characterize the Palm distributions of the superposition of independent point processes, establishing a simple mixture representation depending on the point processes' Palm distributions and moment measures. We explore two statistical applications enabled by our main result. First, we consider minimum contrast estimation for corrupted point processes. Second, we investigate the class of shot noise Cox processes and derive explicit expressions for their higher-order Palm distributions. In the finite case, we further obtain a tractable expression for the Janossy density, which plays the role of a likelihood function and thus can be used for new likelihood-based inference strategies. Extensions to the superposition of multiple point processes and to higher-order Palm distributions are also presented.