The P\'olya sum process: Limit theorems for conditioned random fields

TL;DR

This paper explores the Pólya sum process, a stochastic process similar to the Poisson process, focusing on limit theorems and statistical properties like H-sufficiency, expanding understanding of its probabilistic structure.

Contribution

It extends the theory of the Pólya sum process by analyzing its limit theorems and H-sufficient statistics, providing new insights into its probabilistic and statistical properties.

Findings

Established limit theorems for the Pólya sum process

Characterized H-sufficient statistics for the process

Linked properties with those of the Poisson process

Abstract

In \cite{hZ09}, Zessin constructed the so-called P\'olya sum process via partial integration technique. This process shares some important properties with the Poisson process such as complete randomness and infinite divisibility. This work discusses H-sufficient statistics for the P\'olya sum process as it was done for the Poisson process in \cite{hZ76}.