Functional Limit Theorems for Local Functionals of Dynamic Point Processes

We establish functional limit theorems for local, additive, interaction functions of temporally evolving point processes. The dynamics are those of a spatial Poisson process on the flat torus with points subject to a birth-death mechanism, and which move according to Brownian motion while alive. The...

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Bibliographic Details
Main Authors Onaran, Efe, Bobrowski, Omer, Adler, Robert J
Format Journal Article
LanguageEnglish
Published 26.10.2023
Subjects
Mathematics - Probability
Online AccessGet full text
DOI10.48550/arxiv.2310.17775

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Summary:We establish functional limit theorems for local, additive, interaction functions of temporally evolving point processes. The dynamics are those of a spatial Poisson process on the flat torus with points subject to a birth-death mechanism, and which move according to Brownian motion while alive. The results reveal the existence of a phase diagram describing at least three distinct structures for the limiting processes, depending on the extent of the local interactions and the speed of the Brownian motions. The proofs, which identify three different limits, rely heavily on Malliavin-Stein bounds on a representation of the dynamic point process via a distributionally equivalent marked point process.
DOI:10.48550/arxiv.2310.17775