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Central Limit Theorems for Local Functionals of Dynamic Point Processes

Efe Onaran, Omer Bobrowski, Robert J. Adler

TL;DR

This work derives finite-dimensional central limit theorems for local interaction functionals of a dynamic point process on the torus, where points undergo birth-death events and Brownian motion with variance $\sigma^2$. The authors employ a marked Poisson process representation and Malliavin-Stein theory for $U$-statistics to obtain regime-dependent Gaussian limits: a slow regime yielding a weighted Ornstein-Uhlenbeck (OU) mixture, a moderate regime producing a Gaussian process with covariances involving functions $\zeta_j(\beta)$, and a fast regime where the time-integrated functional converges to Brownian motion. The analysis hinges on detailed covariance decompositions, Mecke-type counting lemmas for intersection patterns, and chaos expansions to control higher-order fluctuations. These results illuminate how birth-death dynamics and particle motion collectively shape motif-like local functionals in dynamic geometric graphs, with potential implications for mobile networks and dynamic percolation models. The paper also suggests a broader phase diagram with potential transitional regimes between the main three, inviting further study.

Abstract

We establish finite-dimensional central 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 type CLTs for $U$-statistics on a representation of the dynamic point process via a distributionally equivalent marked point process.

Central Limit Theorems for Local Functionals of Dynamic Point Processes

TL;DR

This work derives finite-dimensional central limit theorems for local interaction functionals of a dynamic point process on the torus, where points undergo birth-death events and Brownian motion with variance . The authors employ a marked Poisson process representation and Malliavin-Stein theory for -statistics to obtain regime-dependent Gaussian limits: a slow regime yielding a weighted Ornstein-Uhlenbeck (OU) mixture, a moderate regime producing a Gaussian process with covariances involving functions , and a fast regime where the time-integrated functional converges to Brownian motion. The analysis hinges on detailed covariance decompositions, Mecke-type counting lemmas for intersection patterns, and chaos expansions to control higher-order fluctuations. These results illuminate how birth-death dynamics and particle motion collectively shape motif-like local functionals in dynamic geometric graphs, with potential implications for mobile networks and dynamic percolation models. The paper also suggests a broader phase diagram with potential transitional regimes between the main three, inviting further study.

Abstract

We establish finite-dimensional central 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 type CLTs for -statistics on a representation of the dynamic point process via a distributionally equivalent marked point process.
Paper Structure (15 sections, 19 theorems, 182 equations)

This paper contains 15 sections, 19 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Preliminaries
  4. Marked process model
  5. Counting lemma
  6. Proofs
  7. Covariance characterization
  8. Regime independent expressions for the moments
  9. Covariance for the slow regime
  10. Covariance for the moderate and fast regimes
  11. Covariance of the integrated process in the fast regime
  12. Finite dimensional distributions
  13. Wiener Chaos and U-statistics
  14. CLT for U-statistics
  15. Proofs

Key Result

Theorem 2.2

If $\sigma /r\to 0$, then for some $\mathbf{c}$ with $\|\mathbf{c}\|=1$. Furthermore, if $nr^d \to 0$, then $c_k=1$. If $nr^d\to\infty$, then $c_1=1$. If $nr^d\to\gamma\in (0,\infty)$, then $c_j>0$ for all $1\leq j\leq k$.

Theorems & Definitions (43)

  • Example 2.1: Subgraph counts of geometric graphs
  • Theorem 2.2: Slow regime
  • Theorem 2.3: Moderate regime
  • Theorem 2.4: Fast regime
  • Remark 2.5
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • ...and 33 more