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Scaling Limit of a Stochastic Clustering Model on $\mathbb{R}$

Partha S. Dey, S. Rasoul Etesami, Aditya S. Gopalan

TL;DR

The paper analyzes an infinite-dimensional stochastic clustering model on $\mathbb{R}$ where each point moves halfway toward a random neighbor and merged points are collapsed, with the space scaled to unit intensity. It establishes a unique scaling limit for Algorithm 1 when the initial point process is renewal and finite-variance, via a gap-sequence representation and a reverse-time martingale approach, and shows a Palm-shift form of convergence with a non-renewal limit. A duality between the forward gap-sequence dynamics and the time-reversed weight process is derived, along with a joint convergence result that connects the limiting geometry to merger statistics. A time-reversal based conjecture for a random distribution function on $\mathbb{R}$ is proposed, supported by simulations, and a parallel discussion for Algorithm 2 outlines the potential initial-data dependence. Overall, the work provides a rigorous scaling limit for a stochastic clustering dynamic on an infinite dataset and highlights directions for extending the theory to other dynamics and stationary distributions.

Abstract

We consider an infinite-dimensional stochastic clustering model on $\mathbb{R}$. In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random. Co-located points are merged into a single point, and the resulting simple point process is rescaled to unit intensity. We show that, when the point processes are shifted so that there is a point at the origin, there is a unique weak limit of these dynamics when the initial point process is renewal. We also conjecture that for the time-reversed process and with an appropriate scaling in space, there is a limiting (random) distribution function on $\mathbb{R}$, whose corresponding measure assigns to $\mathbb{R}$ a measure corresponding to the gap between consecutive points. Finally, we discuss some relevant research directions.

Scaling Limit of a Stochastic Clustering Model on $\mathbb{R}$

TL;DR

The paper analyzes an infinite-dimensional stochastic clustering model on where each point moves halfway toward a random neighbor and merged points are collapsed, with the space scaled to unit intensity. It establishes a unique scaling limit for Algorithm 1 when the initial point process is renewal and finite-variance, via a gap-sequence representation and a reverse-time martingale approach, and shows a Palm-shift form of convergence with a non-renewal limit. A duality between the forward gap-sequence dynamics and the time-reversed weight process is derived, along with a joint convergence result that connects the limiting geometry to merger statistics. A time-reversal based conjecture for a random distribution function on is proposed, supported by simulations, and a parallel discussion for Algorithm 2 outlines the potential initial-data dependence. Overall, the work provides a rigorous scaling limit for a stochastic clustering dynamic on an infinite dataset and highlights directions for extending the theory to other dynamics and stationary distributions.

Abstract

We consider an infinite-dimensional stochastic clustering model on . In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random. Co-located points are merged into a single point, and the resulting simple point process is rescaled to unit intensity. We show that, when the point processes are shifted so that there is a point at the origin, there is a unique weak limit of these dynamics when the initial point process is renewal. We also conjecture that for the time-reversed process and with an appropriate scaling in space, there is a limiting (random) distribution function on , whose corresponding measure assigns to a measure corresponding to the gap between consecutive points. Finally, we discuss some relevant research directions.

Paper Structure

This paper contains 16 sections, 5 theorems, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions and Organization of this Paper
  4. Model and Main Results
  5. Point Process Model
  6. Gap Sequence Model
  7. Time Reversal of Gap Sequence Model
  8. Main Results
  9. Proofs of Main Results
  10. Proof of Theorem \ref{['thm:gap-sequence-convergence']}
  11. Proof of Theorem \ref{['thm:cluster-size-convergence']}
  12. Proof of Corollary \ref{['cor:duality']}
  13. Proof of Corollary \ref{['cor:joint-convergence']}
  14. A Further Conjecture
  15. Simulations of Conjecture \ref{['thm:weight-sum-convergence']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\overrightarrow{\Xi}^{(0)}$ be a renewal process with finite inter-renewal variance. The following holds: where the limit is independent of $\overrightarrow{\Xi}^{(0)}$. If $\Theta$ is the Palm shift, then

Figures (5)

  • Figure 1: Empirical distribution of point gaps after 20 and 25 iterations, starting from gaps distributed as $\mathrm{Exp}(1)$ and $\mathrm{Unif}(0,2)$.
  • Figure 2: A simulation of the point process model, without space re-scaling.
  • Figure 3: Figure of forward and reverse dynamics.
  • Figure 4: Relationship between the weight sequences $\eta^{(t)}$ and the initial gap sequence $\Gamma^{(0)}$. The orange dashed line, for example, is given by $\frac{1}{2}\overleftarrow{\Gamma}^{(0)}_{-1} + \overleftarrow{\Gamma}^{(0)}_0 + \frac{1}{2}\overleftarrow{\Gamma}^{(0)}_1$; the blue dashed line is given by $\frac{1}{2}\overleftarrow{\Gamma}^{(0)}_0 + \overleftarrow{\Gamma}^{(0)}_1 + \frac{1}{2}\overleftarrow{\Gamma}^{(0)}_2$.
  • Figure 5: Empirical distribution of $\overleftarrow{F}^{(t)}$ across time for two different samples.

Theorems & Definitions (7)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Conjecture 1
  • Lemma 4.1
  • proof