Central limit theorem for a birth-growth model with Poisson arrivals and random growth speed

TL;DR

This paper proves a Gaussian approximation for a birth-growth model with random seed growth speeds, extending previous fixed-speed models and providing bounds on the convergence rate in probabilistic metrics.

Contribution

It introduces a Gaussian limit theorem for a Poisson-based birth-growth model with random speeds, generalizing prior fixed-speed results and offering explicit convergence bounds.

Findings

Gaussian convergence of weighted sums at exposed points

Non-asymptotic bounds in Wasserstein and Kolmogorov metrics

Extension of classical models to random growth speeds

Abstract

We consider Gaussian approximation in a variant of the classical Johnson--Mehl birth-growth model with random growth speed. Seeds appear randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The location, birth time and growth speed of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed, the time distribution and a weight function $h:\mathbb{R}^d \times [0,\infty) \to [0,\infty)$, we prove a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with fixed growth speed. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.