Mass and radius of balls in Gromov-Hausdorff-Prokhorov convergent sequences

TL;DR

This paper discusses properties of sequences of random compact metric spaces converging in the Gromov-Hausdorff-Prokhorov sense, focusing on how the mass and radius of small balls relate to the nature of the limit space.

Contribution

It formalizes properties relating the mass and radius of small balls in convergent sequences, providing justifications where references are lacking.

Findings

Small open balls have small mass if the limit is non-atomic.

Small closed balls have small radius if the limit is fully supported.

Provides formal justifications for known properties without claiming new results.

Abstract

We survey some properties of Gromov--Hausdorff--Prokhorov convergent sequences $(\mathsf{X}_n, d_{\mathsf{X}_n}, \nu_{\mathsf{X}_n})_{n \ge 1}$ of random compact metric spaces equipped with Borel probability measures. We formalize that if the limit is almost surely non-atomic, then for large $n$ each open ball in $\mathsf{X}_n$ with small radius must have small mass. Conversely, if the limit is almost surely fully supported, then each closed ball in $\mathsf{X}_n$ with small mass must have small radius. We do not claim any new results, but justifications are provided for properties for which we could not find explicit references.