Precompactness of sequences of random variables and random curves revisited

TL;DR

This paper revisits conditions for the precompactness of sequences of probability measures and random curves, introducing a more flexible criterion called sequential tightness that broadens existing theoretical frameworks.

Contribution

It introduces the concept of sequential tightness, relaxing assumptions for asymptotic tightness, and extends characterizations of precompactness for random curves in metric spaces.

Findings

Sequential tightness characterizes precompactness of random measures.

The approach generalizes previous annulus crossing conditions.

Elementary probability tools are sufficient for proofs.

Abstract

This paper studies when a sequence of probability measures on a metric space admit subsequential weak limits. A sufficient condition called sequential tightness is formulated, which relaxes some assumptions for asymptotic tightness used in the Prokhorov -- Le Cam theorem. The proof only uses elementary tools from probability theory. Sequential tightness gives means to characterize the precompact collections of random curves on a compact geodesic metric space in terms of an annulus crossing condition, which generalizes the one by Aizenman and Burchard by allowing estimates for annulus crossing probabilities to be non-uniform over the modulus of annuli.