Precompactness of sequences of random variables and random curves revisited
Osama Abuzaid
TL;DR
The paper introduces sequential tightness as a non-uniform relaxion of asymptotic tightness to characterize precompactness of probability measures on metric spaces and proves that sequential tightness suffices to obtain subsequential weak limits using elementary probabilistic tools. It develops a Skorokhod-like coupling construction with consistent finite-set approximations to extract almost sure convergence, and establishes the equivalence between sequential and asymptotic tightness. For random collections of curves on compact geodesic spaces, the authors define regularity via annulus-crossing counts and prove that regularity is equivalent to precompactness, generalizing prior annulus-crossing criteria. In Euclidean settings, the regularity condition is sharpened to quantitative bounds involving $r^{d-1}$ terms, enabled by cube-based coverings, leading to a stronger, geometry-specific precompactness result. Overall, the work offers an elementary yet powerful framework for establishing scaling limits of lattice interfaces and random curve ensembles beyond previous uniform annulus-crossing assumptions.
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.