Precompactness of sequences of random variables and random curves revisited

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.