Precompactness of sequences of random variables and random curves revisited

Abstract: This paper studies when a sequence $(\mu_n)_{n \in \mathbb N}$ of probability measures on a metric space $(\mathcal X, d)$ 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. In the case where $\mathcal X$ is a compact geodesic metric space, sequential tightness gives means to characterize precompactness of collections of random curves on $\mathcal X$ 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.