Precompactness of sequences of random variables and random curves revisited

Abstract

This paper studies when a sequence of probability measures on a metric space admits subsequential weak limits. A sufficient condition called approximate tightness is formulated, which relaxes some assumptions for asymptotic tightness used in the Prokhorov-Le Cam's theorem. The proof is relatively short and only uses elementary tools from probability theory. We also provide examples where approximate tightness is demonstrably easier to check than asymptotic tightness. Approximate tightness gives means to characterize precompact collections of random curves on a locally path-connected metric space $\mathcal{X}$ in terms of annulus crossing probability estimates which may be non-uniform in the modulus of annuli. This vastly generalizes the scope of the precompactness result by Aizenman and Burchard, which assumes $\mathcal{X}$ to be a compact subset of $\mathbb{R}^d$.