On the Hausdorff measure of non-compactness for the parametrized Prokhorov metric

TL;DR

This paper provides explicit formulas and estimates for the Hausdorff measure of non-compactness in various settings, including probability measures, continuous functions, and stochastic processes, thereby quantifying classical theorems.

Contribution

It introduces explicit formulas and bounds for the Hausdorff measure of non-compactness for the parametrized Prokhorov metric and related function spaces, extending classical theorems.

Findings

Explicit formula for HMNC of Prokhorov metric on probability measures.

Upper and lower bounds for HMNC of continuous functions using Jung's Theorem.

Quantitative estimates for the stochastic Arzelà-Ascoli Theorem.

Abstract

We quantify Prokhorov's Theorem by establishing an explicit formula for the Hausdorff measure of non-compactness (HMNC) for the parametrized Prokhorov metric on the set of Borel probability measures on a Polish space. Furthermore, we quantify the Arzel\`a-Ascoli Theorem by obtaining upper and lower estimates for the HMNC for the uniform norm on the space of continuous maps of a compact interval into Euclidean N-space, using Jung's Theorem on the Chebyshev radius. Finally, we combine the obtained results to quantify the stochastic Arzel\`a-Ascoli Theorem by providing upper and lower estimates for the HMNC for the parametrized Prokhorov metric on the set of multivariate continuous stochastic processes.