On the Hausdorff measure of noncompactness for the parameterized Prokhorov metric

Abstract

Abstract We quantify the Prokhorov theorem by establishing an explicit formula for the Hausdorff measure of noncompactness (HMNC) for the parameterized Prokhorov metric on the set of Borel probability measures on a Polish space. Furthermore, we quantify the Arzelà-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 Arzelà-Ascoli theorem by providing upper and lower estimates for the HMNC for the parameterized Prokhorov metric on the set of multivariate continuous stochastic processes.

Keywords

• Arzelà-Ascoli theorem
• Chebyshev radius
• Hausdorff measure of noncompactness
• Jung’s theorem
• parametrized Prokhorov metric
• Prokhorov’s theorem