On the Hausdorff measure of noncompactness for the parameterized Prokhorov metric

Ben Berckmoes

doi:10.1186/s13660-016-1151-8

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 Arzela-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 Arzela-Ascoli theorem by providing upper and lower estimates for the HMNC for the parameterized Prokhorov metric on the set of multivariate continuous stochastic processes.

Highlights

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
1 Introduction and statement of the main results For the basic probabilistic concepts and results, we refer the reader to any standard work on probability theory such as [ ]
Fix N ∈ N and let C be the space of continuous maps x of the compact interval [, ] into Euclidean N -space RN equipped with the uniform topology τ∞, that is, the topology derived from the uniform norm x ∞ = sup x(t), t∈[, ]

Summary

We claim that

∀P ∈ : P S B(y, ) ≤ p + , y∈Y proving the desired inequality. To establish the claim, take P ∈ , and let Q be a probability measure in such that ρλ(P, Q) ≤ p + /. For a bounded set A ⊂ RN , its diameter is given by diam(A) = sup |x – y|, x,y∈A and the Chebyshev radius by r(A) = inf sup |x – y|. Provides a relation between the diameter and the Chebyshev radius of a bounded set in RN. Let L be the RN -valued map on the compact interval [α, β] defined by β–t t–α L(t) = β – α c + β – α c. Let L and M be the RN -valued maps on the compact interval [α, β] defined by β–t t–α L(t) = β – α c + β – α c and β–t t–α M(t) = β – α y + β – α y.

Then choose points

Conclusions