Tightness and Weak Convergence of Probabilities on the Skorokhod Space on the Dual of a Nuclear Space and Applications

TL;DR

This paper develops a comprehensive framework for analyzing tightness and weak convergence of probability measures on Skorokhod spaces over the dual of nuclear spaces, with applications to Lévy processes and Hilbert space-valued processes.

Contribution

It introduces cylindrical random variables and measures on Skorokhod spaces over nuclear duals, extending classical theorems and providing criteria for tightness and weak convergence.

Findings

Established regularization and Minlos theorems for cylindrical objects.

Provided Lévy's continuity theorem analogues for tightness and convergence.

Applied results to Lévy processes and Hilbert space-valued processes.

Abstract

Let $\Phi'_{\beta}$ denotes the strong dual of a nuclear space $\Phi$ and let $D_{T}(\Phi'_{\beta})$ be the Skorokhod space of right-continuous with left limits (c\`{a}dl\`{a}g) functions from $[0,T]$ into $\Phi'_{\beta}$. In this article we introduce the concepts of cylindrical random variables and cylindrical measures on $D_{T}(\Phi'_{\beta})$, and prove analogues of the regularization theorem and Minlos theorem for extensions of these objects to bona fide random variables and probability measures on $D_{T}(\Phi'_{\beta})$ respectively. Later, we establish analogues of L\'{e}vy's continuity theorem to provide necessary and sufficient conditions for uniform tightness of families of probability measures on $D_{T}(\Phi'_{\beta})$ and sufficient conditions for weak convergence of a sequence of probability measures on $D_{T}(\Phi'_{\beta})$. Extensions of the above results to the space $D_{\infty}(\Phi'_{\beta})$ of c\`{a}dl\`{a}g functions from $[0,\infty)$ into $\Phi'_{\beta}$ are also given. Afterwards, we apply our results to study weak convergence of $\Phi'_{\beta}$-valued c\`{a}dl\`{a}g processes and in particular to L\'{e}vy processes. We finalize with an application of our theory to the study of tightness and weak convergence of probability measures on the Skorokhod space $D_{\infty}(H)$ where $H$ is a Hilbert space.