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

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.