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

Abstract

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

To appear in Studia Mathematica