Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications
Volume 254 / 2020
Abstract
Let $\Phi ’_{\beta }$ denote the strong dual of a nuclear space $\Phi $ and let $D_{T}(\Phi ’_{\beta })$ be the Skorokhod space of right-continuous with left limits (càdlàg) functions from $[0,T]$ into $\Phi ’_{\beta }$. 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 $D_{T}(\Phi ’_{\beta })$. Further, we establish analogues of Lévy’s continuity theorem to provide necessary and sufficient conditions for tightness of a family 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 càdlàg functions from $[0,\infty )$ into $\Phi ’_{\beta }$ are also given. Next, we apply our results to the study of weak convergence of $\Phi ’_{\beta }$-valued càdlàg processes and in particular to Lévy processes. Finally, we apply 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.
