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Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications

Volume 254 / 2020

C. A. Fonseca-Mora Studia Mathematica 254 (2020), 109-147 MSC: 60B10, 60B12, 60F17, 60G17. DOI: 10.4064/sm180629-25-11 Published online: 6 March 2020


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.


  • C. A. Fonseca-MoraEscuela de Matemática
    Universidad de Costa Rica
    San José, 11501-2060, Costa Rica

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