PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

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.

Authors

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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image