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Tightness and weak convergence of probabilities on the Skorokhod space on the dual of a nuclear space and applications. (English) Zbl 1444.60008

Summary: 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.

MSC:

60B10 Convergence of probability measures
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F17 Functional limit theorems; invariance principles
60G17 Sample path properties
60G51 Processes with independent increments; Lévy processes
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