English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let $\Phi'_{\beta}$ denotes the strong dual of a nuclear space $\Phi$ and let$D_{T}(\Phi'_{\beta})$ be the Skorokhod space of right-continuous with leftlimits (c\`{a}dl\`{a}g) functions from $[0,T]$ into $\Phi'_{\beta}$. In thisarticle we introduce the concepts of cylindrical random variables andcylindrical measures on $D_{T}(\Phi'_{\beta})$, and prove analogues of theregularization theorem and Minlos theorem for extensions of these objects tobona fide random variables and probability measures on $D_{T}(\Phi'_{\beta})$respectively. Later, we establish analogues of L\'{e}vy's continuity theorem toprovide necessary and sufficient conditions for uniform tightness of familiesof probability measures on $D_{T}(\Phi'_{\beta})$ and sufficient conditions forweak 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 c\`{a}dl\`{a}g functions from $[0,\infty)$ into$\Phi'_{\beta}$ are also given. Afterwards, we apply our results to study weakconvergence of $\Phi'_{\beta}$-valued c\`{a}dl\`{a}g processes and inparticular to L\'{e}vy processes. We finalize with an application of our theoryto the study of tightness and weak convergence of probability measures on theSkorokhod space $D_{\infty}(H)$ where $H$ is a Hilbert space.

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