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November, 1983 Tightness of Probabilities On $C(\lbrack 0, 1 \rbrack; \mathscr{Y}')$ and $D(\lbrack 0, 1 \rbrack; \mathscr{Y}')$
Itaru Mitoma
Ann. Probab. 11(4): 989-999 (November, 1983). DOI: 10.1214/aop/1176993447


Let $C_\mathscr{I}' = C(\lbrack 0, 1\rbrack; \mathscr{I}')$ be the space of all continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$, where $\mathscr{L}'$ is the topological dual of the Schwartz space $\mathscr{I}$ of all rapidly decreasing functions. Let $C$ be the Banach space of all continuous functions on $\lbrack 0, 1\rbrack$. For each $\varphi \in \mathscr{I}, \Pi_\varphi$ is defined by $\Pi_\varphi:x \in C_\mathscr{I}' \rightarrow x. (\varphi) \in C$. Given a sequence of probability measures $\{P_n\}$ on $C_\mathscr{I}'$ such that for each $\varphi \in \mathscr{I}, \{P_n\Pi^{-1}_\varphi\}$ is tight in $C$, we prove that $\{P_n\}$ itself is tight in $C_\mathscr{I}'.$ A similar result is proved for the space of all right continuous mappings of $\lbrack 0, 1\rbrack$ to $\mathscr{I}'$.


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Itaru Mitoma. "Tightness of Probabilities On $C(\lbrack 0, 1 \rbrack; \mathscr{Y}')$ and $D(\lbrack 0, 1 \rbrack; \mathscr{Y}')$." Ann. Probab. 11 (4) 989 - 999, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0527.60004
MathSciNet: MR714961
Digital Object Identifier: 10.1214/aop/1176993447

Primary: 60B10
Secondary: 60B11 , 60G20

Keywords: convergence in law , Nuclear Frechet space , tightness

Rights: Copyright © 1983 Institute of Mathematical Statistics


Vol.11 • No. 4 • November, 1983
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