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Teor. Veroyatnost. i Primenen., 1961, Volume 6, Issue 1, Pages 87–93 (Mi tvp4750)  

This article is cited in 1 scientific paper (total in 1 paper)

Short Communications

Some Results Associated with Bochner's Theorem

Yu. V. Prokhorov, V. V. Sazonov

Moscow

Abstract: Let $\{ P_\alpha\}$ be a family of probability distributions in a separable Hilbert space (or more generally, in a space $X=Y^*$ conjugate to a countably-Hilbert space $Y$) and let $\{\chi_\alpha\}$ be the family of corresponding characteristic functionals. We investigate whether or not there exists a locally convex topology $\mathscr{T}$ with the following property:
The relative compactness of $\{{P_\alpha}\}$ is equivalent to uniform (with respect to $\alpha$) continuity of $\{\chi_\alpha\}$.
We prove that there is no such topology except for the case of the countably-Hilbert nuclear space $Y$.

Full text: PDF file (777 kB)

English version:
Theory of Probability and its Applications, 1961, 6:1, 82–87

Received: 25.10.1960

Citation: Yu. V. Prokhorov, V. V. Sazonov, “Some Results Associated with Bochner's Theorem”, Teor. Veroyatnost. i Primenen., 6:1 (1961), 87–93; Theory Probab. Appl., 6:1 (1961), 82–87

Citation in format AMSBIB
\Bibitem{ProSaz61}
\by Yu.~V.~Prokhorov, V.~V.~Sazonov
\paper Some Results Associated with Bochner's Theorem
\jour Teor. Veroyatnost. i Primenen.
\yr 1961
\vol 6
\issue 1
\pages 87--93
\mathnet{http://mi.mathnet.ru/tvp4750}
\transl
\jour Theory Probab. Appl.
\yr 1961
\vol 6
\issue 1
\pages 82--87
\crossref{https://doi.org/10.1137/1106006}


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    This publication is cited in the following articles:
    1. O. G. Smolyanov, S. V. Fomin, “Measures on linear topological spaces”, Russian Math. Surveys, 31:4 (1976), 1–53  mathnet  crossref  mathscinet  zmath
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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