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

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 $Y$.

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

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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• http://mi.mathnet.ru/eng/tvp/v6/i1/p87

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Citing articles on Google Scholar: Russian citations, English citations
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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