Some Results Associated with Bochner's Theorem

Yu. V. Prokhorov   ; V. V. Sazonov

Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 1, pp. 87-93

Résumé

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 $Y$.