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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Convergence of measurable random functions
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by L. Š. Grinblat PDF
Proc. Amer. Math. Soc. 74 (1979), 322-325 Request permission

Abstract:

Using the theorem of Fréchet and Kolmogorov about compact subsets of the space ${L_p}[0,1]$ and Prohorov’s theorem about the convergence of measures defined on a complete metric space we proved in [2] the following theorem: Let ${\xi _1}(t), \ldots ,{\xi _n}(t), \ldots$ and $\xi (t)$ be measurable random processes $(0 \leqslant t \leqslant 1)$ and suppose that there exist numbers C and $p \geqslant 1$ such that $E|{\xi _n}(t){|^p} \leqslant C$ for all n and t. If $E|{\xi _n}(t){|^p} \to E|\xi (t){|^p}$ for all t and if for any finite set $\{ {t_1}, \ldots ,{t_k}\} \subset [0,1]$ the joint distribution of ${\xi _n}({t_1}), \ldots ,{\xi _n}({t_k})$ converges to the joint distribution of $\xi ({t_1}), \ldots ,\xi ({t_k})$, then the distribution of $f({\xi _n})$ converges to the distribution of $f(\xi )$ for any continuous functional f on ${L_p}[0,1]$. In this paper this theorem is generalized to random measurable functions. The results of the present paper are related to the results of [1], [3], [4].
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 74 (1979), 322-325
  • MSC: Primary 60B10
  • DOI: https://doi.org/10.1090/S0002-9939-1979-0524310-1
  • MathSciNet review: 524310