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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Convergence of measurable random functions


Author: L. Š. Grinblat
Journal: Proc. Amer. Math. Soc. 74 (1979), 322-325
MSC: Primary 60B10
MathSciNet review: 524310
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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\vert{\xi _n}(t){\vert^p} \leqslant C$ for all n and t. If $ E\vert{\xi _n}(t){\vert^p} \to E\vert\xi (t){\vert^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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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0524310-1
PII: S 0002-9939(1979)0524310-1
Keywords: Measurable random functions, convergence of joint distributions, convergence of measures
Article copyright: © Copyright 1979 American Mathematical Society