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Proceedings of the American Mathematical Society

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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].

References [Enhancements On Off] (What's this?)

  • [1] L. Š. Grinblat, Compactifications of spaces of functions and integration of functionals, Trans. Amer. Math. Soc. 217 (1976), 195-223. MR 0407227 (53:11010)
  • [2] -, A limit theorem for measurable random processes and its applications, Proc. Amer. Math. Soc. 61 (1976), 371-376. MR 0423450 (54:11428)
  • [3] -, Convergence of random processes without discontinuities of the second kind and limit theorems for sums of independent random variables, Trans. Amer. Math. Soc. 234 (1977), 361-379. MR 0494376 (58:13250a)
  • [4] -, Convergence of random processes and of probability measures, Sankhyā Ser. A (to appear). MR 546405 (80j:60005)

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Keywords: Measurable random functions, convergence of joint distributions, convergence of measures
Article copyright: © Copyright 1979 American Mathematical Society

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