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].References
- L. Š. Grinblat, Compactifications of spaces of functions and integration of functionals, Trans. Amer. Math. Soc. 217 (1976), 195–223. MR 407227, DOI 10.1090/S0002-9947-1976-0407227-4
- L. Š. Grinblat, A limit theorem for measurable random processes and its applications, Proc. Amer. Math. Soc. 61 (1976), no. 2, 371–376 (1977). MR 423450, DOI 10.1090/S0002-9939-1976-0423450-2
- L. Š. Grinblat, 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), no. 2, 361–379. MR 494376, DOI 10.1090/S0002-9947-1977-0494376-9
- L. Š. Grinblat, Convergence of random processes and of probability measures, Sankhyā Ser. A 40 (1978), no. 2, 144–155. MR 546405
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