Convergence of measurable random functions

Author:
L. Š. Grinblat

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

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Abstract: Using the theorem of Fréchet and Kolmogorov about compact subsets of the space and Prohorov's theorem about the convergence of measures defined on a complete metric space we proved in [**2**] the following theorem: Let and be measurable random processes and suppose that there exist numbers *C* and such that for all *n* and *t*. If for all *t* and if for any finite set the joint distribution of converges to the joint distribution of , then the distribution of converges to the distribution of for any continuous functional *f* on . 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**].

**[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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DOI:
https://doi.org/10.1090/S0002-9939-1979-0524310-1

Keywords:
Measurable random functions,
convergence of joint distributions,
convergence of measures

Article copyright:
© Copyright 1979
American Mathematical Society