A limit theorem for measurable random processes and its applications

Author:
L. Š. Grinblat

Journal:
Proc. Amer. Math. Soc. **61** (1976), 371-376

MSC:
Primary 60B10

MathSciNet review:
0423450

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let the measurable random processes and be defined on . There exists such that for all and we have . The following assertion is valid: if for any finite set of points the joint distribution of converges to the joint distribution of , and if for all , then for any continuous functional on the distribution of converges to the distribution of . This statement immediately implies the convergence of distributions in some limit theorems for the sums of independent random variables (for example, in one of the theorems of P. Erdös and M. Kac) and in some statistical criteria (for example, in the -criterion of Cramér and von Mises).

**[1]**R. H. Cameron and W. T. Martin,*The Wiener measure of Hilbert neighborhoods in the space of real continuous functions*, J. Math. Phys. Mass. Inst. Tech.**23**(1944), 195–209. MR**0011174****[2]**P. Erdös and M. Kac,*On certain limit theorems of the theory of probability*, Bull. Amer. Math. Soc.**52**(1946), 292–302. MR**0015705**, 10.1090/S0002-9904-1946-08560-2**[3]**D. A. Darling,*The Kolmogorov-Smirnov, Cramér-von Mises tests*, Ann. Math. Statist**28**(1957), 823–838. MR**0093870****[4]**I. I. Gikhman and A. V. Skorokhod,*Introduction to the theory of random processes*, Translated from the Russian by Scripta Technica, Inc, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR**0247660****[5]**Kôsaku Yosida,*Functional analysis*, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR**0180824**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60B10

Retrieve articles in all journals with MSC: 60B10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0423450-2

Article copyright:
© Copyright 1976
American Mathematical Society