Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Convergence of random processes without discontinuities of the second kind and limit theorems for sums of independent random variables


Author: L. Š. Grinblat
Journal: Trans. Amer. Math. Soc. 234 (1977), 361-379
MSC: Primary 60B10; Secondary 60F05
DOI: https://doi.org/10.1090/S0002-9947-1977-0494376-9
MathSciNet review: 0494376
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\xi _1}(t), \ldots ,{\xi _n}(t), \ldots $ and $ \xi (t)$ be random processes on the interval [0, 1], without discontinuities of the second kind. A. V. Skorohod has given necessary and sufficient conditions under which the distribution of $ f({\xi _n}(t))$ converges to the distribution of $ f(\xi (t))$ as $ n \to \infty $ for any functional f continuous in the Skorohod metric. In the following we shall consider only stochastically right-continuous processes without discontinuities of the second kind, i.e., processes such that the space X of their sample functions is the space of all right-continuous functions $ x(t)(0 \leqslant t \leqslant 1)$ without discontinuities of the second kind. For a set $ T = \{ {t_1}, \ldots {t_n}, \ldots \} \subset [0,1]$ the metric $ {\rho _T}$ is defined on X as in 2.3. The metric $ {\rho _T}$ defines on the X the minimal topology in which all functional continuous in Skorohod's metric and also the functional $ x({t_1} - 0),x({t_1}), \ldots ,x({t_n} - 0),x({t_n}), \ldots $ are continuous. We will give necessary and sufficient conditions under which the distribution of $ f({\xi _n}(t))$ converges to the distribution of $ f(\xi (t))$ as $ n \to \infty $ for any completely continuous functional f, i.e. for any functional f which is continuous in any of the metrics $ {\rho _T}$ defined in 2.3.


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

  • [1] A. V. Skorohod, Random processes with independent increments, ``Nauka", Moscow, 1964; English transl., Theory of random processes, National Lending Library for Science and Technology, Boston Spa, Yorkshire, England, 1971. MR 31 #6280. MR 0182056 (31:6280)
  • [2] I. I. Gihman and A. V. Skorohod, The theory of stochastic processes. I, ``Nauka", Moscow, 1971 ; English transl., Springer-Verlag, New York, 1974. MR 49 #6287; 49 # 11603. MR 0346882 (49:11603)
  • [3] L. S. Grinblat, Compactifications of spaces of functions and integration of functionals, Trans. Amer. Math. Soc. 217 (1976), 195-223. MR 0407227 (53:11010)
  • [4] H. Schubert, Topology, Teubner, Stuttgart, 1964; English transl., Allyn and Bacon, Boston, Mass., 1968. MR 30 #551; 37 #2160. MR 0226571 (37:2160)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60B10, 60F05

Retrieve articles in all journals with MSC: 60B10, 60F05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0494376-9
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society