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

MathSciNet review:
0494376

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let and 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 converges to the distribution of as 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 without discontinuities of the second kind. For a set the metric is defined on *X* as in 2.3. The metric defines on the *X* the minimal topology in which all functional continuous in Skorohod's metric and also the functional are continuous. We will give necessary and sufficient conditions under which the distribution of converges to the distribution of as for any completely continuous functional *f*, i.e. for any functional *f* which is continuous in any of the metrics defined in 2.3.

**[1]**A. V. Skorohod,*Sluchainye protsessy s nezavisimymi prirashcheniyami*, Izdat. “Nauka”, Moscow, 1964 (Russian). MR**0182056****[2]**Ĭ. Ī. Gīhman and A. V. Skorohod,*The theory of stochastic processes. I*, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by S. Kotz; Die Grundlehren der mathematischen Wissenschaften, Band 210. MR**0346882****[3]**L. Š. Grinblat,*Compactifications of spaces of functions and integration of functionals*, Trans. Amer. Math. Soc.**217**(1976), 195–223. MR**0407227**, 10.1090/S0002-9947-1976-0407227-4**[4]**Horst Schubert,*Topology*, Translated from the German by Siegfried Moran, Allyn and Bacon, Inc., Boston, Mass., 1968. MR**0226571**

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