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