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Transactions of the American Mathematical Society
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
MathSciNet review: 0494376
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0494376-9
PII: S 0002-9947(1977)0494376-9
Article copyright: © Copyright 1977 American Mathematical Society