Convergence of random processes without discontinuities of the second kind and limit theorems for sums of independent random variables
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- by L. Š. Grinblat PDF
- Trans. Amer. Math. Soc. 234 (1977), 361-379 Request permission
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
- A. V. Skorohod, Sluchaĭ nye protsessy s nezavisimymi prirashcheniyami, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0182056
- Ĭ. Ī. Gīhman and A. V. Skorohod, The theory of stochastic processes. I, Die Grundlehren der mathematischen Wissenschaften, Band 210, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by S. Kotz. MR 0346882
- L. Š. Grinblat, Compactifications of spaces of functions and integration of functionals, Trans. Amer. Math. Soc. 217 (1976), 195–223. MR 407227, DOI 10.1090/S0002-9947-1976-0407227-4
- Horst Schubert, Topology, Allyn and Bacon, Inc., Boston, Mass., 1968. Translated from the German by Siegfried Moran. MR 0226571
Additional Information
- © Copyright 1977 American Mathematical Society
- 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