This work is concerned with the study of stochastic processes which are continuous in probability, over various parameter spaces, from the point of view of approximation and extension. A stochastic version of the classical theorem of Mergelyan on polynomial approximation is shown to be valid for subsets of the plane whose boundaries are sets of rational approximation. In a similar vein, one can obtain a version in the context of continuity in probability of the theorem of Arakelyan on the uniform approximation of continuous functions on a closed set by entire functions. Locally bounded processes continuous in probability are characterized via operators from -spaces to spaces of continuous functions. This characterization is utilized in a discussion of the problem of extension of the parameter space.
@article{bwmeta1.element.bwnjournal-article-smv121i1p15bwm,
author = {Leon Brown and Bertram Schreiber},
title = {Stochastic continuity and approximation},
journal = {Studia Mathematica},
volume = {119},
year = {1996},
pages = {15-33},
zbl = {0865.60032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv121i1p15bwm}
}
Brown, Leon; Schreiber, Bertram. Stochastic continuity and approximation. Studia Mathematica, Tome 119 (1996) pp. 15-33. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv121i1p15bwm/
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