Stochastic continuity and approximation

Volume 121 / 1996

Leon Brown, Bertram M. Schreiber Studia Mathematica 121 (1996), 15-33 DOI: 10.4064/sm-121-1-15-33


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 $L^1$-spaces to spaces of continuous functions. This characterization is utilized in a discussion of the problem of extension of the parameter space.


  • Leon Brown
  • Bertram M. Schreiber

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