Chapter 9. Quasi-Banach Spaces of Random Variables and Modeling of Stochastic Processes

$39.50

Oleksandr Mokliachuk
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

Part of the book: Stochastic Processes: Fundamentals and Emerging Applications

Abstract

The results of the study of the problem of modeling of stochastic processes in various spaces with given reliability and accuracy are proposed. In the first part of the chapter, results on modeling of stochastic processes in DV,W spaces of random variables are presented. These quasi-Banach spaces with a probability norm are convenient for estimating the reliability and accuracy of modeling of stochastic processes that do not have moments; for example, processes with L`evy distribution or Cauchy distribution. Note that the classical methods use the values of the moments of stochastic processes for estimating the reliability and accuracy of the constructed model. When using quasi-Banach spaces of random variables, these obstacles may be overcome. In the second part of the chapter properties of stochastic processes in Subϕ(Ω) spaces are described. The Karhunen-Lo’eve representation of stochastic processes with the help of systems of orthogonal polynomials are exploited. In cases where polynomials for this representation cannot be found explicitly, approximations are used for the description of the model. However, each approximation introduces its own errors. The impact of approximations errors on the reliability and accuracy of models of stochastic processes constructed using this method is analyzed.

Keywords: stochastic process, models of stochastic processes, sub-Gaussian spaces of
random variables, Kσ-spaces of random variables, DV,W spaces of random variables,
Karhunen-Lo`eve representation, accuracy and reliability of simulation


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