Kolmogorov Consistency Theorem for Nonstochastic Random Processes

Victor Ivanenko () and Illia Pasichnichenko ()
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Victor Ivanenko: National Technical University of Ukraine KPI
Illia Pasichnichenko: National Technical University of Ukraine KPI

Sankhya A: The Indian Journal of Statistics, 2019, vol. 81, issue 2, No 6, 399-405

Abstract: Abstract Stochastic random phenomena studied in probability theory constitute only a part of all random phenomena, as was pointed out by Borel (1956) and Kolmogorov (1986). The need to study nonstochastic randomness led to new models. In particular, Ivanenko and Labkovsky (Sankhya A 77, 2, 237–248. 2015) defined a set of finitely additive probability measures as a set of accumulation points of a sequence or a net of frequency distributions. Here we prove the existence theorem for a nonstochastic random process described by a system of weak* closed sets of finite-dimensional distributions. Concretely, we show that a corresponding system of random variables can be defined on a probability space with a probability measure determined up to some set of measures, provided that the sets of finite-dimensional distributions are consistent.

Keywords: Kolmogorov extension theorem; Multiple priors; Consistency; Finite-dimensional distribution; Weak-star topology.; Primary 60G05; Secondary 28A33 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s13171-017-0121-7

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