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Kolmogorov Consistency Theorem for Nonstochastic Random Processes

Author

Listed:
  • Victor Ivanenko

    (National Technical University of Ukraine KPI)

  • Illia Pasichnichenko

    (National Technical University of Ukraine KPI)

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.

Suggested Citation

  • Victor Ivanenko & Illia Pasichnichenko, 2019. "Kolmogorov Consistency Theorem for Nonstochastic Random Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(2), pages 399-405, December.
    • Handle: RePEc:spr:sankha:v:81:y:2019:i:2:d:10.1007_s13171-017-0121-7
    DOI: 10.1007/s13171-017-0121-7
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    References listed on IDEAS

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    1. Ivanenko, Victor & Pasichnichenko, Illia, 2017. "Expected utility for nonstochastic risk," Mathematical Social Sciences, Elsevier, vol. 86(C), pages 18-22.
    2. Miller, J. Isaac & Ratti, Ronald A., 2009. "Crude oil and stock markets: Stability, instability, and bubbles," Energy Economics, Elsevier, vol. 31(4), pages 559-568, July.
    3. Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
    4. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
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