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Interval distribution function of a bounded chaotic sequence as a basis of nonaxiomatic probability theory

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Ukrainian Mathematical Journal Aims and scope

We introduce the notion of interval distribution function of random events on the set of elementary events and the notion of interval function of the frequencies of these events. In the limiting case, the interval function turns into the ordinary distribution function and the interval function of frequencies (under certain conditions) turns into the density of distribution of random events. The case of discrete sets of elementary events is also covered. This enables one to introduce the notion of the probability of occurrence of random events as a result of the limit transition.

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References

  1. V. P. Chistyakov, A Course of Probability Theory [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  2. V. S. Korolyuk and A. V. Skorokhod, “Studies in probability theory at the Institute of Mathematics of the Academy of Sciences of Ukrainian SSR for 50 years,” Ukr. Mat. Zh., 36, No. 5, 571–575 (1984).

    MATH  MathSciNet  Google Scholar 

  3. A. N. Kolmogorov, Basic Concepts of Probability Theory [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  4. V. M. Kuntsevich and M. M. Lychak, Guaranteed Estimations, Adaptation, and Robustness in Control Systems, Springer, Berlin (1992).

    Book  Google Scholar 

  5. R. von Mises, Wahrscheinlichkeit, Statistik und Wahrheit, Springer, Berlin (1928).

    MATH  Google Scholar 

  6. Yu. I. Alimov, “Utilitarian logic in the construction of probability theory,” in: Collection of Scientific Works “Semantics and Informatics” [in Russian], Issue 24, VINITI, Moscow (1985), pp. 58–86.

    Google Scholar 

  7. M. M. Lychak, “Elements of the theory of randomness and its applications,” Probl. Upravl. Informat., No. 5, 52–63 (2002).

  8. M. M. Lychak, “Continuous chaotic processes and their interval characteristics,” Probl. Upravl. Informat., No. 3, 82–96 (2004).

  9. M. M. Lychak, “Interval characteristics of chaotic sequences,” Kibernet. Sistem. Anal., No. 5, 58–71 (2004).

    Google Scholar 

  10. J. S. Nicolis, Dynamics of Hierarchical Systems. An Evolutionary Approach, Springer, Berlin (1986).

    MATH  Google Scholar 

  11. H. Haken, Advanced Synergetics. Instability Hierarchies of Self-Organizing Systems and Devices, Springer, Berlin (1983).

    MATH  Google Scholar 

  12. V. S. Charin, Linear Transformations and Convex Sets, Vyshcha Shkola, Kiev (1978).

    Google Scholar 

  13. A. Brónsted, An Introduction to Convex Polytopes, Springer, New York (1983).

    Google Scholar 

  14. A. Rényi, Levelek a Valószinüségröl [in Hungarian], Akadémia Kiadó, Budapest (1969).

    Google Scholar 

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Translated From Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, Pp. 1128–1137, August, 2008.

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Lychak, M.M. Interval distribution function of a bounded chaotic sequence as a basis of nonaxiomatic probability theory. Ukr Math J 60, 1318–1328 (2008). https://doi.org/10.1007/s11253-009-0125-0

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  • DOI: https://doi.org/10.1007/s11253-009-0125-0

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