On the Occurrence Mechanism of of the Random Value with the “Heavy” Polynomial “Tails” in the Stochastic Processes of Gaussian Distributions

Fluid, gas and plasma mechanics


Khatuntseva O. N.

Korolev Rocket and Space Corporation «Energia», 4а, Lenin str., Korolev, Moscow region, 141070, Russia

e-mail: Olga.Khatuntseva@rsce.ru


Most of the economic, biological, physical and other processes are adequately unpredictable. The lack of the determinism in such stochastic processes is related not only to probability distribution of the considered process possible realizations of the, but also with the possible unsteady behavior of the probability density. Stochastic process may gain the additional uncertainty if the probability density of the random value realization can be described at random fr om the set of the feasible functions describing it.

The article considers the problems related to the possible of the probability density non-uniqueness of the random value realization in stochastic processes. It shows that the method for the stochastic processes description for the systems with no selected equilibrium states allows find different probability densities for the stochastic process that may realize at random way the certain range of random values.

In this respect, the following conclusion based on the Lindeberg Central Limit Theorem can be made. The Gaussian distributions with “heavy tails” manifestation may be considered as the consequence of the hidden factors occurrence in the system, affecting the probability density dynamics when random value realization of the considered process causes the variation of the probability density of realization. It confirms the prognosing feasibility of natural and technogenic catastrophes based on the considered random values distributions analysis from the viewpoint of their deviation from the normal (Gaussian) law of distribution. In hydrodynamics, such deviation indicates the appearance of the coherent structures and the possibility for the transition form the laminar flow mode to the turbulent one.

The phase space dimensionality analysis allows determining both stable and unstable branches of the solution for the probability density in such random value realization domains, wh ere a unique solution may be realized.


stochastic processes, hidden Markov model, discontinuous functions, probability density


  1. Tuzikova E.S. Trudy MAI, 2012, no. 52, available at: http://trudymai.ru/eng/published.php?ID=29584

  2. Semakov S.L., Semakov I.S. Trudy MAI, 2018, no. 100, available at: http://trudymai.ru/eng/published.php?ID=93446

  3. Kuznetsov V.S., Shevchenko I.V., Volkov A.S., Solodkov A.V. Trudy MAI, 2017, no. 96, available at: http://trudymai.ru/eng/published.php?ID=85813

  4. Mikhailov V.Yu., Vitomskii E.V. Trudy MAI, 2018, no. 98, available at: http://trudymai.ru/eng/published.php?ID=90426

  5. Krivoruchko D.D., Skrylev A.V., Skorokhod E.P. Trudy MAI, 2017, no. 92, available at: http://trudymai.ru/eng/published.php?ID=76859

  6. Yakimov V.L., Pankratov A.V. Trudy MAI, 2015, no. 83. available at: http://trudymai.ru/eng/published.php?ID=62242

  7. Maslov G.A., Lapushkin V.N. Trudy MAI, 2015, no. 80, available at: http://trudymai.ru/eng/published.php?ID=56918

  8. Vasil’eva O.A. Trudy MAI, 2014, no. 78, available at: http://trudymai.ru/eng/published.php?ID=53684

  9. Khatuntseva O.N. Trudy MAI, 2018, no. 100, available at: http://trudymai.ru/eng/published.php?ID=93311

  10. Khatuntseva O.N. Trudy MAI, 2018, no. 101, available at: http://trudymai.ru/eng/published.php?ID=96567

  11. Bak P. How Nature Works: the Science of Self-Organized Criticality, New York, Springer-Verlag, 1996, 212 p.

  12. Malinetskii G.G., Potapov A.B. Sovremennye problemy nelineinoi dinamiki (Modern Problems of Nonlinear Dynamics), Moscow, Editorial URSS, 2000, 336 p.

  13. Vatanabe S., Ikeda N. Stokhasticheskie differentsial’nye uravneniya i diffuzionnye protsessy (Stochastic differential equations and diffusion processes), Moscow, Mir, 1984, 448 p.

  14. Khatuntseva O.N. Zhurnal estestvennykh i tekhnicheskikh nauk, 2017, no. 11 (113), pp. 255 – 258.

  15. Klimontovich Yu.L. Uspekhi fizicheskikh nauk, 1994, vol. 164, no. 8, pp. 811 – 844.

  16. Ito K Stochastic Differential Equations, Memoirs of the American Mathematical Society, vol. 4, 1951, pp. 1-51, available at: http://dx.doi.org/10.1090/memo/0004

  17. Anishchenko V.S., Vadisova T.E., Shimanskii-Gaier L. Dinamicheskoe i staticheskoe opisanie kolebatel’nykh system (Dynamic and Static Description of Oscillating Systems), Moscow-Izhevsk, NITs “Regulyarnaya i khaoticheskaya dinamika”, Institut komp’yuternykh issledovanii, 2005. 156 c.

  18. Rabiner L.A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition, IEEE Proceedings, 1989, vol. 77, no. 2, pp. 257 – 286.

  19. Momzikova M.P., Velikodnaya O.I., Pinskii M.Ya., Sirotkin A.V., Tulup’ev A.L., Fil’chenkov A.A. Trudy SPIIRAN, 2010, no. 2, pp. 122 – 142.

  20. Lifshits E.M., Pitaevskii L.P. Teoreticheskaya fizika. T.X. Fizicheskaya kinetika (Physical Kinetics), Moscow, Nauka, 2002, 536 p.

  21. Feder E. Fraktaly (Fractals), Moscow, Mir, 1991, 260 p.


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