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.

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


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