Weak convergence of stochastic processes from spaces $F_\psi(\Omega)$

  • Yurii Yuriiovych Mlavets Uzhhorod National University
  • Yurii Vasylovych Kozachenko Taras Shevchenko National University of Kyiv
  • Nataliia Vasylivna Yurchenko Uzhhorod National University
DOI: https://doi.org/10.19139/soic.v6i2.394
Keywords: weak convergence, quasi-metric space, $K_\sigma$-space, $\mathbf{F}_\psi(\Omega)$ space, majorant characteristic, metric massiveness, condition $\mathbf{H}$, stochastic processes, Monte Carlo method

Abstract

This paper is devoted to the investigation of conditions for the  eak convergence in the space $C(T)$ of the stochastic processes from the space $\mathbf{F}_\psi(\Omega)$. Using this conditions the limit theorem for stochastic processes from the space $\mathbf{F}_\psi(\Omega)$ has been obtained. This theorem can be utilized  for gaining the given approximation accuracy  and reliability of integrals depending on parameter by Monte Carlo method.

Author Biographies

Yurii Yuriiovych Mlavets, Uzhhorod National University
Department of Cybernetics and Applied Mathematics, associate professor
Yurii Vasylovych Kozachenko, Taras Shevchenko National University of Kyiv
Department of Probability Theory, Statistics and Actuarial Mathematics, professor
Nataliia Vasylivna Yurchenko, Uzhhorod National University
Department of Algebra, associate professor

References

E. A. Abzhanov, and Y. V. Kozachenko, Some Properties of Random Processes in Banach $K_sigma$-spaces, Ukrainian Mathem. Journal, vol. 37, no. 3, pp. 275-280, 1985.

V. V. Buldygin, and Y. V. Kozachenko, On sub-Gaussian Random Variables, Ukrainian Mathem. Journal, vol. 32, no. 6, pp. 483-489, 2013.

V. V. Buldygin, and Y. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, American Mathematical Society, Providence, Rhode Island, 2000.

M. Dozzi, Y. Kozachenko, Y. Mishura, and K. Ralchenko, Asymptotic Growth of Trajectories of Multifractional Brownian Motion, with Statistical Applications to Drift Parameter Estimation, Statistical Inference for Stochastic Processes, DOI: 10.1007/s11203-016-9147-z, pp. 1-32, 2016.

S. V. Ermakov, and E. I. Ostrovskii, Continuity Conditions, Exponential Estimates, and the Central Limit Theorem for Random Fields, VINITI, Moscow, 1986. (in Russian)

Y. Kozachenko, A. Olenko, and O. Polosmak, Uniform Convergence of Wavelet Expansions of Gaussian Random Processes, Stochastic Analysis and Applications, vol. 29, no. 2, pp. 169-184, 2011.

Y. V. Kozachenko, and Y. Y. Mlavets, The Banach Spaces $mathbf{F}_psi(Omega)$ of Random Variables, Theory Probab. and Math. Statist., vol. 86, pp. 105-121, 2013.

Y. V. Kozachenko, and Y. Y. Mlavets, Stochastic Processes from $mathbf{F}_psi(Omega)$ Spaces, Contemporary Mathematics and Statistics, vol. 2, no. 1, pp. 55-75, 2014.

Y. V. Kozachenko, and Y. Y. Mlavets, Reliability and Accuracy in the Space $L_p(T)$ for the Calculation of Integrals Depending on a Parameter by Monte-Carlo Method, Monte Carlo Methods Appl., vol. 21, no. 3, pp. 233-244, 2015.

Y. V. Kozachenko, Y. Y. Mlavets, and O. M. Moklyachuk, Qasi-Banach Spaces of Random Variables, Carpathians, Uzhhorod, 2015. (in Ukrainian)

Y. Y. Mlavets, $mathbf{F}_psi(Omega)$ -- spaces of Random Variables with Exponential Function $psi$, Bull. Taras Shevchenko Nat. Univ. Kiev., vol. 2, pp. 19-22, 2012. (in Ukrainian)

Y. Y. Mlavets, A Relationship Between the Spaces $mathbf{F}_psi(Omega)$ and Orlicz Spaces Random Variables, Sci. Bull. Uzhhorod Univ., vol. 25, no. 1, pp. 77-84, 2014. (in Ukrainian)

O. I. Vasylyk, Y. V. Kozachenko, and R. E. Yamnenko, $varphi$-sub-Gaussian Stochastic Process, Vydavnycho-Poligrafichnyi Tsentr, Kyivskyi Universytet, Kyiv, 2008. (In Ukrainian)

Published
2018-06-24
How to Cite
Mlavets, Y. Y., Kozachenko, Y. V., & Yurchenko, N. V. (2018). Weak convergence of stochastic processes from spaces $F_\psi(\Omega)$. Statistics, Optimization & Information Computing, 6(2), 266-277. https://doi.org/10.19139/soic.v6i2.394
Issue
Vol 6 No 2 (2018)
Section
Research Articles
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