Estimates for the distribution of Hölder semi-norms of real stationary Gaussian processes with a stable correlation function

Abstract

Complex random variables and processes with a vanishing pseudo-correlation are called proper. There is a class of stationary proper complex random processes that have a stable correlation function. In the present article we consider real stationary Gaussian processes with a stable correlation function. It is shown that the trajectories of stationary Gaussian proper complex random processes with zero mean belong to the Orlich space generated by the function $U(x) = e^{x^2/2}-1$. Estimates are obtained for the distribution of semi-norms of sample functions of Gaussian proper complex random processes with a stable correlation function, defined on the compact $\mathbb{T} = [0,T]$, in Hölder spaces.

Key words: stationary Gaussian processes, proper complex random processes, Orlicz spaces, moduli of continuity, Hölder semi-norms.

Pages of the article in the issue: 25 - 30

Language of the article: Ukrainian

Author Biography

D. V. Zatula, Taras Shevchenko National University of Kyiv
Кафедра обчислювальної математики факультету комп'ютерних наук та кібернетики

References

BULDYGIN, V.V. and KOZACHENKO, I.V. (2000) Metric characterization of random variables and random processes (Vol. 188). American Mathematical Soc.

KOZACHENKO, Yu.V. and ZATULA, D.V. (2015) Lipschitz conditions for stochastic processes in the Banach spaces Fψ(Ω) of random variables. Theory of Probability and Mathematical Statistics. 91. pp.43-60.

KOZACHENKO, Yu.V. and ZATULA, D.V. (2019) Estimates for distributions of Hölder semi-norms of random processes from Fψ(Ω) spaces, defined on the interval [0,∞). Statistics, Optimization & Information Computing. 7 (1). pp.198-210.

DUDLEY, R.M. (1973) Sample functions of the Gaussian processes. The Annals of Probability. 1 (1). pp.3-68.

KOZACHENKO, Yu.V. (1985) Random processes in Orlicz spaces. I. Theory of Probability and Mathematical Statistics. 30. pp.103-117. (in Russian)

ZATULA, D.V. (2013) Modules of continuity of random processes from Orlicz spaces of random variables, defined on the interval. Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics & Mathematics. (2). pp.23-28.

PETRANOVA, M.Yu. and KOZACHENKO, Yu.V. (2017) Vlasni kompleksni vypadkovi procesy. Zbirnyky naukovyx prac' profesors'ko-vykladac'koho skladu DonNU imeni Vasylya Stusa.

KOZACHENKO, Yu.V. and PETRANOVA, M.Yu. (2017) Dijsni stacionarni hausovi procesy zi stijkymy korelyacijnymy funkciyamy. Naukovyj visnyk Uzhhorods'koho universytetu. Seriya «Matematyka i informatyka». 2 (31). pp.90-100.

How to Cite
Zatula, D. V. (1). Estimates for the distribution of Hölder semi-norms of real stationary Gaussian processes with a stable correlation function. Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, (1-2), 25-30. https://doi.org/10.17721/1812-5409.2020/1-2.3
Section
Algebra, Geometry and Probability Theory