# On the increase rate of random fields from space $Sub_{\varphi}(\Omega)$ on unbounded domains

### Abstract

This paper mainly focuses on the estimates for distribution of supremum for the normalized φ-sub-Gaussian random fields defined on the unbounded domain. In particular, we obtain the estimates for distribution of supremum for the normalized solution of the hyperbolic equation of mathematical physics, which will be useful to construct modeless. By using this result, we can approximate the solutions of such equation with given accuracy and reliability in the uniform metric.### References

Kahane J.P. Properties locales des fonctions a series de Fouries aleatories, Studia Math. - 1960. - Vol. 19. - №1, 1-25.

Kahane J.P. Sur la divergence presque sure presque partout de certaines series de Fouries aleatories,

Ann. Univ. Scient., Budapest., Sect. Math., - 1960 - 1961.-Vol. 3 - 4, 101 - 108.

M. A. Krasnosel'skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958. - 271p.; English transl., Noordhof, Gr"{o}ningen, 1961. MRO106412 (21:5144).

R. Giuliano Antonini, Yu. Kozachenko, T. Nikitina,

Space of $varphi$-subgaussian random variables emorie di Mathematica e Applicazioni (Accademia Nazionale delle Scienze detta dei XL). - 2003 - Vol/ XXVII, fasc.1, 95 - 124.

Yu. V. Kozachenko and I. E. Ostrovskii, Banach spaces of random variables of sub-Gaussian type,

Theory Probab. Mathem. Statist. textbf{32} (1985), 42 - 53.

V.V. Buldygin and Yu.V. Kozachenko, Metric Characterization of Random Variables and Random processes, American Mathematical Society, Providence, Rhode (2000).

Ya.A. Kovalchuk and Yu.V. Kozachenko, Boundary value problems with random initial conditions and series

of functions of $Sub_varphi left( Omega right)$, Ukr. Matem. Zh., textbf{50(4)} (1998) 504 - 515.

Yu. Kozachenko, O. Vasylyk and R. Yamnenko, Upper estimate of overrunning by $Sub_varphi left( Omega right)$ random process

the level specified by continuous function, Random Operators and Stochastic Equations, (2005), Vol.13, No.2, 111 - 128.

Yu. V. Kozachenko and G. I. Slyvka, Justification of the Fourier method for hyperbolic equations with random initial conditions, Teor. Imovirnost.

Matem. Statist., textbf{69} (2003), 63 - 78; English transl. in Theory Probab. Mathem. Statist. textbf{69} (2004), 67 - 83.

Yu. V. Kozachenko and G. I. Slyvka, Boundary-value problems for equations of mathematical physics with strictly

random initial conditions Theory of Stochastic Processes, (2004), № 1-2, 60 - 71.

B. V. Dovgay, Yu. V. Kozachenko and G. I. Slyvka-Tylyshchak, "The boundary-value problems of mathematical physics with random factors," "Kyiv university", Kyiv, 2008, 173p. (Unkrainian).

I.V. Dariychuk , Yu.V. Kozachenko, M.M. Perestyuk,

Stochastic processes from Orlicz space - Chernivtci: "Zoloti Lutavru", 2011. - 212p. (Ukrainian)

Slyvka-Tylyshchak A. I. Simulation of vibrations of a rectangular membrane with random initial conditions, Annales Mathematicae and Informaticae. - 2012.-№39, 325-338.

G. N. Polozhiy, Equations of Mathematical Physics,

"Vyshaya shkola", Moscow, 1964. (Russian).

*Statistics, Optimization & Information Computing*,

*2*(2), 79-92. https://doi.org/10.19139/soic.v2i2.45

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).