Stochastic processes in the spaces

Authors:
Yu. V. Kozachenko and O. M. Moklyachuk

Translated by:
Oleg Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **82** (2010).

Journal:
Theor. Probability and Math. Statist. **82** (2011), 43-56

MSC (2010):
Primary 60G07

DOI:
https://doi.org/10.1090/S0094-9000-2011-00826-5

Published electronically:
August 2, 2011

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the spaces of random variables . We study the conditions for the convergence of series and the distribution of the supremum of stochastic processes in these spaces.

**1.**V. V. Buldygin and Yu. V. Kozachenko,*Metric characterization of random variables and random processes*, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR**1743716****2.**Yu. V. Kozachenko,*On the distribution of the supremum of random processes in quasi-Banach 𝐾_{𝜎}-spaces*, Ukraïn. Mat. Zh.**51**(1999), no. 7, 918–930 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**51**(1999), no. 7, 1029–1043 (2000). MR**1727696**, https://doi.org/10.1007/BF02592039**3.**V. V. Buldygin,*\cyr Skhodimost′ sluchaĭnykh èlementov v topologicheskikh prostranstvakh*, “Naukova Dumka”, Kiev, 1980 (Russian). MR**734899****4.**E. A. Abzhanov and Yu. V. Kozachenko,*Some properties of random processes in Banach 𝐾_{𝜎}-spaces*, Ukrain. Mat. Zh.**37**(1985), no. 3, 275–280, 403 (Russian). MR**795565****5.**E. A. Abzhanov and Yu. V. Kozachenko,*Random processes in quasi-Banach 𝐾_{𝜎}-spaces of random variables*, Probabilistic methods for the investigation of systems with an infinite number of degrees of freedom (Russian), Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1986, pp. 4–11, i (Russian). MR**895373****6.**Yu. V. Kozachenko and E. I. Ostrovskiĭ,*Banach spaces of random variables of sub-Gaussian type*, Teor. Veroyatnost. i Mat. Statist.**32**(1985), 42–53, 134 (Russian). MR**882158****7.**Yu. V. Kozachenko,*Random processes in Orlicz spaces. I*, Teor. Veroyatnost. i Mat. Statist.**30**(1984), 92–107, 152 (Russian). MR**800835****8.**Yu. V. Kozachenko,*Random processes in Orlicz spaces. II*, Teor. Veroyatnost. i Mat. Statist.**31**(1984), 44–50, 143 (Russian). MR**816125**

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Additional Information

**Yu. V. Kozachenko**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine

Email:
yvk@univ.kiev.ua

**O. M. Moklyachuk**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine

Email:
omoklyachuk@ukr.net

DOI:
https://doi.org/10.1090/S0094-9000-2011-00826-5

Keywords:
Stochastic processes,
prenorm,
quasinorm,
pre-Banach space,
quasi-Banach space,
the convergence of series

Received by editor(s):
January 21, 2010

Published electronically:
August 2, 2011

Additional Notes:
The first author is grateful to the Department of Mathematics and Statistics of the University “La Trobe”, Melbourne, for support in the framework of the research grant “Stochastic Approximation in Finance and Signal Processing”

Article copyright:
© Copyright 2011
American Mathematical Society