Sample continuity and modeling of stochastic processes from the spaces

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

Translated by:
O. Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **83** (2010).

Journal:
Theor. Probability and Math. Statist. **83** (2011), 95-110

MSC (2010):
Primary 60G07

DOI:
https://doi.org/10.1090/S0094-9000-2012-00844-2

Published electronically:
February 2, 2012

MathSciNet review:
2768851

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Random sequences and stochastic processes belonging to the spaces are studied in the paper. Conditions for the sample continuity of such processes are found. The convergence of series of random variables belonging to the spaces are considered. Models of stochastic processes belonging to the spaces are studied. Several examples of models are given.

**1.**Yu. V. Kozachenko and O. M. Moklyachuk,*Stochastic processes in the spaces*, Teor. Imovir. Mat. Stat.**82**(2010), 56-66; English transl. in Theor. Probability and Math. Statist.**82**(2011), 43-56. MR**2790483 (2011m:60101)****2.**V. V. Buldygin and Yu. V. Kozachenko,*Metric Characterization of Random Variables and Random Processes*, TViMS, Kiev, 1998; English transl., American Mathematical Society, Providence, Rhode Island, 2000. MR**1743716 (2001g:60089)****3.**Yu. V. Kozachenko,*On the distribution of the supremum of random processes in quasi-Banach -spaces*, Ukrain. Mat. Zh.**51**(1999), no. 7, 918-930; English transl. in Ukrainian Math. J.**51**(2000), no. 7, 1029-1043. MR**1727696 (2000k:60066)****4.**V. V. Buldygin,*Convergence of Random Elements in Topological Spaces*, Naukova Dumka, Kiev, 1980. (Russian) MR**734899 (84m:60011)****5.**E. A. Abzhanov and Yu. V. Kozachenko,*Some properties of random processes in Banach -spaces*, Ukrain. Mat. Zh.**37**(1985), no. 3, 275-280; English transl. in Ukr. Math. J.**37**(1986), no. 3, 209-213. MR**795565 (87m:60095)****6.**E. A. Abzhanov and Yu. V. Kozachenko,*Random processes in -spaces of random variables*, Probabilistic Methods for the Investigation of Systems with an Infinite Number of Degrees of Freedom (A. V. Skorokhod, ed.), Institute of Mathematics, Academy of Science of Ukrain. SSR, Kiev, 1986, pp. 4-11, Russian. MR**895373 (88g:60101)****7.**Yu. V. Kozachenko,*Random processes in Orlicz spaces. I*, Teor. Veroyatnost. i Mat. Statist.**30**(1984), 92-107; English transl. in Theory Probab. Math. Statist.**30**(1985), 103-117. MR**800835 (86m:60111)****8.**Yu. V. Kozachenko,*Random processes in Orlicz spaces. II*, Teor. Veroyatnost. i Mat. Statist.**31**(1984), 44-50; English transl. in Theory Probab. Math. Statist.**31**(1985), 51-58. MR**816125 (87b:60063)**

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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/S0094-9000-2012-00844-2

Keywords:
Stochastic processes,
modeling of stochastic processes,
pre-norm,
quasi-norm,
pre-Banach spaces,
quasi-Banach spaces,
spaces $D_{V,W}$

Received by editor(s):
July 19, 2010

Published electronically:
February 2, 2012

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

Article copyright:
© Copyright 2012
American Mathematical Society