Lipschitz conditions for $\operatorname {Sub}_\varphi (\Omega )$-processes and applications to weakly self-similar processes with stationary increments
Authors:
Yuriĭ Kozachenko, Tommi Sottinen and Ol’ga Vasylyk
Translated by:
the authors
Journal:
Theor. Probability and Math. Statist. 82 (2011), 57-73
MSC (2010):
Primary 60G17, 60G18
DOI:
https://doi.org/10.1090/S0094-9000-2011-00827-7
Published electronically:
August 2, 2011
MathSciNet review:
2790484
Full-text PDF Free Access
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Abstract: We study the Lipschitz continuity of generalized sub-Gaussian processes and provide estimates for the distribution of the norms of such processes. The results are applied to the case of weakly self-similar generalized sub-Gaussian processes with stationary increments (the fractional Brownian motion is a particular case of these processes).
References
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- R. M. Dudley, Sample functions of the Gaussian process, Ann. Probability 1 (1973), no. 1, 66–103. MR 346884, DOI https://doi.org/10.1214/aop/1176997026
- R. Giuliano Antonini, Yu. V. Kozachenko, and T. Nikitina, Space of $\varphi$-subGaussian random variables, Rendiconti Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni 121$^{0}$, vol. XXVII, fasc. 1, 2003, pp. 92–124.
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- M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722
References
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TBiMC, Kiev, 1998; English transl., American Mathematical Society, Providence, RI, 2000. MR 1743716 (2001g:60089)
- R. M. Dudley, Sample functions of the Gaussian processes, Ann. Probab. 1 (1973), no. 1, 3–68. MR 0346884 (49:11605)
- R. Giuliano Antonini, Yu. V. Kozachenko, and T. Nikitina, Space of $\varphi$-subGaussian random variables, Rendiconti Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni 121$^{0}$, vol. XXVII, fasc. 1, 2003, pp. 92–124.
- Yu. V. Kozachenko, Random processes in Orlicz spaces. I, Teor. Veroyatnost. i Mat. Statist. 30 (1984), 103–117; English transl. in Theory Probab. Math. Statist. 31 (1984), 51–58. MR 800835 (86m:60111)
- M. A. Krasnosel’skiĭ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958; English transl., Noordhoff, Groningen, 1961. MR 0126722 (23:A4016)
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Additional Information
Yuriĭ 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
Tommi Sottinen
Affiliation:
University of Vaasa, Faculty of Technology, Department of Mathematics and Statistics, P.O. Box 700, FIN-65101 Vaasa, Finland
Email:
tommi.sottinen@uwasa.fi
Ol’ga Vasylyk
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:
ovasylyk@univ.kiev.ua
Keywords:
Self-similarity,
fractional Brownian motion,
generalized sub-Gaussian processes,
Lipschitz continuity
Received by editor(s):
December 16, 2009
Published electronically:
August 2, 2011
Additional Notes:
The first author is indebted to the Department of Mathematics and Statistics of the University “La Trobe”, Melbourne, for the support in the framework of the research grant “Stochastic approximation in finance and signal-processing”.
Article copyright:
© Copyright 2011
American Mathematical Society
