Conditions for the uniform convergence in probability of wavelet decompositions for stochastic processes from the space $\operatorname {Exp}_{\varphi }(\Omega )$
Authors:
Yu. V. Kozachenko and O. V. Polos’mak
Translated by:
O. Klesov
Journal:
Theor. Probability and Math. Statist. 81 (2010), 85-99
MSC (2000):
Primary 60G12, 42C40
DOI:
https://doi.org/10.1090/S0094-9000-2011-00812-5
Published electronically:
January 20, 2011
MathSciNet review:
2667312
Full-text PDF Free Access
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Additional Information
Abstract: Conditions for the uniform convergence in probability on the interval $[0,T]$ of wavelet decompositions of Orlicz stochastic processes of exponential type are found in the paper.
References
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References
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev, 1998; English transl., American Mathematical Society, Providence, RI, 2000. MR 1743716 (2001g:60089)
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- C. K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992. MR 1150048 (93f:42055)
- I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, 1992. MR 1162107 (93e:42045)
- P. Flandrin, Time-Frequency/Time-Scale Analysis, Academic Press, New York, 1999. MR 1681043 (2000e:94014)
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- Yu. V. Kozachenko and O. V. Polosmak, Uniform convergence in probability of wavelet expansions of random processes from $L_2(\Omega )$, Random Oper. Stoch. Equ. 16 (2008), no. 4, 12–37. MR 2494935 (2010c:42081)
- Yu. V. Kozachenko, M. M. Perestyuk, and O. I. Vasylyk, On uniform convergence of wavelet expansion of $\phi$-sub-Gaussian random processes, Random Oper. Stoch. Equ. 14 (2006), no. 3, 209–233. MR 2264363 (2008e:60092)
- Yu. V. Kozachenko, Lectures on Wavelet Analysis, TViMS, Kyiv, 2004. (Ukrainian)
- Yu. V. Kozachenko and G. I. Slivka, Justification of the Fourier method for hyperbolic equations with random initial conditions, Teor. Ĭmovir. Mat. Stat. 69 (2003), 63–78; English transl. in Theory Probab. Math. Statist. 69 (2004), 67–83. MR 2110906 (2005k:60127)
- Yu. V. Kozachenko and M. M. Perestyuk, On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. I, Ukrain. Mat. Zh. 59 (2007), no. 12, 1647–1660; English transl. in Ukrainian Math. J. 59 (2007), no. 12, 1850–1869. MR 2411593 (2009b:60058)
- Yu. V. Kozachenko and M. M. Perestyuk, On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II, Ukrain. Mat. Zh. 60 (2008), no. 6, 759–775; English transl. in Ukrainian Math. J. 60 (2008), no. 6, 876–900. MR 2485176 (2009m:60095)
- Yu. V. Kozachenko and E. V. Turchin, Conditions for the uniform convergence of distributions of $\phi$-sub-Gaussian random processes in systems of functions generated by wavelets, Teor. Ĭmovir. Mat. Stat. 78 (2008), 74–85; English transl. in Theory Probab. Math. Statist. 78 (2009), 83–95. MR 2446851 (2009m:60085)
- E. Barrasa de la Krus and Yu. V. Kozachenko, Boundary-value problems for equations of mathematical physics with stricly Orlicz random initial conditions, Random Oper. Stoch. Equ. 3 (1995), no. 3, 201–220. MR 1354813 (96h:60009)
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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. V. Polos’mak
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:
olgapolosmak@yandex.ru
Keywords:
Stochastic processes,
wavelet decomposition,
convergence in probability,
Orlicz spaces
Received by editor(s):
March 11, 2009
Published electronically:
January 20, 2011
Additional Notes:
This research was supported by the La Trobe University Research Grant #501821
Article copyright:
© Copyright 2011
American Mathematical Society
