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Conditions for the uniform convergence of expansions of $ \varphi$-sub-Gaussian stochastic processes in function systems generated by wavelets


Authors: Yu. V. Kozachenko and E. V. Turchin
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 83-95
MSC (2000): Primary 60G07; Secondary 42C40
DOI: https://doi.org/10.1090/S0094-9000-09-00764-9
Published electronically: August 4, 2009
MathSciNet review: 2446851
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Abstract | References | Similar Articles | Additional Information

Abstract: The expansions with uncorrelated coefficients in function systems generated by wavelets are constructed in the paper for second order stochastic processes. Conditions for the uniform convergence with probability one on a finite interval are found for expansions whose coefficients are independent. Conditions for the uniform convergence in probability on a finite interval are found for expansions of strictly $ \varphi$-sub-Gaussian stochastic processes.


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  • 1. L. Beghin, Yu. V. Kozachenko, E. Orsingher, and L. Sakhno, On the solutions of linear odd-order heat-type equations with random initial conditions, J. Stat. Physics 127 (2007), no. 4, 721-739. MR 2319850 (2008g:60211)
  • 2. V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev, 1999; English transl., AMS, Providence, RI, 2000. MR 1743716 (2001g:60089)
  • 3. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, Pennsylvania, 1992. MR 1162107 (93e:42045)
  • 4. I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Probability Theory and Mathematical Statistics, Vyshcha Shkola, Kiev, 1988. (Russian)
  • 5. R. Giuliano Antonini, Yu. Kozachenko, and T. Nikitina, Spaces of $ \phi$-sub-Gaussian random variables, Rendiconti Accademia Nazionale delle Scienze detta dei XL, Memorie di Matematica e Applicazioni, 121 (XXVII) (2003), no. 1, 95-124. MR 2056414 (2005f:60036)
  • 6. E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press Inc., Boca Rotan, FL, 1996. MR 1408902 (97i:42015)
  • 7. J.-P. Kahane, Some Random Series of Functions, Lexington, MA, 1968. MR 0254888 (40:8095)
  • 8. Yu. V. Kozachenko and Yu. A. Koval'chuk, Boundary value problems with random initial conditions and functional series of $ \operatorname{Sub}_{\varphi}(\Omega)$. I, Ukrain. Mat. Zh. 50 (1998), no. 4, 504-515; English transl. in Ukrainian Math. J. 50 (1999), no. 4, 572-585. MR 1698149 (2000f:60029)
  • 9. Yu. V. Kozachenko, M. M. Perestyuk, and O. I. Vasylyk, On uniform convergence of wavelet expansions of $ \varphi$-sub-Gaussian random processes, Random Oper. Stochastic Equations 14 (2006), no. 3, 209-232. MR 2264363 (2008e:60092)
  • 10. Yu. V. Kozachenko and I. V. Rozora, Accuracy and reliability of models of stochastic processes of the space $ \operatorname{Sub}_{\varphi}(\Omega)$, Teor. Imovir. Mat. Stat. 71 (2005), 93-104; English transl. in Theory Probab. Math. Statist. 71 (2006), 105-117. MR 2144324 (2005m:60077)
  • 11. Yu. V. Kozachenko and G. I. Slivka, Justification of the Fourier method for hyperbolic equations with random initial conditions, Teor. Imovir. Mat. Stat. 69 (2004), 63-78; English transl. in Theory Probab. Math. Statist. 69 (2005), 67-83. MR 2110906 (2005k:60127)
  • 12. Yu. Kozachenko and E. Turchyn, On one Karhunen-Loève-like expansion for stationary random processes, Int. J. Statistics and Management Systems 3 (2008), no. 1-2, 43-55.
  • 13. Yu. V. Kozachenko, I. V. Rozora, and Ye. V. Turchyn, On an expansion of random processes in series, Random Oper. Stochastic Equations 15 (2007), no. 1, 15-33. MR 2316186 (2008a:60131)
  • 14. Yu. V. Kozachenko, Lectures on Wavelet Analysis, TBiMC, Kyiv, 2004. (Ukrainian)
  • 15. M. A. Krasnosel'skiĭand Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958; English transl., Noordhoff, Groningen, 1961. MR 0126722 (23:A4016)
  • 16. G. Walter and J. Zhang, A wavelet-based KL-like expansion for wide-sense stationary random processes, IEEE Trans. Signal Process. 42 (1994), no. 7, 1737-1745.
  • 17. G. Walter and X. Shen, Wavelets and other Orthogonal Systems, Chapman and Hall, CRC, London, 2000. MR 1887929 (2003b:42003)

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Additional Information

Yu. V. Kozachenko
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: yvk@univ.kiev.ua

E. V. Turchin
Affiliation: Department of Higher Mathematics, Faculty for Mechanization of Agriculture, Dnipropetrovs’k State Agriculture University, Voroshilov Street 25, Dnipropetrovs’k, Ukraine
Email: evgturchyn@ukr.net

DOI: https://doi.org/10.1090/S0094-9000-09-00764-9
Keywords: Wavelets, $\varphi $-sub-Gaussian stochastic processes
Received by editor(s): May 17, 2007
Published electronically: August 4, 2009
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society