Convergence in L p ([0, T]) of Wavelet Expansions of φ-Sub-Gaussian Random Processes

  • Yuriy Kozachenko
  • Andriy Olenko
  • Olga Polosmak


The article presents new results on convergence in L p ([0,T]) of wavelet expansions of φ-sub-Gaussian random processes. The convergence rate of the expansions is obtained. Specifications of the obtained results are discussed.


Convergence rate Convergence in probability  Sub-Gaussian random process Wavelets 

AMS 2000 Subject Classifications

60G10 60G15 42C40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Atto A, Berthoumieu Y (2012) Wavelet packets of nonstationary random processes: contributing factors for stationarity and decorrelation. IEEE Trans Inf Theory 58(1):317–330CrossRefMathSciNetGoogle Scholar
  2. Bardet JM, Tudor CA (2010) A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter. Stoch Process Appl 120(12):2331–2362CrossRefzbMATHMathSciNetGoogle Scholar
  3. Buldygin VV, Kozachenko YuV (2000) Metric characterization of random variables and random processes. American Mathematical Society, Providence R.I.Google Scholar
  4. Cambanis S, Masry E (1994) Wavelet approximation of deterministic and random signals: convergence properties and rates. IEEE Trans Inf Theory 40(4):1013–1029CrossRefzbMATHMathSciNetGoogle Scholar
  5. Clausel M, Roueff F, Taqqu MS, Tudor C (2012) Large scale behavior of wavelet coefficients of non-linear subordinated processes with long memory. Appl Comput Harmon Anal 32(2):223–241CrossRefzbMATHMathSciNetGoogle Scholar
  6. Daubechies I (1992) Ten lectures on wavelets. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  7. Didier G, Pipiras V (2008) Gaussian stationary processes: adaptive wavelet decompositions, discrete approximations and their convergence. J Fourier Anal Appl 14:203–234CrossRefzbMATHMathSciNetGoogle Scholar
  8. Giuliano Antonini R, Kozachenko YuV, Nikitina T (2003) Spaces of φ-sub-Gaussian random variables. Mem Mat Appl 121(27)fasc 1:95–124Google Scholar
  9. Hardle W, Kerkyacharian G, Picard D, Tsybakov A (1998) Wavelets, approximation and statistical applications. Springer, New YorkCrossRefGoogle Scholar
  10. Jaffard S (2001) Wavelet expansions, function spaces and multifractal analysis. In: Byrnes JS (ed) Twentieth century harmonic analysis—a celebration. Kluwer Acad Publ, Dordrecht, pp 127–144CrossRefGoogle Scholar
  11. Kozachenko Yu, Kamenshchikova O (2009) Approximation of \(\operatorname {SSub}_{\varphi}(\Omega)\) stochastic processes in the space \(L_{p}(\mathbb {T})\). Theory Probab Math Stat 79:83–88CrossRefMathSciNetGoogle Scholar
  12. Kozachenko Yu, Kovalchuk Yu (1998) Boundary value problems with random initial conditions and functional series from Sub ϕ(Ω). Ukr Math J 50:504–515zbMATHMathSciNetGoogle Scholar
  13. Kozachenko Yu, Ostrovskyi E (1985) Banach spaces of random variables of sub-Gaussian type. Theory Probab Math Stat 32:42–53Google Scholar
  14. Kozachenko Yu, Polosmak O (2008) Uniform convergence in probability of wavelet expansions of random processes from L 2(Ω). Random Oper Stoch Equ 16(4):12–37CrossRefMathSciNetGoogle Scholar
  15. Kozachenko Yu, Olenko A, Polosmak O (2011) Uniform convergence of wavelet expansions of Gaussian random processes. Stoch Anal Appl 29:169–184CrossRefzbMATHMathSciNetGoogle Scholar
  16. Kozachenko Yu, Olenko A, Polosmak O (2013) Convergence rate of wavelet expansions of Gaussian random processes. Commun Stat Theory Methods (to appear, 2013)Google Scholar
  17. Kurbanmuradov O, Sabelfeld K (2008) Convergence of fourier-wavelet models for Gaussian random processes. SIAM J Numer Anal 46(6):3084–3112CrossRefzbMATHMathSciNetGoogle Scholar
  18. Labate D, Weiss G, Wilson E (2013) Wavelets. Notices Amer Math Soc 60(1):66–76CrossRefzbMATHMathSciNetGoogle Scholar
  19. Triebel H (2008) Function spaces and wavelets on domains. European Mathematical Society, ZuürichCrossRefzbMATHGoogle Scholar
  20. Vershynin R (2012) Introduction to the non-asymptotic analysis of random matrices. In: Eldar Y, Kutyniok G (eds) Compressed sensing, theory and applications. Cambridge University Press, Cambridge, pp 210–268CrossRefGoogle Scholar
  21. Yamnenko R (2006) Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion. Theory Stoch Process 12(28)1–2:261–275MathSciNetGoogle Scholar
  22. Zhang J, Waiter G (1994) A wavelet-based KL-like expansion for wide-sense stationary random processes. IEEE Trans Signal Process 42(7):1737–1745CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Yuriy Kozachenko
    • 1
  • Andriy Olenko
    • 2
  • Olga Polosmak
    • 3
  1. 1.Department of Probability Theory, Statistics and Actuarial MathematicsKyiv UniversityKyivUkraine
  2. 2.Department of Mathematics and StatisticsLa Trobe UniversityVICAustralia
  3. 3.Department of Economic CyberneticsKyiv UniversityKyivUkraine

Personalised recommendations