Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 14 (2009) > Paper 94 open journal systems 


Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems

Werner Linde, FSU Jena
Antoine Ayache, UMR CNRS 8524 , Laboratoire Paul Painleve


Abstract
The aim of the present paper is to investigate series representations of the Riemann--Liouville process Rα, α>1/2, generated by classical orthonormal bases in L2 [0,1]. Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of Rα via the trigonometric system possesses the optimal convergence rate if and only if 1/2<α≤ 2. For the Haar system we have an optimal approximation rate if 1/2<α<3/2 while for α>3/2 a representation via the Haar system is not optimal. Estimates for the rate of convergence of the Haar series are given in the cases α>3/2 and α=3/2. However, in this latter case the question whether or not the series representation is optimal remains open. Recently M. A. Lifshits answered this question (cf. [13]). Using a different approach he could show that in the case α= 3/2 a representation of the Riemann­Liouville process via the Haar system is also not optimal.


Full text: PDF

Pages: 2691-2719

Published on: December 21, 2009
Bibliography
  1. Ayache, Antoine; Linde, Werner. Approximation of Gaussian random fields: general results and optimal wavelet representation of the Lévy fractional motion. J. Theoret. Probab. 21 (2008), no. 1, 69--96. MR2384473 (2009c:60127)
  2. Ayache, Antoine; Taqqu, Murad S. Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl. 9 (2003), no. 5, 451--471. MR2027888 (2004m:60084)
  3. Belinsky, Eduard; Linde, Werner. Small ball probabilities of fractional Brownian sheets via fractional integration operators. J. Theoret. Probab. 15 (2002), no. 3, 589--612. MR1922439 (2004d:60092)
  4. Carl, Bernd; Stephani, Irmtraud. Entropy, compactness and the approximation of operators.Cambridge Tracts in Mathematics, 98. Cambridge University Press, Cambridge, 1990. x+277 pp. ISBN: 0-521-33011-4 MR1098497 (92e:47002)
  5. Dzhaparidze, Kacha; van Zanten, Harry. Optimality of an explicit series expansion of the fractional Brownian sheet. Statist. Probab. Lett. 71 (2005), no. 4, 295--301. MR2145497 (2006g:60057)
  6. Fernique, Xavier. Fonctions aléatoires gaussiennes, vecteurs aléatoires gaussiens.(French) [Gaussian random functions, Gaussian random vectors] Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1997. iv+217 pp. ISBN: 2-921120-28-3 MR1472975 (99f:60078)
  7. Gilsing, Hagen; Sottinen, Tommi. Power series expansions for fractional Brownian motions. Theory Stoch. Process. 9 (2003), no. 3-4, 38--49. MR2306058
  8. Iglói, E. A rate-optimal trigonometric series expansion of the fractional Brownian motion. Electron. J. Probab. 10 (2005), no. 41, 1381--1397 (electronic). MR2183006 (2006f:60039)
  9. Kühn, Thomas; Linde, Werner. Optimal series representation of fractional Brownian sheets. Bernoulli 8 (2002), no. 5, 669--696. MR1935652 (2003m:60131)
  10. Ledoux, Michel. Isoperimetry and Gaussian analysis. Lectures on probability theory and statistics (Saint-Flour, 1994), 165--294, Lecture Notes in Math., 1648, Springer, Berlin, 1996. MR1600888 (99h:60002)
  11. Kwapień, Stanisƚaw; Woyczyński, Wojbor A. Random series and stochastic integrals: single and multiple.Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xvi+360 pp. ISBN: 0-8176-3572-6 MR1167198 (94k:60074)
  12. Li, Wenbo V.; Linde, Werner. Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 (1999), no. 3, 1556--1578. MR1733160 (2001c:60059)
  13. Li, W. V.; Shao, Q.-M. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, 533--597, Handbook of Statist., 19, North-Holland, Amsterdam, 2001. MR1861734
  14. Lifshits, M. A. On Haar expansion of Riemann--Liouville process in a critical case. Available at arXiv:0910.2177 (2009).
  15. Lifshits, Mikhail A.; Linde, Werner; Shi, Zhan. Small deviations of Riemann-Liouville processes in $Lsb q$-spaces with respect to fractal measures. Proc. London Math. Soc. (3) 92 (2006), no. 1, 224--250. MR2192391 (2006m:60053)
  16. Lifshits, Mikhail; Simon, Thomas. Small deviations for fractional stable processes. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 725--752. MR2144231 (2006d:60081)
  17. Luschgy, Harald; Pagès, Gilles. High-resolution product quantization for Gaussian processes under sup-norm distortion. Bernoulli 13 (2007), no. 3, 653--671. MR2348745 (2009d:60111)
  18. Luschgy, Harald; Pagès, Gilles. Expansions for Gaussian processes and Parseval frames. Electron. J. Probab. 14 (2009), no. 42, 1198--1221. MR2511282
  19. Malyarenko, Anatoliy. An optimal series expansion of the multiparameter fractional Brownian motion. J. Theoret. Probab. 21 (2008), no. 2, 459--475. MR2391256 (2009j:60082)
  20. Pisier, Gilles. The volume of convex bodies and Banach space geometry.Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989. xvi+250 pp. ISBN: 0-521-36465-5; 0-521-66635-X MR1036275 (91d:52005)
  21. Schack, H. An optimal wavelet series expansion of the Riemann--Liouville process. To appear J. Theoret. Probab.
  22. Zygmund, A. Trigonometric Series. Cambridge Univ. Press, Cambridge, 1950.
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489