Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 14 (2009) > Paper 42 open journal systems 


Expansions for Gaussian Processes and Parseval Frames

Harald Luschgy, University of Trier
Gilles Pagès, Université Paris 6


Abstract
We derive a precise link between series expansions of Gaussian random vectors in a Banach space and Parseval frames in their reproducing kernel Hilbert space. The results are applied to pathwise continuous Gaussian processes and a new optimal expansion for fractional Ornstein-Uhlenbeck processes is derived.


Full text: PDF

Pages: 1198-1221

Published on: June 1, 2009
Bibliography
  1. Ayache, A., Taqqu, M.S., Rate optimality of Wavelet series approximations of fractional Brownian motion, J. Fourier Anal. and Appl., 9 (2003), 451-471. Math. Review 2027888
  2. Bogachev, V.I., Gaussian Measures, AMS, 1998. Math. Review 1642391
  3. Bronski, J.C., Small ball constants and tight eigenvalue asymptotics for fractional Brownsian motions, J. Theoret. Probab. 16 (2003), 87-10 Math. Review 1956822
  4. Christensen, O., An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003. Math. Review 1946982
  5. Dzhaparidze, K., van Zanten, H., A series expansion of fractional Brownian motion, Probab. Theory Relat. Fields, 130 (2004), 39-55. Math. Review 2092872
  6. Dzhaparidze, K., van Zanten, H., Optimality of an explicit series expansion of the fractional Brownian sheet, Statist. Probab. Letters 71 (2005), 295 - 301. Math. Review 2145497
  7. Dzhaparidze, K., van Zanten, H., Krein's spectral theory and the Paley-Wiener expansion of fractional Brownian motion, Ann. Probab. 33 (2005), 620-644. Math. Review 2123205
  8. Kühn, T., Linde, W., Optimal series representation of fractional Brownian sheets, Bernoulli 8 (2002), 669 - 696. Math. Review 1935652
  9. Ledoux, M., Talagrand, M., Probability in Banach Spaces, Springer, Berlin, 1991. Math. Review 1102015
  10. Lukacs, E, Characteristic Functions, second edition, Griffin, London, 1970. Math. Review 0346874
  11. Luschgy, H., Linear estimators and radonifying operators Theory Probab. Appl. 40 (1995), 205-213. Math. Review 1346746
  12. Luschgy, H., Pagès, G., Sharp asymptotics of the functional quantization problem for Gaussian processes, Ann. Probab. 32 (2004), 1574- 1599. Math. Review 2060310
  13. Luschgy, H., Pagès, G., High-resolution product quantization for Gaussian processes under sup-norm distortion, Bernoulli 13 (2007), 653-671. Math. Review 2348745
  14. Meyer, Y., Sellan, F. Taqqu, M.S., Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion, J. Fourier Anal. and Appl. 5 (1999), 465-494. Math. Review 1755100
  15. Neveu, J., Processus Aéatoires Gaussiens, Les Presses de l'Université de Montréal, 1968. Math. Review 0272042
  16. Rosenblatt, M., Some results on the asymptotic behaviour of eigenvalues for a class of integral equations with translation kernel, J. Math. Mech. 12 (1963), 619 - 628. Math. Review 0150551
  17. Vakhania, N.N., Tarieladze, V.I., Chobanyan, S.A., Probability Distributions on Banach Spaces, Kluwer, Boston, 1987. Math. Review 1435288
















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


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

Electronic Journal of Probability. ISSN: 1083-6489