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 Electronic Journal of Probability > Vol. 11 (2006) > Paper 28 open journal systems 


High resolution quantization and entropy coding for fractional Brownian motion

Steffen Dereich, TU Berlin
Michael Scheutzow, TU Berlin


Abstract
We establish the precise asymptotics of the quantization and entropy coding errors for fractional Brownian motion with respect to the supremum norm and Lp [0,1]-norm distortions. We show that all moments in the quantization problem lead to the same asymptotics. Using a general principle, we conclude that entropy coding and quantization coincide asymptotically. Under supremum-norm distortion, our proof uses an explicit construction of efficient codebooks based on a particular entropy constrained coding scheme.


Full text: PDF

Pages: 700-722

Published on: August 27, 2006
Bibliography
  1. Cover, Thomas M.; Thomas, Joy A. Elements of information theory. John Wiley & Sons, Inc., New York, 1991. xxiv+542 pp. ISBN: 0-471-06259-6 MR1122806 (92g:94001)
  2. J.Creutzig. Approximation of Gaussian random vectors in Banach spaces. Ph.D. Dissertation, Friedrich-Schiller-Universitat Jena, 2002.
  3. Dembo, Amir; Kontoyiannis, Ioannis. Source coding, large deviations, and approximate pattern matching. IEEE Trans. Inform. Theory 48 (2002), no. 6, 1590--1615. MR1909475 (2003f:94039)
  4. S.Dereich. High resolution coding of stochastic processes and small ball probabilities. Ph.D. Dissertation, TU Berlin, URL: http://edocs.tu-berlin.de/diss/2003/dereich_steffen.htm, 2003.
  5. Dereich, Steffen. Asymptotic behavior of the distortion-rate function for Gaussian Bull. Sci. Math. 129 (2005), no. 10, 791--803. MR2178943 (2006i:60006)
  6. S.Dereich. The coding complexity of diffusion processes under ${L}^p[0,1]$-norm distortion. Preprint, 2006.
  7. S.Dereich. The coding complexity of diffusion processes under supremum norm distortion. Preprint, 2006.
  8. Dereich, S.; Fehringer, F.; Matoussi, A.; Scheutzow, M. On the link between small ball probabilities and the quantization J. Theoret. Probab. 16 (2003), no. 1, 249--265. MR1956830 (2004e:60013)
  9. Dereich, S.; Lifshits, M. A. Probabilities of randomly centered small balls and quantization in Ann. Probab. 33 (2005), no. 4, 1397--1421. MR2150193 (2006f:60010)
  10. S.Dereich, T.Muller-Gronbach, and K.Ritter. Infinite-dimensional quadrature and quantization. Preprint, arXiv:math.PR/0601240, 2006.
  11. Graf, Siegfried; Luschgy, Harald. Foundations of quantization for probability distributions. Lecture Notes in Mathematics, 1730. Springer-Verlag, Berlin, 2000. x+230 pp. ISBN: 3-540-67394-6 MR1764176 (2001m:60043)
  12. S.Graf, H.Luschgy, and G.Pags. Optimal quantizers for Radon random vectors in a Banach space. Preprint, 2005.
  13. Gray, Robert M.; Neuhoff, David L. Quantization. IEEE Trans. Inform. Theory 44 (1998), no. 6, 2325--2383. MR1658787 (99i:94029)
  14. Kuelbs, James; Li, Wenbo V. Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 (1993), no. 1, 133--157. MR1237989 (94j:60078)
  15. Li, W. V.; Shao, Q.-M. Gaussian processes: inequalities, small ball probabilities and 533--597, Handbook of Statist., 19, North-Holland, Amsterdam, 2001. MR1861734
  16. Luschgy, Harald; Pagès, Gilles. Functional quantization of Gaussian processes. J. Funct. Anal. 196 (2002), no. 2, 486--531. MR1943099 (2003i:60006)
  17. Luschgy, Harald; Pagès, Gilles. Sharp asymptotics of the functional quantization problem for Gaussian Ann. Probab. 32 (2004), no. 2, 1574--1599. MR2060310 (2005d:60036)
  18. G.Pagès and J.Printems. Functional quantization for pricing derivatives. Preprint, Universit de Paris VI, LPMA no. 930, 2004.
















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Electronic Journal of Probability. ISSN: 1083-6489