The Annals of Probability

Sharp asymptotics of the functional quantization problem for Gaussian processes

Harald Luschgy and Gilles Pagès

Full-text: Open access


The sharp asymptotics for the L2-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.

Article information

Ann. Probab., Volume 32, Number 2 (2004), 1574-1599.

First available in Project Euclid: 18 May 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E99: None of the above, but in this section 60G15: Gaussian processes 94A24: Coding theorems (Shannon theory) 94A34: Rate-distortion theory

High-resolution quantization product quantization Shannon entropy Gaussian process


Luschgy, Harald; Pagès, Gilles. Sharp asymptotics of the functional quantization problem for Gaussian processes. Ann. Probab. 32 (2004), no. 2, 1574--1599. doi:10.1214/009117904000000324.

Export citation


  • Berger, T. (1971). Rate Distortion Theory. Prentice-Hall, Englewood Cliffs, NJ.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Binia, J. (1974). On the $\varepsilon$-entropy of certain Gaussian processes. IEEE Trans. Inform. Theory 20 190--196.
  • Bronski, J. C. (2003). Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions. J. Theoret. Probab. 16 87--100.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Dereich, S. (2003). Small ball probabilities around random centers of Gaussian measures and applications to quantization. J. Theoret. Probab. 16 427--449.
  • Dereich, S., Fehringer, F., Matoussi, A. and Scheutzow, M. (2003). On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces. J. Theoret. Probab. 16 249--265.
  • Donoho, D. L. (2000). Counting bits with Kolmogorov and Shannon. Technical Report 38, Stanford Univ.
  • Freedman, D. (1999). On the Bernstein-von Mises theorem with infinite--dimensional parameters. Ann. Statist. 27 1119--1140.
  • Gao, F., Hanning, J. and Torcaso, F. (2003). Integrated Brownian motions and exact $L_2$-small balls. Ann. Probab. 31 1320--1337.
  • Gersho, A. and Gray, R. M. (1992). Vector Quantization and Signal Compression. Kluwer, Boston.
  • Graf, S. and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions. Lecture Notes in Math. 1730. Springer, Berlin.
  • Graf, S., Luschgy, H. and Pagès, G. (2003). Functional quantization and small ball probabilities for Gaussian processes. J. Theoret. Probab. 16 1047--1062.
  • Gray, R. M. and Neuhoff, D. L. (1998). Quantization. IEEE Trans. Inform. Theory 44 2325--2383.
  • Ihara, S. (1970). On $\ve$-entropy of equivalent Gaussian processes. Nagoya Math. J. 37 121--130.
  • Ihara, S. (1993). Information Theory. World Scientific, Singapore.
  • Kolmogorov, A. N. (1956). On the Shannon theory of information transmission in the case of continuous signals. IRE Trans. Inform. Theory 2 102--108.
  • Luschgy, H. and Pagès, G. (2002). Functional quantization of Gaussian processes. J. Funct. Anal. 196 486--531.
  • Papageorgiou, A. and Wasilkowski, G. W. (1990). On the average complexity of multivariate problems. J. Complexity 6 1--23.
  • Ritter, K. (2000). Average-Case Anslysis of Numerical Problems. Lecture Notes in Math. 1733. Springer, Berlin.
  • Rosenblatt, M. (1963). Some results on the asymptotic behavior of eigenvalues for a class of integral equations with translation kernel. J. Math. Mech. 12 619--628.
  • Rosenfeld, A. and Kak, A. C. (1976). Digital Picture Processing. Academic Press, New York.
  • Shannon, C. E. and Weaver, W. (1949). The Mathematical Theory of Communication. Univ. Illinois Press, Urbana.
  • Stein, M. L. (1999). Interpolation of Spatial Data. Springer, New York.