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Entropy-constrained functional quantization of Gaussian processes
Authors:
Siegfried Graf and Harald Luschgy
Journal:
Proc. Amer. Math. Soc. 133 (2005), 3403-3409
MSC (2000):
Primary 60G15, 94A24; Secondary 60B11, 94A34
Posted:
May 2, 2005
MathSciNet review:
2161166
Full-text PDF Free Access
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Additional Information
Abstract: The sharp asymptotics for the entropy-constrained  -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 of the covariance operator.
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problem for Gaussian processes, Ann. Probab. 32
(2004), no. 2, 1574–1599. MR 2060310
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Zador, P.L. (1966): Topics in the asymptotic quantization of continuous random variables. Bell Laboratories Technical Report.
- 1.
- Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Cambridge University Press. MR 0898871 (88i:26004)
- 2.
- Bronski, J.C. (2003). Small ball constants and tight eigenvalue asmptotics for fractional Brownian motions. J. Theoretical Probab. 16, 87-100. MR 1956822 (2004b:60105)
- 3.
- Gao, F., Hanning, J. and Torcaso, F. (2003). Integrated Brownian motions and exact
 -small balls. Ann. Probab. 31, 1320-1337. MR 1989435 (2004k:60104)
- 4.
- Gersho, A. and Gray, R.M. (1992). Vector Quantization and Signal Compression. Kluwer, Boston.
- 5.
- Graf, S. and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions. Lecture Notes in Math. 1730. Springer, Berlin. MR 1764176 (2001m:60043)
- 6.
- Gray, R.M. and Neuhoff, D.L. (1998). Quantization. IEEE Trans. Inform. Theory 44, 2325 - 2383. MR 1658787 (99i:94029)
- 7.
- Gray, R.M., Linder, T. and Li, J. (2002). A Lagrangian formulation of Zador's entropy-constrained quantization theorem. IEEE Trans. Inform. Theory 48, 695-707. MR 1889976 (2003f:94036)
- 8.
- Ihara, S. (1993). Information Theory for Continuous Systems. World Scientific, Singapore. MR 1249933 (94m:94008)
- 9.
- Karol', A.I., Nazarov, A.I. and Nikitin, Ya. Yu. (2003). Tensor products of compact operators and logarithmic -small ball asymptotics for Gaussian random fields. Preprint.
- 10.
- Luschgy, H. and Pagès, G. (2002). Functional quantization of Gaussian processes. J. Funct. Anal. 196, 486-531.MR 1943099 (2003i:60006)
- 11.
- Luschgy, H. and Pagès, G. (2004). Sharp asymptotics of the functional quantization problem for Gaussian processes. Ann. Probab. 32, 1574-1599. MR 2060310
- 12.
- Nazarov, A.I. and Nikitin, Ya. Yu. (2004). Exact
 -small ball behaviour of integrated Gaussian processes and spectral asymptotics of boundary value problems. Probab. Theory Related Fields 129, 469-494. MR 2078979
- 13.
- Nazarov, A.I. and Nikitin, Ya.Yu. (2003). Logarithmic -small ball asymptotics for some fractional Gaussian processes. Preprint.
- 14.
- Ritter, K. (2000). Average-Case Analysis of Numerical Problems. Lecture Notes in Math. 1733. Springer, Berlin.MR 1763973 (2001i:65001)
- 15.
- Rosenblatt, M. (1963). Some results on the asymptotic behaviour of eigenvalues for a class integral equations with translation kernel. J. Math. and Mechanics 12, 619-628. MR 0150551 (27:547)
- 16.
- Zador, P.L. (1963). Development and evaluation of procedures for quantizing multivariate distributions. Ph.D. dissertation, Stanford Univ.
- 17.
- Zador, P.L. (1966): Topics in the asymptotic quantization of continuous random variables. Bell Laboratories Technical Report.
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Additional Information
Siegfried Graf
Affiliation:
Fakultät für Mathematik und Informatik, Universität Passau, D-94030 Passau, Germany
Email:
graf@fmi.uni-passau.de
Harald Luschgy
Affiliation:
FB IV-Mathematik, Universität Trier, D-54286 Trier, Germany
Email:
luschgy@uni-trier.de
DOI:
http://dx.doi.org/10.1090/S0002-9939-05-07888-3
PII:
S 0002-9939(05)07888-3
Keywords:
Functional quantization,
Gaussian process,
entropy,
distortion rate function
Received by editor(s):
December 11, 2003
Received by editor(s) in revised form:
June 14, 2004
Posted:
May 2, 2005
Communicated by:
Richard C. Bradley
Article copyright:
© Copyright 2005 American Mathematical Society
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