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Entropy-constrained functional quantization of Gaussian processes
Author(s):
Siegfried
Graf;
Harald
Luschgy
Journal:
Proc. Amer. Math. Soc.
133
(2005),
3403-3409.
MSC (2000):
Primary 60G15, 94A24;
Secondary 60B11, 94A34
Posted:
May 2, 2005
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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.
References:
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- Gao, F., Hanning, J. and Torcaso, F. (2003). Integrated Brownian motions and exact
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- 5.
- Graf, S. and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions. Lecture Notes in Math. 1730. Springer, Berlin. MR 1764176 (2001m:60043)
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- 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)
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- 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.
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- 16.
- Zador, P.L. (1963). Development and evaluation of procedures for quantizing multivariate distributions. Ph.D. dissertation, Stanford Univ.
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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:
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
Copyright of article:
Copyright
2005,
American Mathematical Society
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