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Asymptotically optimal quantization schemes for Gaussian processes on Hilbert spaces*

Published online by Cambridge University Press:  10 May 2010

Harald Luschgy ,
Gilles Pagès  and
Benedikt Wilbertz
Harald Luschgy
Affiliation:
Universität Trier, FB IV-Mathematik, 54286 Trier, Germany
Gilles Pagès
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Paris 6, Case Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France
Benedikt Wilbertz
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Paris 6, Case Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France
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Abstract

We describe quantization designs which lead to asymptotically and order optimal functional quantizers for Gaussian processes in a Hilbert space setting. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions.

Keywords

Functional quantizationGaussian processBrownian motionRiemann-Liouville processoptimal quantizer

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 14 , 2010 , pp. 93 - 116
Copyright
© EDP Sciences, SMAI, 2010

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Footnotes

*

This work was supported in part by the AMaMeF Exchange Grant 1323 of the ESF.

References

A. Benveniste, P. Priouret and M. Métivier, Adaptive algorithms and stochastic approximations. Springer-Verlag, New York, Inc. (1990).
Cohort, P., Limit theorems for random normalized distortion. Ann. Appl. Probab. 14 (2004) 118–143. CrossRef
S. Dereich, High resolution coding of stochastic processes and small ball probabilities. Ph.D. thesis, TU Berlin (2003).
A. Gersho and R.M. Gray, Vector Quantization and Signal Compression. Kluwer, Boston (1992).
S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions. Lect. Notes Math. 1730. Springer, Berlin (2000).
Graf, S. and Luschgy, H., The point density measure in the quantization of self-similar probabilities. Math. Proc. Cambridge Phil. Soc. 138 (2005) 513–531. CrossRef
Gray, R.M. and Neuhoff, D.L., Quantization. IEEE Trans. Inform. 44 (1998) 2325–2383. CrossRef
H.J. Kushner and G.G. Yin, Stochastic approximation algorithms and applications. First edition, volume 35 of Applications of Mathematics. Springer-Verlag, New York (1997), p. xxii+417.
Lapeyre, B., Pagès, G. and Sab, K., Sequences with low discrepancy. Generalization and application to robbins-monro algorithm. Statistics 21 (1990) 251–272. CrossRef
Luschgy, H. and Pagès, G., Functional quantization of stochastic processes. J. Funct. Anal. 196 (2002) 486–531. CrossRef
Luschgy, H. and Pagès, G., Sharp asymptotics of the functional quantization problem for Gaussian processes. Ann. Probab. 32 (2004) 1574–1599.
Luschgy, H. and Pagès, G., Sharp asymptotics of the kolmorogov entropy for Gaussian measures. J. Funct. Anal. 212 (2004) 89–120. CrossRef
M. Mrad and S. Ben Hamida, Optimal quantization: Evolutionary algorithm vs. stochastic gradient, in JCIS (2006).
Pagès, G., A space vector quantization method for numerical integration. J. Appl. Comput. Math. 89 (1997) 1–38. CrossRef
G. Pagès, H. Pham and J. Printems, Optimal quantization methods and applications to numerical methods and applications in finance, in Handbook of Computational and Numerical Methods in Finance, S. Rachev (Ed.), Birkhäuser (2004), pp. 253–298.
Pagès, G. and Printems, J., Optimal quadratic quantization for numerics: the Gaussian case. Monte Carlo Meth. Appl. 9 (2003) 135–166. CrossRef
Pagès, G. and Printems, J., Functional quantization for numerics with an application to option pricing. Monte Carlo Meth. Appl. 11 (2005) 407–446. CrossRef
G. Pagès and J. Printems, www.quantize.maths-fi.com. Website devoted to quantization (2005). maths-fi.com.
Vu, K.T. and Gorenflo, R., Asymptotics of singular values of volterra integral operators. Numer. Funct. Anal. Optimiz. 17 (1996) 453–461.