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.
Mots clés : functional quantization, gaussian process, brownian motion, Riemann-Liouville process, optimal quantizer
@article{PS_2010__14__93_0,
author = {Luschgy, Harald and Pag\`es, Gilles and Wilbertz, Benedikt},
title = {Asymptotically optimal quantization schemes for gaussian processes on {Hilbert} spaces},
journal = {ESAIM: Probability and Statistics},
pages = {93--116},
publisher = {EDP-Sciences},
volume = {14},
year = {2010},
doi = {10.1051/ps:2008026},
mrnumber = {2654549},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2008026/}
}
TY - JOUR AU - Luschgy, Harald AU - Pagès, Gilles AU - Wilbertz, Benedikt TI - Asymptotically optimal quantization schemes for gaussian processes on Hilbert spaces JO - ESAIM: Probability and Statistics PY - 2010 SP - 93 EP - 116 VL - 14 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2008026/ DO - 10.1051/ps:2008026 LA - en ID - PS_2010__14__93_0 ER -
%0 Journal Article %A Luschgy, Harald %A Pagès, Gilles %A Wilbertz, Benedikt %T Asymptotically optimal quantization schemes for gaussian processes on Hilbert spaces %J ESAIM: Probability and Statistics %D 2010 %P 93-116 %V 14 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2008026/ %R 10.1051/ps:2008026 %G en %F PS_2010__14__93_0
Luschgy, Harald; Pagès, Gilles; Wilbertz, Benedikt. Asymptotically optimal quantization schemes for gaussian processes on Hilbert spaces. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 93-116. doi : 10.1051/ps:2008026. http://www.numdam.org/articles/10.1051/ps:2008026/
[1] , and , Adaptive algorithms and stochastic approximations. Springer-Verlag, New York, Inc. (1990). | Zbl
[2] , Limit theorems for random normalized distortion. Ann. Appl. Probab. 14 (2004) 118-143. | Zbl
[3] , High resolution coding of stochastic processes and small ball probabilities. Ph.D. thesis, TU Berlin (2003).
[4] and , Vector Quantization and Signal Compression. Kluwer, Boston (1992). | Zbl
[5] and , Foundations of Quantization for Probability Distributions. Lect. Notes Math. 1730. Springer, Berlin (2000). | Zbl
[6] and , The point density measure in the quantization of self-similar probabilities. Math. Proc. Cambridge Phil. Soc. 138 (2005) 513-531. | Zbl
[7] and , Quantization. IEEE Trans. Inform. 44 (1998) 2325-2383. | Zbl
[8] and , Stochastic approximation algorithms and applications. First edition, volume 35 of Applications of Mathematics. Springer-Verlag, New York (1997), p. xxii+417. | Zbl
[9] , and , Sequences with low discrepancy. Generalization and application to robbins-monro algorithm. Statistics 21 (1990) 251-272. | Zbl
[10] and , Functional quantization of stochastic processes. J. Funct. Anal. 196 (2002) 486-531. | Zbl
[11] and , Sharp asymptotics of the functional quantization problem for Gaussian processes. Ann. Probab. 32 (2004) 1574-1599. | Zbl
[12] and , Sharp asymptotics of the kolmorogov entropy for Gaussian measures. J. Funct. Anal. 212 (2004) 89-120. | Zbl
[13] and , Optimal quantization: Evolutionary algorithm vs. stochastic gradient, in JCIS (2006).
[14] , A space vector quantization method for numerical integration. J. Appl. Comput. Math. 89 (1997) 1-38. | Zbl
[15] , and , 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. | Zbl
[16] and , Optimal quadratic quantization for numerics: the Gaussian case. Monte Carlo Meth. Appl. 9 (2003) 135-166. | Zbl
[17] and , Functional quantization for numerics with an application to option pricing. Monte Carlo Meth. Appl. 11 (2005) 407-446. | Zbl
[18] and , www.quantize.maths-fi.com. Website devoted to quantization (2005). maths-fi.com.
[19] and , Asymptotics of singular values of volterra integral operators. Numer. Funct. Anal. Optimiz. 17 (1996) 453-461. | Zbl
Cité par Sources :
