@article {GaussianProcessQuantization,
title = {Functional quantization of Gaussian processes},
journal = {Journal of Functional Analysis},
volume = {196},
number = {2},
year = {2002},
month = {December},
pages = {486{\textendash}531},
publisher = {Academic Press},
abstract = {Quantization consists in studying the $L^r$ -error induced by the approximation of a random vector $X$ by a vector (quantized version) taking a finite number $n$ of values. For $R^m$-valued random vectors the theory and practice is quite well established and in particular, the asymptotics as $n\to \infty$ of the resulting minimal quantization error for nonsingular distributions is well known: it behaves like $c(X,r,m)n^{-1/m}$. This paper is a transposition of this problem to random vectors in an infinite dimensional Hilbert space and in particular, to stochastic processes $(X_t)_{t \in [0,1]}$ viewed as $L^2([0,1],dt)$-valued random vectors. For Gaussian vectors and the $L^2$-error we present detailed results for stationary and optimal quantizers. We further establish a precise link between the rate problem and Shannon{\textendash}Kolmogorov{\textquoteright}s entropy of $X$ : This allows us to compute the exact rate of convergence to zero of the minimal $L^2$-quantization error under rather general conditions on the eigenvalues of the covariance operator. Typical rates are $O(\log(n)^{-a}) , a > 0$. They are obtained, for instance, for the fractional Brownian motion and the fractional Ornstein{\textendash}Uhlenbeck process. The exponent a is closely related with the $L^2$-regularity of the process.
},
keywords = {fractional Brownian motion, Gaussian process, quantization of probability distribution, Shannon{\textendash}Kolmogorov entropy, stationary processes},
issn = {0022-1236},
author = {Harald Luschgy and Gilles Pag{\`e}s}
}