Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Functional quantization of Gaussian processes: a first approach

Auteur(s):

Code(s) de Classification MSC:

• 60E99 None of the above but in this section
• 60G15 Gaussian processes
• 94A24 Coding theorems (Shannon theory)
• 94A34 Rate-distortion theory

Résumé: 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 $\real^m$-valued random vectors the theory and practice is quite well established and in particular, the asymptotics as $n \rightarrow \infty$ 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 Kolmogorov'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-Uhlenbeck process. The exponent $a$ is closely related with the $L^2$-regularity of the process.

Mots Clés: Quantization of probability distribution ; Gaussian process ; Kolmogorov entropy ; Fractional Brownian Motion ; stationary processes

Date: 2001-12-06

Prépublication numéro: PMA-700

Pdf file : PMA-700.pdf