Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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