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Functional quantization of Gaussian processes

TitleFunctional quantization of Gaussian processes
Publication TypeJournal Article
Year of Publication2002
AuthorsHarald Luschgy, and Gilles Pagès
JournalJournal of Functional Analysis
Volume196
Pagination486–531
Date PublishedDecember
ISSN0022-1236
Keywordsfractional Brownian motion, Gaussian process, quantization of probability distribution, Shannon–Kolmogorov entropy, stationary processes
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–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.