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
Date PublishedDecember
Keywordsfractional Brownian motion, Gaussian process, quantization of probability distribution, Shannon–Kolmogorov entropy, stationary processes

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.