Functional quantization of Gaussian processes

Quantization consists in studying the Lr-error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. For Rm-valued random vectors the theory and practice is quite well established and in particular, the asymptotics as n→∞ 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 (Xt)t∈[0,1] viewed as L2([0,1],dt)-valued random vectors. For Gaussian vectors and the L2-error we present detailed results for stationary and optimal quantizers. We further establish a precise link between the rate problem and Shannon–Kolmogorovʹs entropy of X. This allows us to compute the exact rate of convergence to zero of the minimal L2-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 L2-regularity of the process.