Functional Quantization and Small Ball Probabilities for Gaussian Processes

Siegfried Graf (), Harald Luschgy () and Gilles Pagès ()
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Siegfried Graf: Universität Passau
Harald Luschgy: Universität Trier
Gilles Pagès: Université Paris 6

Journal of Theoretical Probability, 2003, vol. 16, issue 4, 1047-1062

Abstract: 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. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer(4) and Dereich et al. (3) relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L r -quantization error from logarithmic small ball asymptotics and vice versa.

Keywords: optimal quantizer; asymptotic quantization error; small ball probability; Komogorov entropy; Gaussian process (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1023/B:JOTP.0000012005.32667.9d

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