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Functional Quantization and Small Ball Probabilities for Gaussian Processes

Author

Listed:
  • Siegfried Graf

    (Universität Passau)

  • Harald Luschgy

    (Universität Trier)

  • Gilles Pagès

    (Université Paris 6)

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.

Suggested Citation

  • Siegfried Graf & Harald Luschgy & Gilles Pagès, 2003. "Functional Quantization and Small Ball Probabilities for Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 16(4), pages 1047-1062, October.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:4:d:10.1023_b:jotp.0000012005.32667.9d
    DOI: 10.1023/B:JOTP.0000012005.32667.9d
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    References listed on IDEAS

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    1. Wenbo V. Li & Qi-Man Shao, 1999. "Small Ball Estimates for Gaussian Processes under Sobolev Type Norms," Journal of Theoretical Probability, Springer, vol. 12(3), pages 699-720, July.
    2. S. Dereich & F. Fehringer & A. Matoussi & M. Scheutzow, 2003. "On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces," Journal of Theoretical Probability, Springer, vol. 16(1), pages 249-265, January.
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