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On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces

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  • S. Dereich
  • F. Fehringer
  • A. Matoussi
  • M. Scheutzow

Abstract

Let μ be a centered Gaussian measure on a separable Banach space E and N a positive integer. We study the asymptotics as N→∞ of the quantization error, i.e., the infimum over all subsets ℰ of E of cardinality N of the average distance w.r.t. μ to the closest point in the set ℰ. We compare the quantization error with the average distance which is obtained when the set ℰ is chosen by taking N i.i.d. copies of random elements with law μ. Our approach is based on the study of the asymptotics of the measure of a small ball around 0. Under slight conditions on the regular variation of the small ball function, we get upper and lower bounds of the deterministic and random quantization error and are able to show that both are of the same order. Our conditions are typically satisfied in case the Banach space is infinite dimensional.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:1:d:10.1023_a:1022242924198
    DOI: 10.1023/A:1022242924198
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    References listed on IDEAS

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    1. V. Li, Wenbo, 2001. "Small ball probabilities for Gaussian Markov processes under the Lp-norm," Stochastic Processes and their Applications, Elsevier, vol. 92(1), pages 87-102, March.
    2. David M. Mason & Zhan Shi, 2001. "Small Deviations for Some Multi-Parameter Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 14(1), pages 213-239, January.
    3. 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.
    4. Thomas Dunker, 2000. "Estimates for the Small Ball Probabilities of the Fractional Brownian Sheet," Journal of Theoretical Probability, Springer, vol. 13(2), pages 357-382, April.
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    Cited by:

    1. Steffen Dereich, 2003. "Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization," Journal of Theoretical Probability, Springer, vol. 16(2), pages 427-449, April.
    2. S. Dereich & C. Vormoor, 2011. "The High Resolution Vector Quantization Problem with Orlicz Norm Distortion," Journal of Theoretical Probability, Springer, vol. 24(2), pages 517-544, June.
    3. 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.

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