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Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

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  • Steffen Dereich

    (Institut für Mathematik)

Abstract

Let μ be a centered Gaussian measure on a separable Hilbert space (E, ∥ ⋅ ∥). We are concerned with the logarithmic small ball probabilities around a μ-distributed center X. It turns out that the asymptotic behavior of −log μ(B(X,ε)) is a.s. equivalent to that of a deterministic function φ R (ε). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.(8)

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:2:d:10.1023_a:1023578812641
    DOI: 10.1023/A:1023578812641
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    References listed on IDEAS

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    1. 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.
    2. Jared C. Bronski, 2003. "Small Ball Constants and Tight Eigenvalue Asymptotics for Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 16(1), pages 87-100, January.
    Full references (including those not matched with items on IDEAS)

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