论文信息 - Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

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

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 φR(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)

S. Dereich

[1] de Ng Dick Bruijn. Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform. , 1959 .

[2] R. Dudley,et al. On the Lower Tail of Gaussian Seminorms , 1979 .

[3] A. Acosta. Small Deviations in the Functional Central Limit Theorem with Applications to Functional Laws of the Iterated Logarithm , 1983 .

[4] J. Kuelbs,et al. The Gaussian measure of shifted balls , 1994 .

[5] Amir Dembo,et al. Large Deviations via Parameter Dependent Change of Measure, and an Application to the Lower Tail of Gaussian Processes , 1995 .

[6] M. Lifshits. Gaussian Random Functions , 1995 .

[7] Mikhail Lifshits,et al. On the lower tail probabilities of some random series , 1997 .

[8] Amir Dembo,et al. Large Deviations Techniques and Applications , 1998 .

[9] S. Graf,et al. Foundations of Quantization for Probability Distributions , 2000 .

[10] Amir Dembo,et al. Source coding, large deviations, and approximate pattern matching , 2001, IEEE Trans. Inf. Theory.

[11] Gilles Pagès,et al. Functional quantization of Gaussian processes , 2002 .

[12] Jan Hannig,et al. Integrated Brownian motions and exact L2-small balls , 2003 .

[13] Jared C. Bronski,et al. Small Ball Constants and Tight Eigenvalue Asymptotics for Fractional Brownian Motions , 2003 .

[14] Michael Scheutzow,et al. On the Link Between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces , 2003 .