Electronic Journal of Probability

Small Deviations of Gaussian Random Fields in $L_q$-Spaces

Mikhail Lifshits, Werner Linde, and Zhan Shi

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We investigate small deviation properties of Gaussian random fields in the space $L_q(R^N,\mu)$ where $\mu$ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby "thin" measures $\mu$, i.e., those which are singular with respect to the $N$--dimensional Lebesgue measure; the so-called self-similar measures providing a class of typical examples. For a large class of random fields (including, among others, fractional Brownian motions), we describe the behavior of small deviation probabilities via numerical characteristics of $\mu$, called mixed entropy, characterizing size and regularity of $\mu$. For the particularly interesting case of self-similar measures $\mu$, the asymptotic behavior of the mixed entropy is evaluated explicitly. As a consequence, we get the asymptotic of the small deviation for $N$-parameter fractional Brownian motions with respect to $L_q(R^N,\mu)$-norms. While the upper estimates for the small deviation probabilities are proved by purely probabilistic methods, the lower bounds are established by analytic tools concerning Kolmogorov and entropy numbers of Holder operators.

Article information

Electron. J. Probab., Volume 11 (2006), paper no. 46, 1204-1233.

Accepted: 8 December 2006
First available in Project Euclid: 31 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 28A80: Fractals [See also 37Fxx] 47B06: Riesz operators; eigenvalue distributions; approximation numbers, s- numbers, Kolmogorov numbers, entropy numbers, etc. of operators

Gaussian random fields fractional Brownian motion fractal measures Kolmogorov numbers metric entropy

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Lifshits, Mikhail; Linde, Werner; Shi, Zhan. Small Deviations of Gaussian Random Fields in $L_q$-Spaces. Electron. J. Probab. 11 (2006), paper no. 46, 1204--1233. doi:10.1214/EJP.v11-379. https://projecteuclid.org/euclid.ejp/1464730581

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