Small deviations of gaussian random fields in lq–spaces

Mikhail Lifshits, Werner Linde, Zhan Shi

Research output

4 Citations (Scopus)


We investigate small deviation properties of Gaussian random fields in the space Lq(ℝN, μ) where μ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby “thin” measures μ, 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 μ, called mixed entropy, characterizing size and regularity of μ. For the particularly interesting case of self–similar measures μ, 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 Lq(ℝN, μ)– 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 Hölder operators.

Original languageEnglish
Pages (from-to)1204-1233
Number of pages30
JournalElectronic Journal of Probability
Publication statusPublished - 1 Jan 2006

Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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