Abstract
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.
Highlights
The aim of the present paper is the investigation of the small deviation behavior of Gaussian random fields in the Lq–norm taken with respect to a rather arbitrary measure on RN
For a Gaussian random field (X(t), t ∈ RN ), for a measure μ on RN, and for any q ∈ [1, ∞) we are interested in the behavior of the small deviation function φq,μ(ε) := − log P
In this article we are mostly interested in the behavior of the N –parameter fractional Brownian motion, an essential part of our estimates is valid for much more general processes
Summary
One has to introduce some kind of entropy of μ taking into account the size of its support as well as the distribution of the mass on [0, 1] This is done in the following way. − log P{ WH Lq([0,1],μ) < ε} ε−1/(H+ν) · log(1/ε)β/(H+ν) There is another quantity, a kind of inner mixed entropy, equivalent to σμ(H,q)(n) in the one–parameter case. We did not find in the literature the notions of outer and inner mixed entropy as defined above, some similar objects do exist: cf the notion of weighted Hausdorff measures investigated in [22], pp. The quantitative properties of Hausdorff dimension and entropy of a set seem to have almost nothing in common (think of any countable set — its Hausdorff dimension is zero while the entropy properties can be quite non–trivial)
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