Small Deviations of Gaussian Random Fields in Lq -spaces Mikhail Lifshits, Werner Linde and Zhan Shi 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 H¨older operators.
Small Deviations of Gaussian Random Fields in Lq-spaces Mikhail ...
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 âthinâ measures µ, i.e., those which are singular with respect to the Nâ dimensional Lebesgue measure; the soâcalled selfâsimilar ...
Curse of Dimensionality in Approximation of Random Fields. Mikhail Lifshits and Ekaterina Tulyakova. Consider a random field of tensor product type X(t),t â [0 ...
F. Aurzada, I. Ibragimov, M. Lifshits, and J.H. van Zanten. We investigate the small deviation probabilities of a class of very smooth stationary Gaussian ...
In 1986 V. M. Zolotarev announced an explicit description of the behavior of P(â. â j=1 Ï(j)ξ2 j ⤠r) with Ï decreasing and logarithmically convex. We show that ...
discrete white noise type initial data, and describe functional large deviations for ... tion interval, momentum, energy, etc, our results extend those of. Ryan and ...
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Our statistics of interest are the maximum of a random field G, resulting from the .... by a postdoctoral fellowship from the ISM-CRM, Montreal, Quebec, Canada. 1.
We propose here to adapt the FieldSim package to conditional simulations. Definitions and ..... Anisotropic analysis of some Gaussian models. Journal of Fourier ...
The fractional Brownian motion (fBm), introduced by Kolmogorov (1940) (and developed by. Mandelbrot and Van Ness 1968) is nowadays widely used to model ...
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Statistical Parametric Mapping (SPM) for these two situations be developed separately. In ... Mapping (SPM) approach based on Random Field Theory (RFT).
synthetic data set to discuss the conditions under which higher order features ..... In our experiment, we used the Automatic Content Extraction (ACE) data [9], ...
on a segmentation approach to be (1) robust to noise, (2) able to handle large variances ... cation [24]. Recent years have seen the emergence of Conditional Random Fields .... The Dice index measures the degree of spatial overlap between ...