Small ball estimates for the fractional brownian sheet

We consider the d-dimensional fractional Brownian sheet, which is the centered Gaussian process $(B_x)_{x\in [0,1]^d$ over the d-dimensional unit cube with covariance ${\bf E} B_x B_y =\prod_{j=1}^d (|x_j|^\gamma + |y_j|^\gamma - |x_j-y_j|^\gamm)/2$ where the parameter $\gamma$ takes values in the interval $(0,2)$. We investigate the asymptotic behaviour of the logarithm of their small ball probabilities under different norms. For H\"older-type and Orlicz norms we obtain upper bounds and we cite known lower bounds both of polynomial order. The gap that still remains is of logarithmic order and does not depend on the dimension d. This improves known results. The key point of our method is the link between t he asymptotic behavior of the small ball probabilities of Gaussian processes and the metric entropy of the unit balls of their reproducing kernel Hilbert spaces which has been discovered by J.~Kuelbs and W.~V.~Li in 1993. In addition we obtain estimates for Kolmogorov and entropy numbers of embeddings of Bessel potential spaces with dominating mixed smoothness. For $d=2$ we present an easier proof for a result of M.~Talagrand which is used to show lower bounds for the small ball beha vior of the Brownian sheet under the sup-norm.

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Dunker, Thomas: Small ball estimates for the fractional brownian sheet. 1999.

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