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Small deviations for some multi-parameter Gaussian processes. (English) Zbl 0982.60024

Given \(\alpha=(\alpha_1,\ldots,\alpha_d)\) with \(0<\alpha_j<2\), there exists a centered Gaussian process \((X^\alpha(t))_{t\in[0,1]^d}\) with continuous paths and covariance function \[ \mathbb E X^\alpha(t) X^\alpha(s) = \prod_{j=1}^d\frac{ s_j^{\alpha_j}+ t_j^{\alpha_j} -|s_j-t_j|^{\alpha_j}}{2} . \tag{1} \] As usual, this process is called \(\alpha\)-fractional Brownian sheet. One of the most interesting (widely open) questions is to determine the small ball behaviour of \(X^\alpha\) w.r.t. the uniform norm, i.e. the behaviour of \(-\log \mathbb P(\sup_{t\in[0,1]^d}|X^\alpha(t)|<\varepsilon)\) as \(\varepsilon\to 0\). For example, if \(d=2\) and \(\alpha=(1,1)\), M. Talagrand [Ann. Probab. 22, No. 2, 1331-1354 (1994; Zbl 0835.60031)] proved that (1) behaves like \(\varepsilon^{-2}(\log(1/\varepsilon))^3\). The aim of the present paper is to investigate the behaviour of (1) for \(\alpha\)’s possessing a unique minimum \(\gamma\). Then it turns out that the small ball behaviour of \(X^\alpha\) coincides with that of the (one-dimensional) \(\gamma\)-fractional Brownian motion, i.e. (1) behaves in this case like \(\varepsilon^{-2/\gamma}\). The authors show by example how to use such results to compute the Hausdorff dimension of some exceptional sets determined by maximal increments.
Reviewer’s remark: In the forthcoming paper “Small ball probabilities of fractional Brownian sheets via fractional integration operators” by E. Belinsky and the reviewer an alternative proof of the main result in the present article is given. Moreover, Talagrand’s theorem for \(d=2\) and \(\alpha=(1,1)\) could be extended to \(\alpha=(\gamma,\gamma)\) with \(0<\gamma<2\).

MSC:

60G15 Gaussian processes
60G60 Random fields
28A78 Hausdorff and packing measures

Citations:

Zbl 0835.60031
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