Estimates for the small ball probabilities of the fractional Brownian sheet
Given a stochastic process \(X=(X(t))_{t\in T}\) over an index set \(T\) which is a.s.~bounded, the small ball problem for \(X\) (in the log-level) is to determine the behaviour of \(-\log(\sup_{t\in T}|X(t)|<e)\) as \(e\to 0\). Let \(B_d^\gamma\), \(0<\gamma<2\), be the \(d\)-dimensional fractional Brownian sheet indexed by the unit cube \([0,1]^d\) in \(\mathbb R^d\). Only few results are known for the exact small ball behaviour of this process. \textit{M. Talagrand} [Ann. Probab. 22, No. 3, 1331-1354 (1994; Zbl 0835.60031)] described the small ball behaviour of \(B_2^1\) while the general two-dimensional case, i.e.~for \(B_d^\gamma\), \(0<\gamma<2\), was recently solved by \textit{E. Belinsky} and the reviewer [to appear J. Theor.~Probab.]. The aim of the paper under review is to prove lower and upper estimates of small ball probabilities for certain Gaussian fields on \([0,1]^d\), including \(B_d^\gamma\). These estimates are quite sharp and differ only by some power of \(\log(1/e)\). This extends former results by \textit{T. Dunker, W. Linde, T. Kühn} and \textit{M. A. Lifshits} [J. Approximation Theory 101, No. 1, 63-77 (1999; Zbl 0945.47022)] in the case of \(B_d^1\). Unfortunately, the proof of Proposition 1 in the present paper rests upon a wrong argument. But it may be corrected by putting some additional assumptions onto the correlation functions of the Gaussian process (correction will appear soon). Since these additional assumptions hold for \(B_d^\gamma\), this does not influence the main estimates for the Brownian sheet.
scientific article; zbMATH DE number 1512524