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The small ball problem for the Brownian sheet. (English) Zbl 0835.60031

Let \(B_{s,t}\), \(s,t \in \mathbb{R}^+\), be a Brownian sheet, i.e. the centered Gaussian process with continuous realizations and covariance \(EB_{s,t} B_{s',t'} = \min (s,s') \min (t,t')\). Denote by \(\lambda\) the Lebesgue measure on \([0,1]^2\), by \(|\cdot |_2\) the usual norm in \(L^2 (\lambda)\), and by \(|\cdot |_{\psi_\alpha}\) the Orlicz norm \[ |f |_{\psi_\alpha} = \inf \Bigl \{ c > 0 : \int \exp \bigl \{ |f |^\alpha/c \bigr \} d \lambda \leq 2 \Bigr\}, \quad \alpha \geq 2. \] The following results are obtained: For a universal constant \(C\) and \(0 < \varepsilon \leq 1/2\), \[ \exp \left \{ - {C \over \varepsilon^2} \left( \log {1 \over \varepsilon} \right)^3 \right \} \leq P \Bigl[ \sup_{0 \leq s,t \leq 1} |B_{s,t} |\leq \varepsilon \Bigr] \leq \exp \left \{ - {1 \over C \varepsilon^2} \left( \log {1 \over \varepsilon} \right)^3 \right \}, \]
\[ P \bigl[ |B_{s,t} |_2 \leq \varepsilon \bigr] \leq \exp \left \{ - {C \over \varepsilon^2} \left( \log {1 \over \varepsilon} \right)^2 \right\}, \] and, given \(2 \leq \alpha < \infty\), there exists a constant \(K (\alpha)\) such that for all \(0 < \varepsilon \leq 1/2\), \[ \left \{ - {K (\alpha) \over \varepsilon^2} \left( \log {1 \over \varepsilon} \right)^{3 - 2/ \alpha} \right \} \leq P \bigl[ |B_{s,t} |_{\psi_\alpha} \leq \varepsilon \bigr] \leq \left \{- {1 \over K (\alpha) \varepsilon^2} \left( \log {1 \over \varepsilon} \right)^{3 - 2/ \alpha} \right\}. \] The most interesting part of the article, namely the proof of upper bounds for sup and Orlicz norms, is based on some combinatorial estimates for linear operations formed by means of a special orthogonal system in \(L^2 (\lambda)\) similar to Haar functions. The results for sup norm are also generalized to the case of Gaussian processes \(B^\mu_{s,t}\) such that \[ EB^\mu_{s,t} B^\mu_{s',t'} = \mu \biggl \{ \bigl[ 0, \min (s,s') \bigr] \times \bigl[ 0, \min (t,t') \bigr] \biggr\}, \] where \(\mu\) is a positive measure on \([0,1]^2\).

MSC:

60G17 Sample path properties
60G15 Gaussian processes
60E15 Inequalities; stochastic orderings
60B11 Probability theory on linear topological spaces
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46A35 Summability and bases in topological vector spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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