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Small tails for the supremum of a Gaussian process. (English) Zbl 0641.60044
Let T be a compact metric space. Let \((X_ t)_{t\in T}\) be a Gaussian process with continuous covariance. Assume that the variance has a unique maximum at some point \(\tau\) and that \(X_ t\) has a.s. bounded sample paths. We prove that \[ \lim_{u\to \infty}P(Sup X_ t>u)/P(X_{\tau}>u)=1 \text{ if and only if } \lim_{h\to 0} h^{- 1}E(_{t\in T_ h}(X_ t-X_{\tau}))=0 \] where \(T_ h=\{t\in T\); \(E(X_ tX_{\tau})\geq E(X^ 2_{\tau})-h^ 2\}\).

MSC:
60G15 Gaussian processes
60G17 Sample path properties
60E15 Inequalities; stochastic orderings
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