Small tails for the supremum of a Gaussian process. (English) Zbl 0641.60044

Talagrand, Michel

Ann. Inst. Henri Poincaré, Probab. Stat. 24, No. 2, 307-315 (1988).

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

Keywords:

Borell’s inequality; Gaussian process with continuous covariance; bounded sample paths