The supremum of a particular Gaussian field.(English)Zbl 0541.60034

Let X(t), $$t=(t_ 1,t_ 2)$$, be a real valued, two-parameter, homogeneous Gaussian random field with zero mean and covariance function given by $$R_{\tau}(s,t)=E\{X(s)X(t)\}= (1-| t_ 1-s_ 1| /| \tau_ 1|)^+(1-| t_ 2-S_ 2| /| \tau_ 2|)^+,\quad M(X,T)=\max \{X(t):0\leq t_ i\leq T_ i, i=1,2\}$$. The author gives an upper bound for the probability $$P\{M(X,T)>u\}$$, which is more exact than the one by E. M. Cabaña and W. Wschebor [J. Appl. Probab. 18, 536-541 (1981; Zbl 0459.60045)]. In the case $$T=\tau =(1,1)$$ a lower bound for the same probability is also given.
Reviewer: A.Gushchin

MSC:

 60G15 Gaussian processes 60G60 Random fields 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 60G17 Sample path properties

Zbl 0459.60045
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