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

Adler, Robert J.

Ann. Probab. 12, 436-444 (1984).

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

Keywords:

two-parameter, homogeneous Gaussian random field

Citations:

Zbl 0459.60045