Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s))²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D₁ be the unit cube in ^N and for 0<k, D_k= {x^N: k^–1xD₁}, Z(k)=sup{X(t),tD_k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk)^1/20 as k a.s.