Summary
Let X={X(t), t∈ℝ N} be a centred Gaussian random field with covariance ℰX(t)X(s)=r(t−s) continuous on ℝN×ℝN and r(0)=1. Let σ(t,s)=(ℰ(X(t)−X(s)) 2)1/2; σ(t,s) is a pseudometric on ℝN. Assume X is σ-separable. Let D 1 be the unit cube in ℝN and for 0<k∈ℝ, D k= {x∈ℝN: k −1 x∈D1}, Z(k)=sup{X(t),t∈D k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦→∼8 then Z(k)-(2Nlogk) 1/2→0 as k→∞ a.s.
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Ortega, J. Asymptotic behaviour of Gaussian random fields. Z. Wahrscheinlichkeitstheorie verw Gebiete 59, 169–177 (1982). https://doi.org/10.1007/BF00531741
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DOI: https://doi.org/10.1007/BF00531741