Extremes of multidimensional stationary Gaussian random fields

2016/10/10 by Soja-Kukieła, Natalia
#60G15 (Primary) #60G60 (Secondary) #60G70 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1610.02888

Abstract

Let \X(t):t=(t₁, t₂, …, t_d)∈[0,∞)^d\ be a centered stationary Gaussian field with almost surely continuous sample paths, unit variance and correlation function r satisfying conditions r(t)<1 for every t≠ 0 and r(t)=1-∑_i=1^d |t_i|^α_i + o(∑_i=1^d |t_i|^α_i), as t→0, with constants α₁, α₂, …, α_d ∈(0,2]. The main result of this contribution is the description of the asymptotic behaviour of P(sup\X(t): t\inJ^x_m \\leqslant u), as u→∞, for some Jordan-measurable sets J^x_m of volume proportional to P(sup\X(t):t∈[0,1]^d\>u)^-1(1+o(1)).

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