Sojourns of Vector-Valued Stationary Gaussian Random Fields

2025/10/08 by Dębicki, Krzysztof, Hashorva, Enkelejd, Michna, Zbigniew
#FOS: Mathematics #Primary 60G15 #Probability (math.PR) #secondary 60G70

paper · doi:10.48550/arxiv.2510.06930

Abstract

For a centered, homogeneous R^d-valued Gaussian random field X(t), t in R^k, with covariance matrix function R(s,t) = E[X(s) X(t)^T], we investigate the exact asymptotics of kappa_u(x) = P( theta(u) * integral over [0,T]^k of 1X(t) > u b dt > x ), where b = (b1, ..., bd)^T, as u -> infinity, with x >= 0 and T > 0, and theta(u) is a scaling function related to the expansion of R(s,t) around (0,0). To approximate kappa_u(x), we extend both Berman's original approach and the uniform double-sum method to the multivariate setting. Furthermore, we derive the exact asymptotics for the supremum of X, thus extending several recent results in the literature.

Citations

• High-level moving excursions for spatiotemporal Gaussian random fields with long range dependence
• Sojourns of fractional Brownian motion queues: transient asymptotics
• On Berman functions
• Extremes of Vector-Valued Gaussian Processes
• Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view
• Basic properties of the Multivariate Fractional Brownian Motion

Related

• Sojourn times of Gaussian related random fields
• Tail Asymptotics for the Extremes of Bivariate Gaussian Random Fields
• Extremes of homogeneous Gaussian random fields
• Extremes of multidimensional stationary Gaussian random fields
• Asymptotic covariances for functionals of weakly stationary random fields