Sojourns of Vector-Valued Stationary Gaussian Random Fields
Probability
2025-10-09 v1
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 1{X(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.
Cite
@article{arxiv.2510.06930,
title = {Sojourns of Vector-Valued Stationary Gaussian Random Fields},
author = {Krzysztof Dębicki and Enkelejd Hashorva and Zbigniew Michna},
journal= {arXiv preprint arXiv:2510.06930},
year = {2025}
}