Sojourns of Vector-Valued Stationary Gaussian Random Fields
Krzysztof Dębicki, Enkelejd Hashorva, Zbigniew Michna
TL;DR
This paper develops exact asymptotics for the sojourn times of centered, vector-valued Gaussian random fields by extending Berman's approach to the multivariate setting and coupling it with a multivariate uniform double-sum method. Under regularity on the covariance structure, a local-variance representation, and the Savage condition, the authors identify a limiting Berman-type function $\mathcal{B}_{\mathbf{Y},\mathbf{w}}(x)$ that governs the tail of the scaled sojourn $\theta(u)L_u(T)$, with $\mathbf{w}=\Sigma^{-1}\mathbf{b}$ and $\mathbf{Y}$ a limit GRF with covariance $R_V$. They provide both a single-limit representation and a double-sum representation for $\mathcal{B}_{\mathbf{Y},\mathbf{w}}(x)$, and establish the corresponding exact asymptotics for the supremum of $\mathbf{X}$. The results generalize univariate high-exceedance theory to the vector-valued case and yield tractable constants (Berman functions) that capture the dependence structure of the field. This advances theoretical understanding of high-threshold sojourns in multi-parameter Gaussian fields with practical implications for occupation-time-type problems in finance and engineering.
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.