Asymptotic Behavior of Path Functionals for Vector-Valued Gaussian Processes at High Levels
We study precise asymptotics for high-level exceedance probabilities of path functionals of continuous vector-valued Gaussian processes. The probabilities have the form $$ \mathbb{P}\{\Gamma_{[0,T]}(\check{\boldsymbol{u}}(\boldsymbol{X}-u\boldsymbol{b}))>L_u\}, \qquad u\to\infty, $$ where $\boldsymbol{X}$ is a centered $\mathbb R^d$-valued Gaussian process and $\Gamma$ belongs to a broad class satisfying natural monotonicity, scaling, no-atom, and continuity assumptions. The class covers classical sojourn times, Choquet-type sojourn integrals, area-under-the-curve functionals, and, through a local-footprint extension, shrinking-window Parisian persistence functionals. We obtain exact asymptotics in the stationary case and in the non-stationary case when the inverse generalized variance has a unique minimizer at the boundary point. The non-stationary theorem covers the regimes $\alpha<\beta$, $\alpha=\beta$, and $\alpha>\beta$, which lead respectively to Pickands-type, Piterbarg-type, and deterministic limiting constants. We also derive conditional limit laws for the first exceedance time. The main claims are stated in the body of the paper, while the proofs and auxiliary estimates are collected in the appendices.