Asymptotic Behavior of Path Functionals for Vector-Valued Gaussian Processes at High Levels
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Understanding how long a stochastic system stays in a “safe” region is a core question in risk management, queueing and reliability. In this project we study high exceedance probabilities of the form \(\mathbb{P} \{ \Gamma_{[0,T]} ( \hat{\boldsymbol{u}} ( \boldsymbol{X} - u \boldsymbol{b} ) ) > L_u \},\) as $u \to \infty$, where $\Gamma_{[0,T]}$ is a functional of a continuous $d$-dimensional Gaussian process $\mathbf X(t)$ on $[0,T]$, and $L_u$ is some sequence of thresholds, chosen appropriately for each $\Gamma$. The class of functionals we treat is quite broad, including functionals of the form \(\Gamma_E ( \boldsymbol{f} ) = \int_{E} G ( \boldsymbol{f} ( t ) ) \, d t \quad \text{and} \quad \Gamma_{E \times F} ( \boldsymbol{f} ) = \sup_{t \in E} \inf_{s \in F} \min_{i = 1, \dots, d} f_i ( t, s ),\) where $G$ is some function satisfying additional assumptions. In particular, this class includes the classical sojourn time, Parisian (moving-window infimum) functional, area under the curve, as well as compositions of those with continuous but not necessarily linear operators. Regarding the class of the Gaussian processes, we study both stationary and non-stationary cases under the assumptions similar to those of Dębicki-Hashorva-Wang (2019).
Key technical contributions include the extension of Pickands-type arguments to these vector-valued settings and general functionals, supported by lemmas detailing conditional process behaviour, uniform convergence, and properties of the functionals themselves. The presentation will outline the main theorems, discuss the crucial assumptions, and illustrate the framework’s applicability with examples. This work provides a unified approach to understanding extreme sojourns for a broad class of Gaussian models.