AI 中文总结
研究连续向量值高斯过程路径泛函高水平超越概率渐近性，给出平稳及特定非平稳情况精确渐近性，推导首次超越时间条件极限定律，涵盖多种泛函类型。
AI 中文摘要
我们研究连续向量值高斯过程路径泛函高水平超越概率的精确渐近性。这些概率具有形式\(\mathbb{P}\{\Gamma_{[0,T]}(\check{\boldsymbol{u}}(\boldsymbol{X}-u\boldsymbol{b}))>L_u\}\)，当\(u\to\infty\)时，其中\(\boldsymbol{X}\)是中心\(\mathbb{R}^d\)值高斯过程，\(\Gamma\)属于满足自然单调性、缩放、无原子和连续性假设的广泛类。该类涵盖经典逗留时间、Choquet型逗留积分、曲线下面积泛函等。我们在平稳情况以及逆广义方差在边界点有唯一极小值的非平稳情况下获得精确渐近性。还推导了首次超越时间的条件极限定律。主要结论在正文中陈述，证明和辅助估计在附录中收集。
英文摘要
We study precise asymptotics for high-level exceedance probabilities of path functionals of continuous vector-valued Gaussian processes. The probabilities have the form $$ \mathbb{P}\{Γ_{[0,T]}(\check{\boldsymbol{u}}(\boldsymbol{X}-u\boldsymbol{b}))>L_u\}, \qquad u\to\infty, $$ where $\boldsymbol{X}$ is a centered $\mathbb R^d$-valued Gaussian process and $Γ$ 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 $α<β$, $α=β$, and $α>β$, 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.
