AI 中文总结
研究紧致度量空间上中心高斯过程的高极小值，证明在\(M > u\)条件下，缩放超调量\(u(M - u)\)收敛到指数随机变量，且极小值点条件律的弱子序列极限是最优协方差能量测度，通过多种过程举例说明。
AI 中文摘要
设\(X(t)\)，\(t\in K\)，是在紧致度量空间\(K\)上具有连续样本路径的中心高斯过程，令\(M = \min_{t\in K}X(t)\)。设\(\sigma_*^2\)表示与\(X\)相关的最小协方差能量，假设\(\sigma_*^2>0\)。受[参考文献]中关于光滑高斯过程结果的启发，我们证明，在\(M > u\)的条件下，当\(u\to\infty\)时，缩放后的超调量\(u(M - u)\)收敛到均值为\(\sigma_*^2\)的指数随机变量。此外，\(X\)的可测极小值点的条件律的每个弱子序列极限都是最优协方差能量测度。特别是，如果该测度是唯一的，那么条件律弱收敛到它。平稳高斯过程、分数布朗运动和分数布朗片说明了这些结果。
英文摘要
Let $X(t)$, $t\in K$, be a centred Gaussian process with continuous sample paths on a compact metric space $K$, and let $M=\min_{t\in K}X(t)$. Let $σ_*^2$ denote the minimum covariance energy associated with $X$, and assume that $σ_*^2>0$. Motivated by the results of \cite{chakrabarty2018asymptotic} for smooth Gaussian processes, we show that, conditionally on $M>u$, the scaled overshoot $u(M-u)$ converges, as $u\to\infty$, to an exponential random variable with mean $σ_*^2$. Moreover, every weak subsequential limit of the conditional law of a measurable minimizer of $X$ is an optimal covariance-energy measure. In particular, if this measure is unique, then the conditional law converges weakly to it. The results are illustrated by stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet.
