论文标题
多维高斯过程的极端间隔
Extrema of multi-dimensional Gaussian processes over random intervals
论文作者
论文摘要
本文研究了多维高斯过程的超极端的关节尾部渐近级，以随机的间隔定义为 $ P（U）：= \ Mathbb {p} \ left \ {\ cap_ {i = 1}^n \ left（\ sup_ {\ sup_ {t \ in [0，\ Mathcal {t} _i]} u \ to \ infty，$$ 其中$ x_i（t），t \ ge0 $，$ i = 1,2，\ cdots，n，$是独立的以平稳增量为中心的高斯流程，$ \ boldsymbol {\ Mathcal {t Mathcal {t}} =（\ Mathcal {\ Mathcal {\ Mathcal {t}组件，独立于高斯进程，而$ c_i \ in \ mathbb {r} $，$ a_i> 0 $，$ i = 1,2，\ cdots，n $。我们的结果表明，$ p（u）$的渐近学结构取决于漂移$ c_i $的标志。我们还讨论了相关的多维再生模型，并得出相应的毁灭概率。
This paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as $$ P(u):=\mathbb{P}\left\{\cap_{i=1}^n \left(\sup_{t\in[0,\mathcal{T}_i]} ( X_{i}(t) +c_i t )>a_i u \right)\right\}, \ \ \ u\to\infty, $$ where $X_i(t), t\ge0$, $i=1,2,\cdots,n,$ are independent centered Gaussian processes with stationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \cdots, \mathcal{T}_n)$ is a regularly varying random vector with positive components, which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$, $a_i>0$, $i=1,2,\cdots,n$. Our result shows that the structure of the asymptotics of $P(u)$ is determined by the signs of the drifts $c_i$'s. We also discuss a relevant multi-dimensional regenerative model and derive the corresponding ruin probability.
