论文标题
反映高斯流程在离散网格上的极端
Extremes of Reflecting Gaussian Processes on Discrete Grid
论文作者
论文摘要
对于$ \ {x（t），t \ ing_Δ\} $ sentary增量和a.s的集中的高斯过程示例路径在离散网格上$g_Δ= \ {0，δ，2δ，... \} $，其中$δ> 0 $，我们研究了固定反射过程$$ q_ {δ，x}（x}（t）= \ sup \ sup \ sup \ limits_ in [t，\ infty） x（s）-x（t）-c（s-t）\ big），\ t \ ing_Δ$$，$ c> 0 $。我们得出了$$ \ mathbb p \ {\ sup \ limits_ {t \ in [0，t] \ capg_Δ} q_ {δ，x}（x}（x}（x}> u \} \ quad \ qued \ quat pect {and} \ quad \ quad \ quad \ fimin can） g_Δ} q_ {δ，x}（t）> u \}，$$ as $ u \ to \ infty $，$ t> 0 $。看来$φ= \ lim_ {u \ to \ infty} \ frac {σ^2（u）} {u} $确定渐近线，导致三个定性不同的方案：$φ= 0 $，$φ\ in（in（0，\ infty）$和$φ= \ uftty $。
For $\{X(t), t \in G_δ\}$ a centered Gaussian process with stationary increments and a.s. sample paths on a discrete grid $G_δ=\{0,δ,2δ, ...\}$, where $δ>0$, we investigate the stationary reflected process $$Q_{δ,X}(t) = \sup\limits_{s\in [t,\infty)\cap G_δ}\big( X(s)-X(t)-c(s-t)\big), \ t \in G_δ$$ with $c>0$. We derive the exact asymptotics of $$\mathbb P\{\sup\limits_{t\in [0,T]\cap G_δ} Q_{δ,X}(t)>u\} \quad \text{and} \quad \mathbb P\{\inf\limits_{t\in [0,T]\cap G_δ} Q_{δ,X}(t)>u\},$$ as $u\to\infty$, with $T>0$. It appears that $φ=\lim_{u\to\infty} \frac{σ^2(u)}{u}$ determines the asymptotics, leading to three qualitatively different scenarios: $φ=0$, $φ\in(0,\infty)$ and $φ=\infty$.
