arXiv 1907.076182019-07math.PRmath.STstat.TH

Ornstein-Uhlenbeck过程采样最大值的截断现象

Cut-off phenomenon for the maximum of a sampling of Ornstein-Uhlenbeck processes

Gerardo Barrera

备注：: 12 pages
发表信息：: Statistics and Probability Letters 2021, Volume 168, 108954
DOI：: 10.1016/j.spl.2020.108954

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AI总结：

本文研究由α指数稳定过程驱动的n个遍历Ornstein-Uhlenbeck过程采样的最大值过程的截断现象，证明高斯情形下存在轮廓截断、重尾情形下无截断。

摘要：

本文研究了当$n\to \infty$时，以$n\in \mathbb{N}$为索引的连续时间随机过程族在全变差距离下的所谓截断现象，该过程族为\\[ \left( \mathcal{Z}^{(n)}_t= \max\limits_{j\in \{1,\ldots,n\}}{X^{(j)}_t}:t\geq 0\right), \\]其中$X^{(1)},\ldots,X^{(n)}$是由指数为$α$的稳定过程驱动的$n$个遍历Ornstein-Uhlenbeck过程的采样。不难看出，对每个$n\in \mathbb{N}$，当$t$趋于无穷时，$\mathcal{Z}^{(n)}_t$在全变差距离下收敛于极限分布$\mathcal{Z}^{(n)}_\infty$。利用极值渐近理论，在高斯情形下我们证明，$\mathcal{Z}^{(n)}_t$的分布与其极限分布$\mathcal{Z}^{(n)}_\infty$之间的全变差距离，在截断时间附近的恒定时间窗内收敛于一个通用函数，这一事实在随机过程语境中被称为轮廓截断。另一方面，在重尾情形下我们证明不存在截断现象。

英文摘要：

In this article we study the so-called cut-off phenomenon in the total variation distance when $n\to \infty$ for the family of continuous-time stochastic processes indexed by $n\in \mathbb{N}$, \[ \left( \mathcal{Z}^{(n)}_t= \max\limits_{j\in \{1,\ldots,n\}}{X^{(j)}_t}:t\geq 0\right), \] where $X^{(1)},\ldots,X^{(n)}$ is a sampling of $n$ ergodic Ornstein-Uhlenbeck processes driven by stable processes of index $α$. It is not hard to see that for each $n\in \mathbb{N}$, $\mathcal{Z}^{(n)}_t$ converges in the total variation distance to a limiting distribution $\mathcal{Z}^{(n)}_\infty$ as $t$ goes by. Using the asymptotic theory of extremes; in the Gaussian case we prove that the total variation distance between the distribution of $\mathcal{Z}^{(n)}_t$ and its limiting distribution $\mathcal{Z}^{(n)}_\infty$ converges to a universal function in a constant time window around the cut-off time, a fact known as profile cut-off in the context of stochastic processes. On the other hand, in the heavy-tailed case we prove that there is not cut-off.

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