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

Abstract

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 $\alpha$. 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.