Cutoff phenomenon for the maximum of a sampling of Ornstein–Uhlenbeck processes

In this article we study the so-called cut-off phenomenon in total variation distance when $n\to \infty$ for 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 $n$ ergodic Ornstein-Uhlenbeck driven stable index $\alpha$. It not hard to see that each $\mathcal{Z}^{(n)}_t$ converges limiting distribution $\mathcal{Z}^{(n)}_\infty$ as $t$ goes by. Using asymptotic theory extremes; Gaussian case prove between and its universal function constant time window around time, fact known profile context processes. On other hand, heavy-tailed there cut-off.