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

In this article we study the so-called cutoff phenomenon in the total variation distance when n→∞ for the family of continuous-time stochastic processes indexed by n∈N, Ztn≔maxj∈{1,…,n}Xtjt≥0,where X1,…,Xn is a sample of n ergodic Ornstein–Uhlenbeck processes driven by stable noise of index α. It is not hard to see that for each n∈N, Ztn converges in the total variation distance to a limiting distribution Z∞n, 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 Ztn and its limiting distribution Z∞n converges to a universal function in a constant time window around the cutoff time, a fact known as profile cutoff in the context of stochastic processes. On the other hand, in the heavy-tailed case we prove that there is not cutoff.