On the cut-off phenomenon for the maximum of a sampling of Ornstein-Uhlenbeck processes
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In the article we study the so-called cut-off phenomenon in the total variation distance when n tends to infinity for the family of continuous-time stochastic processes indexed by n∈N, ( Z^(n)_t= _j∈{1,...,n}X^(j)_t:t≥ 0), where X^(1),...,X^(n) is a sampling of n ergodic Ornstein-Uhlenbeck processes driven by an α-stable processes. It is not hard to see that for each n∈N, the random variable Z^(n)_t 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 Z^(n)_t and Z^(n)_∞ 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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