Abstract 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 , Z t n ≔ max j ∈ { 1 , … , n } X t j t ≥ 0 , where X 1 , … , X n 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 , Z t n 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 Z t n 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.

中文翻译：

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Ornstein-Uhlenbeck 过程采样最大值的截止现象

摘要 在本文中，我们研究了由 n ∈ N , Z tn ≔ max j ∈ { 1 , … , n } X 索引的连续时间随机过程族在 n → ∞ 时总变异距离中的所谓截止现象tjt ≥ 0 ，其中 X 1 , … , X n 是由指数 α 的稳定噪声驱动的 n 遍历 Ornstein-Uhlenbeck 过程的样本。不难看出，对于每个 n ∈ N，随着 t 的推移，Z tn 在总变化距离上收敛到一个极限分布 Z ∞ n 。使用极值渐近理论；在高斯情况下，我们证明 Z tn 的分布与其极限分布 Z ∞ n 之间的总变化距离在截止时间附近的恒定时间窗口中收敛到一个通用函数，这一事实在随机情况下称为剖面截止过程。另一方面，