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

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  • Barrera, Gerardo

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, 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.

Suggested Citation

  • Barrera, Gerardo, 2021. "Cutoff phenomenon for the maximum of a sampling of Ornstein–Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 168(C).
  • Handle: RePEc:eee:stapro:v:168:y:2021:i:c:s0167715220302571
    DOI: 10.1016/j.spl.2020.108954
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    References listed on IDEAS

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    1. Barrera, Javiera & Lachaud, Béatrice & Ycart, Bernard, 2006. "Cut-off for n-tuples of exponentially converging processes," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1433-1446, October.
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