Cut-off and hitting times of a sample of Ornstein-Uhlenbeck processes and its average

Abstract

A cut-off phenomenon is shown to occur in a sample of n independent, identically distributed Ornstein-Uhlenbeck processes and its average. Their distributions stay far from equilibrium before a certain O(log(n)) time, and converge exponentially fast after. Precise estimates show that the total variation distance drops from almost 1 to almost 0 over an interval of time of length O(1) around log(n)/(2α), where α is the viscosity coefficient of the sampled process. The distribution of the hitting time of 0 by the average of the sample is computed. As n tends to infinity, the hitting time becomes concentrated around the cut-off instant, and its tails match the estimates given for the total variation distance.