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.

Thermalisation for small random perturbations of dynamical systems

“…Ycart [24] and P. Diaconis [34] for an account. We refer to the book of D. Levin et al ([16], Chapter 18) for an introduction of the subject in the Markov chain setting, L. Saloff-Coste [28] provides an extensive list of random walks for which the cut-off phenomenon holds, P. Diaconis [34] for a review on the finite Markov chain case, S. Martínez and B. Ycart [41] for the case of Markov chains with countably infinite state space, G. Chen and L. Saloff-Coste [23] for Brownian motions on a compact Riemann manifold, B. Lachaud [8] and G. Barrera [21] for Ornstein-Uhlenbeck processes on the line and G. Barrera and M. Jara [22] for stochastic small perturbations of one-dimensional dynamical systems.…”

On interweaving relations

“…twice the spectral gap of L β,σ ). Corollary 16 is also relevant for small β > 0, since one cannot hope for an estimate so simple via (27).…”

“…We now turn to another application of interweaving which allows to identify the cut-off phenomena for degenerate hypoelliptic Ornstein-Uhlenbeck semigroups. To this end, let α ≔ (α 1 , α 2 , ... [27] has shown that this family has a cut-off at the time…”

Change detection in a distillation column using non‐linear auto‐regressive moving average with exogenous input model and Hellinger distance

“…, where μ 0 , μ 1 are the means and σ 0 , σ 1 are the standard deviations for p 0 (ɛ) and p 1 (ɛ), respectively. In this case, the HD may be written as [21]…”