The Cutoff Phenomenon For Ornstein-Uhlenbeck Processes

The word cutoff was coined by Diaconis and Aldous in 1983 to characterizethe property of many Markov chains,usually with a high degree of symmetry,toconverge very abruptly to their stationary distribution. The definition of cutoffphenomenon was given by Diaconis and Saloff-Coste, it describles the property ofsteep convergence to equilibrium of certain Markov processes. In the sense of thetotal variation distance, for a certain Markov process of cutoff phenomenon, beforethe so called cutoff instant, the total variation distance between the distribution ofthe process and its asymptotic distribution tends to 1; after that instant, the totalvariation distance tends to 0 abruptly.The aim of the present paper is to investigate the cutoff phenomenon and hittingtimes for n-sample Ornstein-Uhlenbeck process and its average process, the sampleprocess is made up of n independent and identically distributed Ornstein-Uhlenbeckprocesses, and compare the asymptotic tail behaviors of the hitting time with whathappens on the left and on the right of the cutoff instant for the convergence ofboth the n-sample and the average process. At last, we use Kolmogonorv-Smirnovtest to detects the cutoff instant,it appears that the larger the size of the sample,thesteeper the transition to equilibrium.