Cut-off Phenomenon for Converging Processes in the Sense of α-Divergence Measures

The cut-off phenomenon is observed in many processes, among which several Markov chains and random walks. This particular behaviour of some converging processes is observed, around a certain instant where the distance to equilibrium decreases abruptly either from infinity or from a strictly positive constant to 0. Several previous studies have characterized the phenomenon with particular probability metrics: total variation, Hellinger, chi-square and Kullback-Leibler distances. In this article we propose on the one hand a new characterization which takes into account the case where the left limit does not exist, on the other hand we extend some important previous results using generalized distances: some families of α-divergence measures. First we give a result of existence of the cut-off for a sequence of n-tuples of independent processes having exponential convergence rate to their equilibrium distributions. We then give results about explicit expressions of asymptotic distances around cut-off instant. Finally, simulation results are given to illustrate the phenomenon.