The geometric convergence rate of the classical change-point estimate

An analytic method for obtaining optimum rates of convergence for the total variation distances is established. The distance engages the distribution of the maximum likelihood estimate of the change-point in a fixed sample size time ordered data and the distribution of the maximum likelihood estimate of the change-point in an infinite time series data. The method is based purely on probabilistic arguments. Various ideas from random walks, supermartingales, ladder heights, and a recursive sequence of embedded random walks are considered. The optimum rate factor involves the probability of the first passage time to the positive axis.