Submultiplicativity and Stopping of a Coupling Markov Chain for the VKF Method

Abstract

In this paper, we investigate a coupling Markov chain for the VKF method, where an excessively long run of the chain is stopped if the number of its steps exceeds the total length of the trajectories preliminarily computed for the chain. For this version of the algorithm, a submultiplicativity lemma is proved. In addition, a theorem on the probability with which the length of the trajectory exceeds a certain threshold is proved by reasoning a la Fekete’s lemma. Finally, we prove that the probabilities of the results yielded by the ordinary and stopped coupling Markov chains differ in terms of the total variation distance by an exponentially small quantity with respect to the number of preliminary runs.