Bounds in Total Variation Distance for Discrete-time Processes on the Sequence Space

Let \(\mathbb {P}\) and \(\widetilde {\mathbb {P}}\) be the laws of two discrete-time stochastic processes defined on the sequence space \(S^{\mathbb N}\), where S is a finite set of points. In this paper we derive a bound on the total variation distance \(\mathrm {d}_{\text {TV}}(\mathbb {P},\widetilde {\mathbb {P}})\) in terms of the cylindrical projections of \(\mathbb {P}\) and \(\widetilde {\mathbb {P}}\). We apply the result to Markov chains with finite state space and random walks on \(\mathbb {Z}\) with not necessarily independent increments, and we consider several examples. Our approach relies on the general framework of stochastic analysis for discrete-time obtuse random walks and the proof of our main result makes use of the predictable representation of multidimensional normal martingales. Along the way, we obtain a sufficient condition for the absolute continuity of \(\widetilde {\mathbb {P}}\) with respect to \(\mathbb {P}\) which is of interest in its own right.