Bounds in total variation distance for discrete-time processes on the sequence space

Let P and P' be the laws of two discrete-time stochastic processes defined on the sequence space S, where S is a finite set of points. In this paper we derive a bound on the total variation distance dTV(P,P') in terms of the cylindrical projections of P and P'. We apply the result to Markov chains with finite state space and random walks on 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 P' with respect to P which is of interest in its own right.