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

Abstract

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.