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

Potential Analysis(2020)

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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.
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Key words
Total variation distance,Markov chains,Random walks,Normal martingales,Obtuse random walks,60J10,60J05,60G50,60G42,15A69
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