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

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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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Acknowledgments

This research was supported by the Singapore MOE Tier 2 Grant MOE2016-T2-1-036.

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Correspondence to Ian Flint.

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Flint, I., Privault, N. & Torrisi, G.L. Bounds in Total Variation Distance for Discrete-time Processes on the Sequence Space. Potential Anal 52, 223–243 (2020). https://doi.org/10.1007/s11118-018-9744-0

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Keywords

  • Total variation distance
  • Markov chains
  • Random walks
  • Normal martingales
  • Obtuse random walks

Mathematics Subject Classification (2010)

  • 60J10
  • 60J05
  • 60G50
  • 60G42
  • 15A69