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Bounds in total variation distance for discrete-time processes on the sequence space. (English) Zbl 1480.60204

Summary: 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 \(\text{d}_{\text{TV}}(\mathbb{P},\widetilde{\mathbb{P}})\) 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.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J05 Discrete-time Markov processes on general state spaces
60G50 Sums of independent random variables; random walks
60G42 Martingales with discrete parameter
15A69 Multilinear algebra, tensor calculus
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References:

[1] Attal, S.; Deschamps, J.; Pellegrini, C., Complex obtuse random walks and their continuous-time limits, Probab. Theory Relat. Fields, 165, 65-116 (2016) · Zbl 1406.60067 · doi:10.1007/s00440-015-0627-7
[2] Attal, S.; Émery, M., Équations de structure pour des martingales vectorielles, Lecture Notes in Mathematics, 256-278 (1994), Berlin, Heidelberg: Springer Berlin Heidelberg, Berlin, Heidelberg · Zbl 0814.60040
[3] Baur, E.; Bertoin, J., Elephant random walks and their connection to pólya-type urns, Phys. Rev. E., 94, 5, 2016 (1134)
[4] Billingsley, P.: Probability and Measure, vol. 245 of Wiley Series in Probability and Statistics, 3rd edn. Wiley, Hoboken (1995) · Zbl 0822.60002
[5] Brémaud, P., Markov Chains, vol. 31 of Texts in Applied Mathematics (1999), New York: Springer, New York · Zbl 0949.60009
[6] Coletti, Cf; Gava, R.; Schütz, Gm, Central limit theorem and related results for the elephant random walk, J. Math. Phys., 58, 053303 (2017) · Zbl 1375.60086 · doi:10.1063/1.4983566
[7] Franz, U.; Hamdi, T., Stochastic analysis for obtuse random walks, J. Theor. Probab., 28, 619-649 (2015) · Zbl 1323.60062 · doi:10.1007/s10959-013-0522-z
[8] Kumar, N.; Harbola, U.; Lindenberg, K., Memory-induced anomalous dynamics: emergence of diffusion, subdiffusion, and superdiffusion from a single random walk model, Phys. Rev. E., 82, 2, 2010 (1101)
[9] Privault, N., Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. vol. 1982 of Lecture Notes in Mathematics (2009), Berlin: Springer, Berlin · Zbl 1185.60005
[10] Schütz, Gm; Trimper, S., Elephants can always remember: exact long-range memory effects in a non-Markovian random walk, Phys. Rev. E., 70, 045101 (2004) · doi:10.1103/PhysRevE.70.045101
[11] Song, J., Gao, Y., Wang, H., An, B.: Measuring the distance between finite Markov decision processes. In: Thangarajah, J., Tuyls, K., Jonker, C., Marsella, S. (eds.) Proceedings of the 15th International Conference on Autononous Agents and Multiagents Systems, Singapore (2016)
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