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On the spectral analysis of second-order Markov chains
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 3, pp. 573-621.

Second order Markov chains which are trajectorially reversible are considered. Contrary to the reversibility notion for usual Markov chains, no symmetry property can be deduced for the corresponding transition operators. Nevertheless and even if they are not diagonalizable in general, we study some features of their spectral decompositions and in particular the behavior of the spectral gap under appropriate perturbations is investigated. Our quantitative and qualitative results confirm that the resort to second order Markov chains is an interesting option to improve the speed of convergence to equilibrium.

On considère des chaînes de Markov du second ordre, supposées réversibles trajectoriellement. Contrairement à la notion de réversibilité pour les chaînes de Markov usuelles, les opérateurs de transition correspondants ne vérifient pas de propriété de symétrie et ne sont parfois même pas diagonalisables. On étudie néanmoins certains aspects de leurs décompositions spectrales et en particulier le comportement de leurs trous spectraux sous des perturbations appropriées. Les résultats quantitatifs et qualitatifs obtenus confirment que le recours aux chaînes de Markov du second ordre peut se révéler intéressant pour améliorer la vitesse de convergence à l’équilibre.

Published online:
DOI: 10.5802/afst.1383
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     author = {Persi Diaconis and Laurent Miclo},
     title = {On the spectral analysis of second-order {Markov} chains},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {573--621},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 22},
     number = {3},
     year = {2013},
     doi = {10.5802/afst.1383},
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     mrnumber = {3113027},
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     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1383/}
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Persi Diaconis; Laurent Miclo. On the spectral analysis of second-order Markov chains. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 3, pp. 573-621. doi : 10.5802/afst.1383. https://afst.centre-mersenne.org/articles/10.5802/afst.1383/

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