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.
@article{AFST_2013_6_22_3_573_0,
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, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 22},
number = {3},
year = {2013},
doi = {10.5802/afst.1383},
mrnumber = {3113027},
zbl = {06299049},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1383/}
}
TY - JOUR AU - Persi Diaconis AU - Laurent Miclo TI - On the spectral analysis of second-order Markov chains JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2013 SP - 573 EP - 621 VL - 22 IS - 3 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1383/ DO - 10.5802/afst.1383 LA - en ID - AFST_2013_6_22_3_573_0 ER -
%0 Journal Article %A Persi Diaconis %A Laurent Miclo %T On the spectral analysis of second-order Markov chains %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2013 %P 573-621 %V 22 %N 3 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1383/ %R 10.5802/afst.1383 %G en %F AFST_2013_6_22_3_573_0
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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