On the spectral analysis of second-order Markov chains

Persi Diaconis (1) , Laurent Miclo (2)

1 Department of Statistics [Stanford]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: second order Markov chains; trajectorial reversibility; spectrum of non-reversible Markov kernels; spectral gap; improvement of speed of convergence; discrete position-speed Markov chains; homogeneous and Abelian second order Markov chains.; homogeneous and Abelian second order Markov chains

Domaines: Probabilités [math.PR]

Persi Diaconis, Laurent Miclo. On the spectral analysis of second-order Markov chains. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2013, 22 (3), pp.573-621. ⟨hal-00719047⟩