Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

Persi Diaconis (1) , Laurent Miclo (2)

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

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. Perron-Frobenius theory ensures the existence of a corresponding positive eigenvector $\varphi$. The goal of the paper is to give bounds on the amplitude $\max \varphi/\min\varphi$. Two approaches are proposed: one using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen in [7].

Mots clés: first Dirichlet eigenvector; finite absorbing Markov process; entrance boundary; denumerable absorbing birth and death process; spectral estimates; path method

Domaines: Probabilités [math.PR]

Persi Diaconis, Laurent Miclo. Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks. Science China Mathematics, 2016, 59 (2), pp.205-226. ⟨10.1007/s11425-015-5085-2⟩. ⟨hal-01150142v2⟩