V-geometrical ergodicity of Markov kernels via finite-rank approximations

Loïc Hervé (1, 2) , James Ledoux (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 INSA Rennes - Institut National des Sciences Appliquées - Rennes

Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the V-geometrical ergodicity of a general Markov kernel P , which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of P by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in V-geometrical ergodicity.

Mots clés: Rate of convergence; Geometric ergodicity; Spectral gap; Minorization condition; Drift condition; AMS subject classification : 60J05

Domaines: Probabilités [math.PR]

Loïc Hervé, James Ledoux. V-geometrical ergodicity of Markov kernels via finite-rank approximations. Electronic Communications in Probability, 2020, 25 (23), pp.1-12. ⟨10.1214/20-ECP303⟩. ⟨hal-02504691v2⟩