Asymptotic of products of Markov kernels. Application to deterministic and random forward/backward products

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

The asymptotic of products of general Markov/transition kernels is investigated using Doeblin's coefficient. We propose a general approximating scheme as well as a convergence rate in total variation of such products by a sequence of positive measures. These approximating measures and the control of convergence are explicit from the two parameters in the minorization condition associated with the Doeblin coefficient. This allows us to extend the well-known forward/backward convergence results for stochastic matrices to general Markov kernels. A new result for forward/backward products of random Markov kernels is also established.

Mots clés: 60J05; 60F99; 60B20; Non-homogeneous Markov chains; Doeblin's coefficient

Domaines: Probabilités [math.PR]

Loïc Hervé, James Ledoux. Asymptotic of products of Markov kernels. Application to deterministic and random forward/backward products. Statistics and Probability Letters, 2021, 179 (12), pp.article n°109204. ⟨10.1016/j.spl.2021.109204⟩. ⟨hal-02354594v3⟩