Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity

2011/10/14 by Denis Guibourg, Guibourg, Denis, Loı̈c Hervé +3
Mathematics · #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1110.3240

Abstract

Let P be a Markov kernel on a measurable space \X and let V:\X\r[1,+∞). We provide various assumptions, based on drift conditions, under which P is quasi-compact on the weighted-supremum Banach space (\cB_V,‖⋅‖_V) of all the measurable functions f : \X\r\C such that ‖f‖_V := sup_{x∈ \X} |f(x)|/V(x) < ∞. Furthermore we give bounds for the essential spectral radius of P. Under additional assumptions, these results allow us to derive the convergence rate of P on \cB_V, that is the geometric rate of convergence of the iterates Pⁿ to the stationary distribution in operator norm. Applications to discrete Markov kernels and to iterated function systems are presented.

Citations

• Quasi-compactness and absolutely continuous kernels, applications to Markov chains

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