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Dependent couplings and convergence speed of general space Markov chains

Abstract

The rate at which an ergodic $\varphi$-irreducible Markov chain converges to its invariant measure is generally studied by using drift conditions and coupling arguments. Couplings are possible when two independent paths of the process meet at a pseudo atom obtained via a minorization condition. The speed at which the paths may couple depends on the constants appearing at the minorization condition and on auxiliary Foster-Lyapunov functions also known as drifts. Drifts are used basically to control the return times to the pseudo atom. The current project focuses on the use of dependent coupling constructions, aimed at minimising the coupling time. These couplings should lead to tighter bounds for the speed of convergence. Several dependent couplings and state dependent drifts have appeared since, however, none of these have directly addressed the convergence speed problem in the general measurable case. (AU)