Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances

We present a framework for obtaining explicit bounds on the rate of convergence to equilibrium of a Markov chain on a general state space, with respect to both total variation and Wasserstein distances. For Wasserstein bounds, our main tool is Steinsaltz's convergence theorem for locally contractive random dynamical systems. We describe practical methods for finding Steinsaltz's "drift functions" that prove local contractivity. We then use the idea of "one-shot coupling" to derive criteria that give bounds for total variation distances in terms of Wasserstein distances. Our methods are applied to two examples: a two-component Gibbs sampler for the Normal distribution and a random logistic dynamical system.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ238 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Random-walk interpretations of classical iteration methods

AbstractWe give a simple framework for computing relative convergence rates for relaxation methods with discrete Laplace operators (five point or nine point). This gives relations between the convergence rate for Jacobi, point Gauss Seidel, and various block relaxation strategies, essentially by inspection. The framework is a random walk interpretation of Jacobi relaxation that extends to these other relaxation methods