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Let $(X_n)_{n=0}^\infty$ denote a Markov chain on a Polish space that has astationary distribution $\varpi$. This article concerns upper bounds on theWasserstein distance between the distribution of $X_n$ and $\varpi$. Inparticular, an explicit geometric bound on the distance to stationarity isderived using generalized drift and contraction conditions whose parametersvary across the state space. These new types of drift and contraction allow forsharper convergence bounds than the standard versions, whose parameters areconstant. Application of the result is illustrated in the context of anon-linear autoregressive process and a Gibbs algorithm for a random effectsmodel.

Geometric convergence bounds for Markov chains in Wasserstein distance based on generalized drift and contraction conditions