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

## TL;DR

This paper derives explicit geometric bounds on the Wasserstein distance for Markov chains using generalized drift and contraction conditions, enabling sharper convergence estimates than traditional methods.

## Contribution

It introduces new variable-parameter drift and contraction conditions that improve convergence bounds for Markov chains in Wasserstein distance.

## Key findings

- Sharper convergence bounds than standard methods.
- Application to non-linear autoregressive processes.
- Application to Gibbs algorithms for random effects models.

## Abstract

Let $(X_n)_{n=0}^\infty$ denote a Markov chain on a Polish space that has a stationary distribution $\varpi$. This article concerns upper bounds on the Wasserstein distance between the distribution of $X_n$ and $\varpi$. In particular, an explicit geometric bound on the distance to stationarity is derived using generalized drift and contraction conditions whose parameters vary across the state space. These new types of drift and contraction allow for sharper convergence bounds than the standard versions, whose parameters are constant. Application of the result is illustrated in the context of a non-linear autoregressive process and a Gibbs algorithm for a random effects model.