Efficient Coupling for Random Walk with Redistribution

TL;DR

This paper investigates the convergence to stationarity of a finite state Markov chain resembling a lazy nearest neighbor random walk on an interval, using coupling methods to provide probabilistic insights and bounds.

Contribution

It introduces an efficient coupling approach to analyze convergence rates, offering probabilistic interpretation and tight bounds for a diffusion-like Markov chain.

Findings

Coupling identifies bottlenecks affecting convergence

Provides tight bounds on total variation distance

Offers probabilistic interpretation of convergence rates

Abstract

What can one say on convergence to stationarity of a finite state Markov chain that behaves "locally" like a nearest neighbor random walk on ${\mathbb Z}$ ? The model we consider is a version of nearest neighbor lazy random walk on the state space $ \{0,\dots,N\}$: the probability for staying put at each site is $\frac 12$, the transition to the nearest neighbors, one on the right and one on the left, occurs with probability $\frac14$ each, where we identify two sites, $J_0$ and $J_N$ as, respectively, the neighbor of $0$ from the left and the neighbor of $N$ from the right (but $0$ is not a neighbor of $J_0$ and $N$ is not neighbor of $J_N$). This model is a discrete version of diffusion with redistribution on an interval studied by several authors in recent past, and for which the the exponential rates of convergence to stationarity were computed analytically, but had no intuitive or probabilistic interpretation, except for the case where the jumps from the endpoints are identical (or more generally have the same distribution). We study convergence to stationarity probabilistically, by finding an efficient coupling. The coupling identifies the "bottlenecks" responsible for the rates of convergence and also gives tight computable bounds on the total variation norm of the process between two starting points. The adaptation to the diffusion case is straightforward.