Markov processes on a circular lattice

TL;DR

This paper introduces a Markov process framework for discrete circular distributions, providing explicit kernels, moments, and convergence results, with applications to von Mises and wrapped Cauchy distributions.

Contribution

It develops a novel Markov process approach for discrete circular data, including explicit transition kernels and methods to achieve specific stationary distributions.

Findings

Explicit transition kernels for discrete circular Markov processes

Construction of reversible chains with prescribed stationary distributions

Processes with stationary laws like discrete von Mises and wrapped Cauchy distributions

Abstract

We develop a Markov process viewpoint for discrete circular distributions motivated by directional-statistics settings where angles are observed on a finite grid and evolve over time. On the $m$-point discrete circle, the cycle graph, we study diffusion-generated families, obtaining an explicit transition kernel, exact trigonometric moments, and convergence to uniformity. We present a simple approach to construct reversible nearest-neighbour chains with any prescribed strictly positive stationary pmf $\pi$, providing discrete analogues of Markov processes on the continuous circle. We construct processes whose stationary laws are the discrete von Mises and wrapped Cauchy distributions with closed-form normalizers and exact moments.