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Markov processes on a circular lattice

Sourav Majumdar

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 $π$, 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.

Markov processes on a circular lattice

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 -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 , 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.
Paper Structure (11 sections, 11 theorems, 85 equations)

This paper contains 11 sections, 11 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Notation and conventions.
  4. Diffusion semigroups on the cycle
  5. Markov interpretation.
  6. Transition kernel
  7. Exact trigonometric moments
  8. Mixing rates
  9. Nearest-neighbour chains with a prescribed stationary distribution
  10. Discrete von Mises process
  11. Discrete wrapped Cauchy process

Key Result

Theorem 1

For each $k\in\mathbb{Z}_m$, $\varphi_k$ is an eigenfunction of $L$ with eigenvalue Consequently, for $\beta\in(0,1]$ and $t\ge 0$, Moreover, $P_t^{(\beta)}(r,s)$ depends only on $s-r\pmod m$ (translation invariance).

Theorems & Definitions (19)

  • Theorem 1: Explicit transition kernel
  • Corollary 1: Convergence to uniformity
  • proof
  • Proposition 1
  • Corollary 2: A one-parameter concentration summary
  • Theorem 2: Total-variation bound
  • Proposition 2: A reversible nearest-neighbour construction
  • Corollary 3: von Mises stationary law
  • Theorem 3: Normalizing constant for discrete von Mises process
  • Corollary 4: Exact trigonometric moments
  • ...and 9 more