Cutoff for a class of auto-regressive models with vanishing additive noise

TL;DR

This paper investigates the convergence behavior of a family of auto-regressive Markov chains with vanishing additive noise, revealing a cutoff phenomenon as the noise diminishes with increasing system size.

Contribution

It introduces a new analysis of convergence rates for auto-regressive Markov chains with vanishing noise, connecting to Bayesian schemes and demonstrating a cutoff phenomenon.

Findings

Convergence rates are characterized for large system sizes.

A cutoff phenomenon occurs as noise vanishes.

The model relates to Bayesian analysis of exchangeable data.

Abstract

We analyze the convergence rates for a family of auto-regressive Markov chains $(X^{(n)}_k)_{k\geq 0}$ on $\mathbb R^d$, where at each step a randomly chosen coordinate is replaced by a noisy damped weighted average of the others. The interest in the model comes from the connection with a certain Bayesian scheme introduced by de Finetti in the analysis of partially exchangeable data. Our main result shows that, when $n$ gets large (corresponding to a vanishing noise), a cutoff phenomenon occurs.