Cutoffs for product chains

TL;DR

This paper introduces a new approach to analyze the cutoff phenomenon in product Markov chains by relating total variation and Hellinger distances, providing criteria and examples for understanding mixing times.

Contribution

It presents a novel inequality linking total variation and Hellinger distances, enabling the identification of cutoffs in product chains and offering new criteria for their analysis.

Findings

Established a new inequality relating total variation and Hellinger distance.

Provided criteria for cutoff detection in product chains.

Illustrated results with examples like two-state chains and cycles.

Abstract

In this article, we consider products of ergodic Markov chains and discuss their cutoffs in the total variation. Through a new inequality relating the total variation and the Hellinger distance, we may identify the total variation cutoffs with cutoffs in the Hellinger distance. This provides a new scheme to study the total variation mixing of Markov chains, in particular, product chains. In the theoretical framework, a series of criteria are introduced to examine cutoffs and a comparison of mixing between the product chain and its coordinate chains is made in detail. For illustration, we consider products of two-state chains, cycles and other typical examples.