On some possible combinations of mixing rates for strictly stationary, reversible Markov chains

TL;DR

This paper constructs examples of reversible, strictly stationary Markov chains demonstrating that their mixing rates for different conditions can vary widely within certain limits, highlighting the complexity of their dependence structures.

Contribution

It provides explicit examples showing the possible combinations of mixing rates for reversible Markov chains, revealing the flexibility and constraints of their dependence properties.

Findings

Mixing rates for $ ho$-mixing and $eta$-mixing can be arbitrarily related within restrictions.

Constructed examples show the independence of different mixing conditions.

Reversible Markov chains can exhibit a wide range of dependence behaviors.

Abstract

A class of examples is constructed to show that for strictly stationary Markov chains that are reversible, the simultaneous mixing rates for the $\rho$-mixing and strong mixing ($\alpha$-mixing) conditions can be fairly arbitrary, within certain unavoidable tight restrictions. The examples constructed here have the added property that the mixing rate for the absolute regularity ($\beta$-mixing) condition is within a constant factor of that for strong mixing.