Constructing processes with prescribed mixing coefficients

TL;DR

This paper demonstrates the construction of processes with virtually any prescribed $\eta$-mixing coefficients, highlighting the flexibility in the decay rates of dependencies in stochastic processes.

Contribution

It provides a method to construct measures and processes with arbitrary $\eta$-mixing coefficients and slow mixing rates, advancing understanding of dependency decay in stochastic processes.

Findings

Existence of measures with arbitrary $\eta$-mixing coefficients

Construction of processes with arbitrarily slow mixing rates

Implications for strong laws of large numbers and concentration inequalities

Abstract

The rate at which dependencies between future and past observations decay in a random process may be quantified in terms of mixing coefficients. The latter in turn appear in strong laws of large numbers and concentration of measure results for dependent random variables. Questions regarding what rates are possible for various notions of mixing have been posed since the 1960's, and have important implications for some open problems in the theory of strong mixing conditions. This paper deals with $\eta$-mixing, a notion defined in [Kontorovich and Ramanan], which is closely related to $\phi$-mixing. We show that there exist measures on finite sequences with essentially arbitrary $\eta$-mixing coefficients, as well as processes with arbitrarily slow mixing rates.