Mixing Coefficients Between Discrete and Real Random Variables: Computation and Properties

TL;DR

This paper investigates the computation and properties of mixing coefficients between discrete and real-valued random variables, providing new formulas, complexity results, and consistency proofs for empirical estimation.

Contribution

It offers a closed-form formula for the phi-mixing coefficient, NP-completeness results for alpha-mixing, and consistency proofs for empirical estimates in the real-valued case.

Findings

Closed-form formula for phi-mixing coefficient

NP-completeness of alpha-mixing threshold decision

Consistency of empirical estimates with percentile binning

Abstract

In this paper we study the problem of estimating the alpha-, beta- and phi-mixing coefficients between two random variables, that can either assume values in a finite set or the set of real numbers. In either case, explicit closed-form formulas for the beta-mixing coefficient are already known. Therefore for random variables assuming values in a finite set, our contributions are two-fold: (i) In the case of the alpha-mixing coefficient, we show that determining whether or not it exceeds a prespecified threshold is NP-complete, and provide efficiently computable upper and lower bounds. (ii) We derive an exact closed-form formula for the phi-mixing coefficient. Next, we prove analogs of the data-processing inequality from information theory for each of the three kinds of mixing coefficients. Then we move on to real-valued random variables, and show that by using percentile binning and allowing the number of bins to increase more slowly than the number of samples, we can generate empirical estimates that are consistent, i.e., converge to the true values as the number of samples approaches infinity.