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

In this paper we study the problem of estimating the mixing coefficients between two random variables. Three different mixing coefficients are studied, namely alpha-mixing, beta-mixing and phi-mixing coefficients. The random variables can either assume values in a finite set or the set of real numbers. At present, closed-form expressions are available for the beta-mixing coefficient. However, for discrete random variables, available methods for the computation of both the alpha-mixing and the phi-mixing coefficients require an exponential number of computation in the sizes of the alphabets. This is the motivation for the present paper. We derive upper and lower bounds for both the alpha-mixing and the phi-mixing coefficients. Moreover, in case the marginal distributions of the two random variables are uniform, an exact expression is given for the phi-mixing coefficient. This situation arises when empirically generated samples are binned using percentile binning. We also 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 a naive method for estimating mixing coefficients from empirical data does not work. However, 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 converge to the true values.