  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.
