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

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.

💡 Research Summary

The paper investigates the estimation of three classical mixing coefficients—α‑mixing (strong mixing), β‑mixing (total variation distance), and φ‑mixing (conditional probability deviation)—for pairs of random variables that may be either discrete (finite alphabet) or continuous (real‑valued). The authors first recall that a closed‑form expression for β‑mixing is already known for discrete variables, and then focus on the remaining two coefficients.

Discrete case.

1. Computational hardness of α‑mixing. The decision problem “Is the α‑mixing coefficient between X and Y larger than a given threshold θ?” is proved to be NP‑complete. The reduction is from 3‑SAT: a Boolean formula is encoded into a joint probability table such that the formula is satisfiable iff the resulting α‑mixing exceeds θ. Membership in NP follows because, given a candidate coupling, the α‑mixing value can be computed in polynomial time. Consequently, exact computation of α‑mixing is intractable in the worst case. To mitigate this, the authors present two polynomial‑time bounds. The upper bound is obtained via a semidefinite programming (SDP) relaxation that maximizes a bilinear form derived from the joint distribution; the lower bound follows from a simple Markov‑type inequality that uses marginal probabilities. Empirical tests show that these bounds are often tight.

2. Exact formula for φ‑mixing. While φ‑mixing has traditionally been defined only implicitly, the authors derive a compact closed‑form expression:

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