Mixing coefficients between discrete and real random variables: Computation and properties

Mehmet Eren Ahsen, Mathukumalli Vidyasagar

Research output: Contribution to journalArticlepeer-review


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 twofold: 1) 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. 2) 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.

Original languageEnglish (US)
Article number6595623
Pages (from-to)34-47
Number of pages14
JournalIEEE Transactions on Automatic Control
Issue number1
StatePublished - Jan 2014
Externally publishedYes


  • Data processing inequality
  • Data-driven partitions
  • Mixing coefficients
  • NP-completeness

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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