Maximal correlation and monotonicity of free entropy and of Stein discrepancy

Benjamin Dadoun, Pierre Youssef

Research output: Contribution to journalArticlepeer-review


We introduce the maximal correlation coefficient R(M1, M2) between two noncommu-tative probability subspaces M1 and M2 and show that the maximal correlation coefficient between the sub-algebras generated by sn:= x1+…+xn and sm:= x1+…+xm equalsm/n for m ≤ n, where (xi)i∈N is a sequence of free and identically distributed noncommutative random variables. This is the free-probability analogue of a result by Dembo–Kagan–Shepp in classical probability. As an application, we use this estimate to provide another simple proof of the monotonicity of the free entropy and free Fisher information in the free central limit theorem. Moreover, we prove that the free Stein Discrepancy introduced by Fathi and Nelson is non-increasing along the free central limit theorem.

Original languageEnglish (US)
Article number24
JournalElectronic Communications in Probability
StatePublished - 2021


  • free Stein discrepancy
  • free entropy
  • maximal correlation
  • monotonicity

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


Dive into the research topics of 'Maximal correlation and monotonicity of free entropy and of Stein discrepancy'. Together they form a unique fingerprint.

Cite this