We introduce the maximal correlation coefficient between two noncommutative probability subspaces and and show that the maximal correlation coefficient between the sub-algebras generated by and equals for , where 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.
The authors would like to thank the anonymous referees for their numerous generous suggestions which greatly improved the manuscript. For instance, the monotonicity of free Stein discrepancy and its proof was suggested by one of the referees. The authors are grateful to Roland Speicher for helpful comments and for bringing to their attention the recent preprint . The second named author is thankful to Marwa Banna for helpful discussions.
"Maximal correlation and monotonicity of free entropy and of Stein discrepancy." Electron. Commun. Probab. 26 1 - 10, 2021. https://doi.org/10.1214/21-ECP391