Maximal correlation and monotonicity of free entropy and Stein discrepancy

TL;DR

This paper introduces a noncommutative maximal correlation coefficient, establishes its value for certain subalgebras, and uses it to prove monotonicity properties of free entropy, free Fisher information, and free Stein discrepancy in free probability.

Contribution

It defines a maximal correlation coefficient in free probability and applies it to prove monotonicity of free entropy, free Fisher information, and free Stein discrepancy.

Findings

Maximal correlation coefficient equals b7 for specific subalgebras.

Monotonicity of free entropy and free Fisher information in the free central limit theorem.

Free Stein Discrepancy is non-increasing along the free CLT.

Abstract

We introduce the maximal correlation coefficient $R(M_1,M_2)$ between two noncommutative probability subspaces $M_1$ and $M_2$ and show that the maximal correlation coefficient between the sub-algebras generated by $s_n:=x_1+\ldots +x_n$ and $s_m:=x_1+\ldots +x_m$ equals $\sqrt{m/n}$ for $m\le n$, where $(x_i)_{i\in \mathbb{N}}$ 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.