Abstract
We introduce the maximal correlation coefficient R(M1,M2) between two noncommutative probability subspaces M1and M2 and show that the maximal correlation coefficient between the sub-algebras generated by sn:=x1+…+xnand sm:=x1+…+xm equals m∕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.
Highlights
Introduction and main resultPearson’s correlation coefficient ρ(X1, X2) := cov(X1, X2)/σ(X1) σ(X2) is a standard measure of dependency
We introduce the maximal correlation coefficient R(M1, M2) between two noncommutative 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 equals m/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. 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
Summary
Pearson’s correlation coefficient ρ(X1, X2) := cov(X1, X2)/σ(X1) σ(X2) is a standard measure of (bivariate) dependency. The idea of using the maximal correlation inequality in this context goes back to Courtade [5] who used the result of Dembo, Kagan, and Shepp [7] to provide an alternative proof of the monotonicity of entropy in the classical setting. It would be interesting to see if the argument in this paper based on maximal correlation could be extended to cover non-integer exponents Another consequence of Theorem 1.1 concerns the monotonicity of the free Stein discrepancy along the free central limit theorem. As an application of our maximal correlation inequality, we obtain the following corollary extending the aforementioned monotonicity of Stein discrepancy obtained in [6] to the free setting.
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