## Abstract

We introduce the maximal correlation coefficient R(M_{1}, M_{2}) between two noncommu-tative probability subspaces M_{1} and M_{2} and show that the maximal correlation coefficient between the sub-algebras generated by s_{n}:= x_{1}+…+x_{n} and s_{m}:= x_{1}+…+x_{m} equals^{√}m/n for m ≤ n, where (x_{i})_{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 language | English (US) |
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Article number | 24 |

Journal | Electronic Communications in Probability |

Volume | 26 |

DOIs | |

State | Published - 2021 |

## Keywords

- free Stein discrepancy
- free entropy
- maximal correlation
- monotonicity

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty