# Maximal correlation and the rate of Fisher information convergence in the Central Limit Theorem

## TL;DR

This paper investigates how the Fisher information of scaled sums of i.i.d. variables converges in the CLT, linking it to maximal correlation eigenvalues and establishing convergence rates under certain spectral conditions.

## Contribution

It introduces a novel connection between Fisher information convergence in the CLT and the spectral properties of maximal correlation eigenvalues, providing new convergence rate results.

## Key findings

- Fisher information of scaled sums converges at an O(1/n) rate under spectral conditions.
- A relationship between Fisher information behavior and the second-largest eigenvalue of maximal correlation.
- Monotonicity of Fisher information is strengthened assuming eigenvalue inequalities.

## Abstract

We consider the behaviour of the Fisher information of scaled sums of independent and identically distributed random variables in the Central Limit Theorem regime. We show how this behaviour can be related to the second-largest non-trivial eigenvalue associated with the Hirschfeld--Gebelein--R\'{e}nyi maximal correlation. We prove that assuming this eigenvalue satisfies a strict inequality, an $O(1/n)$ rate of convergence and a strengthened form of monotonicity hold.