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

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--Rényi 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.

References in corpus (3)

• Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem
• Functional inequalities for Gaussian convolutions of compactly supported measures: explicit bounds and dimension dependence
• Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations