  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.
