Monotonicity of Entropy and Fisher Information: A Quick Proof via Maximal Correlation

TL;DR

This paper presents a straightforward proof demonstrating the monotonic behavior of entropy and Fisher information in sums of i.i.d. random variables, utilizing maximal correlation characterization.

Contribution

It introduces a simple proof method for entropy and Fisher information monotonicity based on maximal correlation, simplifying previous approaches.

Findings

Entropy increases monotonically with sum size.

Fisher information decreases monotonically with sum size.

The proof leverages maximal correlation properties.

Abstract

A simple proof is given for the monotonicity of entropy and Fisher information associated to sums of i.i.d. random variables. The proof relies on a characterization of maximal correlation for partial sums due to Dembo, Kagan and Shepp.