On the Correlation between Boolean Functions of Sequences of Random Variables

TL;DR

This paper introduces a new inequality linking effective length and maximum correlation of Boolean functions on correlated sequences, providing a tighter upper bound useful across disciplines involving common information.

Contribution

It presents a novel inequality that incorporates effective length into correlation bounds, extending Witsenhausen's maximum-correlation bound for Boolean functions.

Findings

Derived a new upper-bound on Boolean function output correlation

Extended Witsenhausen's maximum-correlation bound to include effective length

Applicable to fields dealing with common information

Abstract

In this paper, we establish a new inequality tying together the effective length and the maximum correlation between the outputs of an arbitrary pair of Boolean functions which operate on two sequences of correlated random variables. We derive a new upper-bound on the correlation between the outputs of these functions. The upper-bound is useful in various disciplines which deal with common-information. We build upon Witsenhausen's bound on maximum-correlation. The previous upper-bound did not take the effective length of the Boolean functions into account.