On the Statistics of the Number of Fixed-Dimensional Subcubes in a Random Subset of the n-Dimensional Discrete Unit Cube

TL;DR

This paper explores the statistical properties of fixed-dimensional subcubes within random subsets of the n-dimensional cube, deriving explicit moments and proving asymptotic normality of their count.

Contribution

It introduces a symbolic computation approach to derive moments and provides a rigorous proof of the asymptotic normality of the number of fixed-dimensional subcubes.

Findings

Explicit expressions for moments of subcube counts

Evidence suggesting asymptotic normality

Rigorous proof of asymptotic normality

Abstract

This paper consists of two independent, but related parts. In the first part we show how to use symbolic computation to derive explicit expressions for the first few moments of the number of implicants that a random Boolean function has, or equivalently the number of fixed-dimensional subcubes contained in a random subset of the $n$-dimensional cube. These explicit expressions suggest, but do not prove, that these random variables are always asymptotically normal. The second part presents a full, human-generated proof, of this asymptotic normality, first proved by Urszula Konieczna.