On the limit distribution of the normality measure of random binary sequences

TL;DR

This paper proves the existence of a limit distribution for the normalized normality measure of random binary sequences, confirming a conjecture by approximating it with exit probabilities of a multidimensional Wiener process.

Contribution

It establishes the limit distribution for the normality measure of random binary sequences, using a novel approximation with a multidimensional Wiener process.

Findings

Confirmed the conjecture on the limit distribution.

Connected the normality measure to Wiener process exit probabilities.

Provided a new probabilistic approach to analyze binary sequence normality.

Abstract

We prove the existence of a limit distribution for the normalized normality measure $\mathcal{N}(E_N)/\sqrt{N}$ (as $N \to \infty$) for random binary sequences $E_N$, by this means confirming a conjecture of Alon, Kohayakawa, Mauduit, Moreira and R{\"o}dl. The key point of the proof is to approximate the distribution of the normality measure by the exiting probabilities of a multidimensional Wiener process from a certain polytope.