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

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{ö}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.

Second version. Some minor changes. To appear in the Bulletin of the London Mathematical Society