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

We prove the existence of a limit distribution for the normalized normality measure $\mathcal{N}(E_N)\mathrm{/}\sqrt{N}$ (as $N\to \mathrm{\infty }$) 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.