  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.
