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

We prove the existence of a limit distribution for the normalized normality measureN(EN)/√N(asN→∞) for random binary sequences EN, by this means confirming a conjecture of Alon,Kohayakawa, Mauduit, Moreira and Rödl. The key point of the proof is to approximate thedistribution of the normality measure by the exiting probabilities of a multidimensional Wienerprocess from a certain polytope