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 N ( E N ) / N (as N → ∞ ) for random binary sequences E N , by this means confirming a conjecture of Alon, Kohayakawa, Mauduit, Moreira and Rödl. 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.