Three measures of pseudorandomness of finite binary sequences were introduced
by Mauduit and S\'ark\"ozy in 1997 and have been studied extensively since
then: the normality measure, the well-distribution measure, and the correlation
measure of order r. Our main result is that the correlation measure of order r
for random binary sequences converges strongly, and so has a limiting
distribution. This solves a problem due to Alon, Kohayakawa, Mauduit, Moreira,
and R\"odl. We also show that the best known lower bounds for the minimum
values of the correlation measures are simple consequences of a celebrated
result due to Welch, concerning the maximum nontrivial scalar products over a
set of vectors.Comment: 19 pages, this version contains small changes taking into account
referee comment
