  Gyarmati, Mauduit and S\'ark\"ozy introduced the cross-correlation measure
$\Phi_k(G)$ of order $k$ to measure the level of pseudorandom properties of
families of finite binary sequences. In an earlier paper we estimated the
cross-correlation measure of a random family of binary sequences. In this
paper, we sharpen these earlier results by showing that for random families,
the cross-correlation measure converges strongly, and so has limiting
distribution. We also give sharp bounds to the minimum values of the
cross-correlation measure, which settles a problem of Gyarmati, Mauduit and
S\'ark\"ozy nearly completely.
