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