  We consider a Wigner-type ensemble, i.e. large hermitian $N\times N$ random
matrices $H=H^*$ with centered independent entries and with a general matrix of
variances $S_{xy}=\mathbb E|H_{xy}|^2$. The norm of $H$ is asymptotically given
by the maximum of the support of the self-consistent density of states. We
establish a bound on this maximum in terms of norms of powers of $S$ that
substantially improves the earlier bound $2\| S\|^{1/2}_\infty$ given in
[arXiv:1506.05098]. The key element of the proof is an effective Markov chain
approximation for the contributions of the weighted Dyck paths appearing in the
iterative solution of the corresponding Dyson equation.
