  We analyze the empirical spectral distribution of random periodic band
matrices with correlated entries. The correlation structure we study was first
introduced in 2015 by Hochst\"attler, Kirsch and Warzel, who named their setup
"almost uncorrelated" and showed convergence to the semicircle distribution in
probability. We strengthen their results which turn out to be also valid almost
surely. Moreover, we extend them to band matrices. Sufficient conditions for
convergence to the semicircle distribution both in probability and almost
surely are provided. In contrast to convergence in probability, almost sure
convergence seems to require a minimal growth rate for the bandwidth. Examples
that fit our general setup include Curie-Weiss distributed, correlated
Gaussian, and as a special case, independent entries.
