  In this article we investigate high-dimensional banded sample covariance
matrices under the regime that the sample size $n$, the dimension $p$ and the
bandwidth $d$ tend simultaneously to infinity such that $$n/p\to 0 \ \
\text{and} \ \ 2d/n\to y>0.$$ It is shown that the empirical spectral
distribution of those matrices almost surely converges weakly to some
deterministic probability measure which is characterized by its moments.
Certain restricted compositions of natural numbers play a crucial role in the
evaluation of the expected moments of the empirical spectral distribution.
