  Kolo\u{g}lu, Kopp and Miller compute the limiting spectral distribution of a
certain class of real random matrix ensembles, known as $k$-block circulant
ensembles, and discover that it is exactly equal to the eigenvalue distribution
of an $k \times k$ Gaussian unitary ensemble. We give a simpler proof that
under very general conditions which subsume the cases studied by
Kolo\u{g}lu-Kopp-Miller, real-symmetric ensembles with periodic diagonals
always have limiting spectral distribution equal to the eigenvalue distribution
of a finite Hermitian ensemble with Gaussian entries which is a `complex
version' of a $k \times k$ submatrix of the ensemble. We also prove an
essentially algebraic relation between certain periodic finite Hermitian
ensembles with Gaussian entries, and the previous result may be seen as an
asymptotic version of this for real-symmetric ensembles. The proofs show that
this general correspondence between periodic random matrix ensembles and finite
complex Hermitian ensembles is elementary and combinatorial in nature.
