  We study two specific symmetric random block Toeplitz (of dimension $k \times
k$) matrices: where the blocks (of size $n \times n$) are (i) matrices with
i.i.d. entries, and (ii) asymmetric Toeplitz matrices.
  Under suitable assumptions on the entries, their limiting spectral
distributions (LSDs) exist (after scaling by $\sqrt{nk}$) when (a) $k$ is fixed
and $n \to\infty$ (b) $n$ is fixed and $k\rightarrow \infty$ (c) $n$ and $k$ go
to $\infty$ simultaneously. Further the LSD's obtained in (a) and (b) coincide
with those in (c) when $n$ or respectively $k$ tends to infinity. This limit in
(c) is the semicircle law in case (i). In Case (ii) the limit is related to the
limit of the random symmetric Toepiltz matrix as obtained by Bryc et al.(2006)
and Hammond and Miller(2005).
