  This article deals with the limiting spectral distributions (LSD) of
symmetric Toeplitz and Hankel matrices with dependent entries. For any fixed
positive integer $m$, we consider these $n \times n$ matrices with entries
$\{Y^{(m)}_j / \sqrt n \ ;{j\geq 0}\}$, where $Y^{(m)}_j= \sum_{r=-m}^{m}
X_{j+r}$ and $\{X_k\}$ are i.i.d. with mean zero and variance one. We provide
an explicit expression for the moment sequences of the LSDs. As a special case,
this article provides an alternate proof for the LSDs of these matrices when
the entries are i.i.d. with mean zero and variance one. The method is based on
the moment method. The idea of proof can also be applied to other patterned
random matrices, namely reverse circulant and symmetric circulant matrices.
