  Consider random symmetric Toeplitz matrices $T_{n}=(a_{i-j})_{i,j=1}^{n}$
with matrix entries $a_{j}, j=0,1,2,...,$ being independent real random
variables such that
  \be \mathbb{E}[a_{j}]=0, \ \ \mathbb{E}[|a_{j}|^{2}]=1 \ \ \textrm{for}\,\ \
j=0,1,2,...,\ee (homogeneity of 4-th moments)
\be{\kappa=\mathbb{E}[|a_{j}|^{4}],}\ee
  \noindent and further (uniform boundedness)\be\sup\limits_{j\geq 0}
\mathbb{E}[|a_{j}|^{k}]=C_{k}<\iy\ \ \ \textrm{for} \ \ \ k\geq 3.\ee
  Under the assumption of $a_{0}\equiv 0$, we prove a central limit theorem for
linear statistics of eigenvalues for a fixed polynomial with degree $\geq 2$.
Without the assumption, the CLT can be easily modified to a possibly non-normal
limit law. In a special case where $a_{j}$'s are Gaussian, the result has been
obtained by Chatterjee for some test functions. Our derivation is based on a
simple trace formula for Toeplitz matrices and fine combinatorial analysis. Our
method can apply to other related random matrix models, including Hankel
matrices and product of several Toeplitz matrices in a flavor of free
probability theory etc. Since Toeplitz matrices are quite different from the
Wigner and Wishart matrices, our results enrich this topic.
