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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 randomvariables 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 forlinear statistics of eigenvalues for a fixed polynomial with degree $\geq 2$.Without the assumption, the CLT can be easily modified to a possibly non-normallimit law. In a special case where $a_{j}$'s are Gaussian, the result has beenobtained by Chatterjee for some test functions. Our derivation is based on asimple trace formula for Toeplitz matrices and fine combinatorial analysis. Ourmethod can apply to other related random matrix models, including Hankelmatrices and product of several Toeplitz matrices in a flavor of freeprobability theory etc. Since Toeplitz matrices are quite different from theWigner and Wishart matrices, our results enrich this topic.

Fluctuation of Eigenvalues for Random Toeplitz and Related Matrices