consider the input sequence of the form $\{x_i: i\geq 0\}$ and the symmetric circulant matrix is defined as $$SC_n=\left(\begin{array}{cccccc}
x_0 & x_1 & x_2 & \cdots & x_{2} & x_1 \\
x_1 & x_0 & x_1 & \cdots & x_{3} & x_{2}\\
x_{2} & x_1 & x_0 & \cdots & x_{4} & x_{3}\\
\vdots & \vdots & {\vdots} & \ddots & {\vdots} & \vdots \\
x_1 & x_2 & x_3 & \cdots & x_1 & x_0
\end{array}\right).$$ For $j=1,2,\ldots, n-1$, its $(j+1)$-th row is obtained by giving its $j$-th row a right circular shift by one positions and the (i,j)-th element of the matrix is $x_{\frac{n}{2}-|\frac{n}{2}-|i-j||}$. Also note that the symmetric circulant matrix is a Toeplitz matrix with the restriction that $x_{n-j}=x_j$.

Now we consider linear eigenvalue statistics as defined in for $SC_n$ with test function $f(x)=x^{p}$, $p\geq 2$. Therefore $$\sum_{k=1}^{n}f(\lambda_k)= \sum_{k=1}^{n}(\lambda_k)^{p}= \Tr(SC_n)^{p},$$ where $\lambda_1,\lambda_2,\ldots,\lambda_n$ are the eigenvalues of $SC_n$. We scale and centre $\Tr(SC_n)^{p}$ to study its fluctuation, and define $$\label{eqn:SCw_p}
w_p := \frac{1}{\sqrt{n}} \bigl\{ \Tr(SC_n)^{p} - \E[\Tr(SC_n)^{p}]\bigr\}.$$ For a given real polynomial $$Q(x)=\sum_{k=1}^da_kx^{k}$$ with degree $d$ where $d\geq 2$, we define $$\label{eqn:RCw_Q}
w_Q := \frac{1}{\sqrt{n}} \bigl\{ \Tr(Q(SC_n)) - \E[\Tr(Q(SC_n))]\bigr\}.$$

Note that $w_Q$ and $w_p$ depends on $n$. But 