\stackrel{d}{\longrightarrow} N(0,\sigma_{p}^2),\end{aligned}$$ where $$\sigma_{Q}^2= \sum_{\ell=1}^d \sum_{k=1}^d a_{\ell} a_k \sigma_{\ell,k}, \ \  \sigma^2_{p}= \sigma_{p,p}$$ and $\sigma_{p,q}$ is as given in (\[eqn:sigma\_p,q\]).

In the above theorems we have considered the fluctuation of $w_p$ for $p\geq2$. For $p=0$, $$\begin{aligned}
w_0 = \frac{1}{\sqrt{n}} \bigl\{ \Tr(I) - \E[\Tr(I)]\bigr\} = \frac{1}{\sqrt{n}}[n-n] = 0\end{aligned}$$ and hence it has no fluctuation. For $p=1$, $$\begin{aligned}
w_1 = \frac{1}{\sqrt{n}} \bigl\{ \Tr(SC_n) - \E[\Tr(SC_n)]\bigr\} = \frac{1}{\sqrt{n}} \big[n \frac{X_0}{\sqrt n} - \E(n\frac{X_0}{\sqrt n})\big] = X_0,\end{aligned}$$ as $\E(X_0)=0$. So $w_1$ is distributed as $X_0$ and its distribution does not depend on $n$. So we ignore these two cases, for $p=0$ and $p=1$.

In Section \[sec:cov\] we prove Theorem \[thm:symcircovar\]. We derived trace formula and state some results which will be used to prove Theorem \[thm:symcircovar\]. In Section \[sec:poly\] we prove Theorem \[thm:symcirpoly\]. We use method of moments and Wick’s formula to prove Theorem \[thm:symcirpoly\].

Proof of Theorem \[thm:symcircovar\] {#sec:cov}
====================================

We first define some notation which will be used in the proof of Theorem \[thm:symcirpoly\]. $$\begin{aligned}
 \label{def:A_p_SC}
A_{p}&=\{(j_1,\ldots,j_{p})\suchthat \sum_{i=1}^{p}\epsilon_i j_i=0\; \mbox{(mod n)}, \epsilon_i\in\{+1,-1\}, 1\le j_1,\ldots,j_{p}\le \frac{n}{2}\}, \\
\tilde{A}_{k}&=\{(j_1,\ldots,j_{k})\suchthat \sum_{i=1}^{k}\epsilon_i j_i=0\; \mbox{(mod } \frac{n}{2}) \mbox{ and } \sum_{i=1}^{k}\epsilon_i j_i \neq 0\; \mbox{(mod }n),  \epsilon_i\in\{+1,-1\}, 1\le j_1,\ldots,j_{k}\le \frac{n}{2}\}, \nonumber
\\A_p^{(k)}&=\{(j_1,\ldots,j_p)\in A_p\suchthat j_1+\cdots+j_k - j_{k+1}-\cdots -j_p=0 