\sum_{r=0}^{ \min\{ \frac{p-1}{2},\frac{q-1}{2} \} }  \frac{b_r}{2^{\frac{p+q-4r-2}{2}}  } \sum_{s=0}^{2r+1}\binom{2r+1}{s}^2 s!(2r+1-s)! \ h_{2r+1}(s) \\          
    \displaystyle   + pq \binom{p-1}{(p-1)/2}\binom{q-1}{(q-1)/2} \l((p-1)/2\r)! \l((q-1)/2\r)!\frac{1}{2^{(\frac{p+q}{2}-1)}}   & \text{if}\ p, \mbox{} q \mbox{ both are odd},\\\\
            0 & \text{otherwise}.        
              \end{array}\right.                  
    \end{aligned}$$ Now combining both the Steps, from (\[eqn:cov\_odd\]) and (\[eqn:cov\_even,2\]), we get that for both the odd and even value of $n, \lim_{n\to\infty}  \Cov\big(w_p,w_q\big)$ is same. Therefore $$\begin{aligned}
 \label{eqn:cov}
    &\lim_{n\to\infty}  \Cov\big(w_p,w_q\big) \nonumber \\ 
    &=  \left\{\begin{array}{ll}     
            \displaystyle \frac{a_1}{2^{\frac{p+q-4}{2}}  } (\E X^4_1- 1) + \sum_{r=2}^{ \min\{ \frac{p}{2},\frac{q}{2} \} } \frac{a_r}{2^{\frac{p+q-4r}{2}}  } \sum_{s=0}^{2r}\binom{2r}{s}^2 s!(2r-s)! \ h_{2r}(s) & \text{if}\ p, \mbox{} q \mbox{ both are even},\\\\
        \displaystyle   \sum_{r=0}^{ \min\{ \frac{p-1}{2},\frac{q-1}{2} \} }  \frac{b_r}{2^{\frac{p+q-4r-2}{2}}  } \sum_{s=0}^{2r+1}\binom{2r+1}{s}^2 s!(2r+1-s)! \ h_{2r+1}(s) \\          
    \displaystyle   + pq \binom{p-1}{(p-1)/2}\binom{q-1}{(q-1)/2} \l((p-1)/2\r)! \l((q-1)/2\r)!\frac{1}{2^{(\frac{p+q}{2}-1)}}   & \text{if}\ p, \mbox{} q \mbox{ both are odd},\\\\
            0 & \text{otherwise}.        
              \end{array}\right.                  
    \end{aligned}$$ This complete the proof of the theorem \[thm:symcircovar\].

Proof of Theorem \[thm:symcirpoly\] {#sec:poly}
===================================

First we begin with some notation and definitions. Recall $A_{p}$ and $\tilde{A}_p$ from in Section \[sec:poly\], $$\begin{aligned}
A_{p} & =\{(j_1,\ldots,j_{p})\suchthat \sum_{i=1}^{p}\epsilon_i j_i=0\; \mbox{(mod n)}, \epsilon_i\in\{+1,-1\}, 1\le j_1,\ldots,j_{p}\le \frac{n}{2}\}, \\
\tilde{A}_{k} &=\{(j_1,\ldots,j_{k})\suchthat \sum_{i=1}^{k}\epsilon_i j_i=0\; \mbox{(mod } \frac{n}{2}) \mbox{ and } \sum_{i=1}^{k}\epsilon_i j_i \neq 0\; \mbox{(mod }n),  \epsilon_i\in\{+1,-1\}, 1\le j_1,\ldots,j_{k}\le \frac{n}{2}\}.\end{aligned}$$

For a vector $ J = (j_1, j_2, \ldots, j_{p})\in A_{p} 