O(n^{\frac{k+\ell}{2}-1}) & \text{if}\ (k-m) \mbox{ and } (\ell-m) \mbox{ both are even,}\\\\
            o(n^{\frac{k+\ell}{2}-1}) & \text{otherwise}.        
              \end{array}\right.         
     \end{aligned}$$

Now from (\[eqn:gamma\_1\]) and (\[card\_B\_k\_l\]), we get $$\begin{aligned}
 \label{card_T_k,l}
    |T^m_{k,\ell}|  & = \left\{\begin{array}{ll}     
             O(n^{\frac{k+\ell}{2}-1}) & \text{if}\ k, \ell \mbox{ and } m \mbox{ all are even or }  k, \ell \mbox{ and } m \mbox{ all are odd},\\\\
            o(n^{\frac{k+\ell}{2}-1}) & \text{otherwise}.        
              \end{array}\right.    
    \end{aligned}$$ On using (\[eqn:T\_k,l\_SC\]) and (\[card\_T\_k,l\]), we get that $T^m_{k,\ell}$ has non-zero contribution in (\[eqn:T\_k,l\_SC\]) only when $k=p$ and $\ell=q$. In fact $T^m_{k,\ell}$ has non-zero contribution only when either $p,q, m$ all are even or $p,q, m$ all are odd. So, if we use (\[card\_T\_k,l\]) in (\[eqn:T\_k,l\_SC\]), we get $$\begin{aligned}
 \label{eqn:lim_cov_SC}
    &\lim_{n\to\infty}  \Cov\big(w_p,w_q\big) \nonumber \\
    &= \displaystyle\lim_{n\to\infty} \frac{1}{n^{\frac{p+q}{2}-1}} \sum_{m=1}^{ \min \{p,q\} } \sum_{I_m} \Big\{\E[X_{i_1} \cdots X_{i_{p}} X_{j_1} \cdots X_{j_{q}} ] -\E[ X_{i_1} \cdots X_{i_{p}} ] \E[ X_{j_1} \cdots X_{j_{q}} ]  \Big\}  \nonumber \\
    &=  \left\{\begin{array}{ll}     
            \displaystyle\lim_{n\to\infty} \frac{1}{n^{\frac{p+q}{2}-1}} \sum_{r=1}^{ \min\{ \frac{p}{2},\frac{q}{2} \} } \sum_{I_{2r}} \big( \E[X_{i_1} \cdots X_{i_{p}} X_{j_1} \cdots X_{j_{q}} ]\\
    \quad -\E[ X_{i_1} \cdots X_{i_{p}} ] \E[ X_{j_1} \cdots X_{j_{q}} ]  \big) & \text{if}\ p,  q \mbox{ both are even,}\\\\
             \displaystyle\lim_{n\to\infty}\frac{1}{n^{\frac{p+q}{2}-1}} \sum_{r=0}^{ \min\{ \frac{p-1}{2},\frac{q-1}{2} \} } \sum_{I_{2r+1}} \big( \E[X_{i_1} \cdots X_{i_{p}} X_{j_1} \cdots X_{j_{q}} ]\\
    \qquad -\E[ X_{i_1} \cdots X_{i_{p}} 