edge, then $${\mathbb{E}}\left(E_{Q_1(\bf{i_1},\bf{j_1})}E_{Q_2(\bf{i_2},\bf{j_2})}\right)={\mathbb{E}}\left(E_{Q_1(\bf{i_1},\bf{j_1})}\right){\mathbb{E}}\left(E_{Q_2(\bf{i_2},\bf{j_2})}\right)=0~,$$ so the contribution to ${\mathop{\text{Var}}}(m_k(A_T))$ is also zero.

Now, suppose $Q=Q_1\bigcup Q_2$ contains no single edges, then there is at least one edge in $Q$ with multiplicity greater than or equal to 4, so the number of non-coincident I-vertices in $Q$ is at least $t_1+t_2-1$ and J-vertices is $s_1+s_2-1$. Since $t_1+s_1-1\leq2k$ and $t_2+s_2-1\leq2k$, we have $$\begin{aligned}
&\quad~{\mathop{\text{Var}}}(m_k(A_T))\nonumber\\
&=\frac{1}{p^2T^{4k}}\sum_{\bf{i_1},\bf{j_1},\bf{i_2},\bf{j_2}}\left[{\mathbb{E}}\left(E_{Q_1(\bf{i_1},\bf{j_1})}E_{Q_2(\bf{i_2},\bf{j_2})}\right)-{\mathbb{E}}\left(E_{Q_1(\bf{i_1},\bf{j_1})}\right){\mathbb{E}}\left(E_{Q_2(\bf{i_2},\bf{j_2})}\right)\right]\\
&=O\left(\frac{1}{p^2T^{4k}}T^{t_1+t_2-1}p^{s_1+s_2-1}\right)\\
&=O(p^{-2})~,\end{aligned}$$ which is .

[**Carleman condition.**]{} In [@li], the density function that corresponds to the Stieltjes transform $s(z)$ in has been derived and it has compact support $[a,b]$, where $$\begin{aligned}
&a=\frac 18\left(-1+20y+8y^2-(1+8y)^{3/2}\right)\cdot\mathbbm{1}_{\{y\geq 1\}}~,\nonumber\\
&b=\frac 18\left(-1+20y+8y^2+(1+8y)^{3/2}\right)~.\label{b}\end{aligned}$$ Therefore, we have $$\begin{aligned}
\label{mkb}
m_k=\sum_{i=0}^{k-1}\frac 1k \begin{pmatrix}
                               2k \\
                               i \\
                             \end{pmatrix}\begin{pmatrix}
                                            k \\
                                            i+1 \\
                                          \end{pmatrix}y^{2k-1-i}\leq b^k~.\end{aligned}$$ From this, it is easy to see that the Carleman condition is satisfied.

The proof of Theorem \[th1\] is complete.

For the verification of Carleman condition, it would be enough to use the Stirling’s formula in $m_k$ to derive a less sharp bound $m_k\leq A^k$ for some $A\geq b$. Since the sharp bound will also be used in Section \[largest\], its early introduction is thus preferred.

\[re1\] The explanation of (5) in is that in the original $Q$-graph, each vertical edge is repeated exactly 