characteristic sequence, $2(t-1)$ positions have been taken to place the “1” and “-1”, there leaves $4k-2(t-1)-2/2$ (the sequence starts and ends with a zero, so we should exclude the two positions at the beginning and at the end, and also “-1” appears in the odd positions, so we should divide it by two) positions to place the remaining “-1”, whose number is $2k-t-s+1$, so the choice is bounded by $$\begin{pmatrix}
  2k-t \\
  2k-t-s+1 \\
\end{pmatrix}.$$

Let $m$ be the number of noncoincident vertical edges, which is no less than $t+s-1$, see Lemma \[lemmabai\], then the expectation $$\begin{aligned}
{\mathbb{E}}[\varepsilon_{j_1\,i_1}\varepsilon_{j_1\,i_2}\varepsilon_{j_2\,s+i_2}\varepsilon_{j_2\,s+i_3}\varepsilon_{j_3\,i_3}\varepsilon_{j_3\,i_4}\varepsilon_{j_4\,s+i_4}\varepsilon_{j_4\,s+i_5}\cdots \varepsilon_{j_{2k}\,s+i_{2k}}\varepsilon_{j_{2k}\,s+i_1}]\end{aligned}$$ is bounded by $M^{4k-2m}\leq M^{4k-2(t+s-1)}$. Finally, the contribution to is bounded by: $$\begin{aligned}
\label{22}
&\quad \quad\frac{1}{pT^{2k}}\sum_{s}\sum_{t}M^{4k-2(t+s-1)}f_{t-1}(k)T^tp^s\begin{pmatrix}
                                                              2k-t \\
                                                              s-1 \\
                                                            \end{pmatrix}\nonumber\\
&=\frac{1}{pT^{2k}}\sum_{t=1}^kM^{4k-2t+2}f_{t-1}(k)T^t\sum_{s=1}^{2k-t}\left(\frac{p}{M^2}\right)^s\begin{pmatrix}
                                                                                                      2k-t \\
                                                                                                      s-1 \\
                                                                                                    \end{pmatrix}\nonumber\\
&=\frac{1}{pT^{2k}}\sum_{t=1}^kM^{4k-2t+2}f_{t-1}(k)T^t\left(\frac{p}{M^2}\right)^{2k-t}\sum_{s=1}^{2k-t}\left(\frac{p}{M^2}\right)^{s-2k+t}\begin{pmatrix}
                                                                                                      2k-t \\
                                                                                                      s-1 \\
                                                                                                    \end{pmatrix}\nonumber\\
&=\frac{1}{pT^{2k}}\sum_{t=1}^kM^{4k-2t+2}f_{t-1}(k)T^t\left(\frac{p}{M^2}\right)^{2k-t}\sum_{l=0}^{2k-t-1}\left(\frac{M^2}{p}\right)^{l}\begin{pmatrix}
                                                                                                      2k-t \\
                                                                                                      l+1 \\
                                                                                                    \end{pmatrix}\nonumber\\
&\leq\frac{1}{pT^{2k}}\sum_{t=1}^kM^{4k-2t+2}f_{t-1}(k)T^t\left(\frac{p}{M^2}\right)^{2k-t}\sum_{l=0}^{2k-t-1}\left(\frac{M^2}{p}\right)^{l}(2k-t)^{l+1}\nonumber\\
&=O\left(\frac{k}{pT^{2k}}\sum_{t=1}^kM^{4k-2t+2}f_{t-1}(k)T^t\left(\frac{p}{M^2}\right)^{2k-t}\right)\nonumber\\
&=O\left(\frac{k\delta^2}{\sqrt p}\sum_{t=0}^{k-1}\frac 1k \begin{pmatrix}
                                                             2k \\
                                                             t \\
                                                           \end{pmatrix}\begin{pmatrix}
                                                                          k \\
                                                                          t+1 \\
                                                                        \end{pmatrix}y_T^{2k-t-1}~.
\right)\end{aligned}$$ Combine , with the fact that $k=(\log p)^{1.01}$ leads to the fact that the contribution of $Q_3$ to is $$\begin{aligned}
\sum_{t=0}^{k-1}\frac 1k \begin{pmatrix}
                                                             2k \\
                                                             t \\
                                                           \end{pmatrix}\begin{pmatrix}
                                                                          k \\
                                                                          t+1 \\
                                                                        \end{pmatrix}y_T^{2k-t-1}\cdot o_k(1)~.\end{aligned}$$ The proof of Theorem \[largek\] is complete.

Derivation of some master 