which equals the total number of “-1”;

5.  The sequences are made with the following subsequence structure: $$\begin{aligned}
    \label{sub}
    & 1\underbrace{00}_{\text{two}}-1\nonumber\\
    &1\underbrace{000000}_{\text{six}}-1\nonumber\\
    &1\underbrace{0000000000}_{\text{ten}}-1\nonumber\\
    &\quad\quad\cdots\nonumber\\
    &1\underbrace{0000\cdots 00000}_{4k-6}-1~.\end{aligned}$$

Denote $f_{t-1}(k)$ as the number of $Q$-graphs whose $M$-pillar satisfies the above condition (1)-(5) (here, we have two index: $t-1$, which is the number of up innovations and $k$, which is a quarter of the total length of the sequence), then we have the contribution of Category 1 to : $$\begin{aligned}
\label{m1}
&\quad~\frac{1}{pT^{2k}}\cdot\sum_{t=1}^kT(T-1)\cdots (T-t+1)p(p-1)\cdots (p-s+1)f_{t-1}(k)\nonumber\\
&=\sum_{t=1}^k y_T^{2k-t}f_{t-1}(k)+O(\frac 1p)~.\end{aligned}$$

Category 3 consists two situations, see the following lemma.

\[lemmabai\] Denote the coincident multiplicities of the $l$-th noncoincident vertical edge by $a_l$, $l=1,2,\cdots, m$, where $m$ is the number of noncoincident vertical edges. If $Q \in Q_3$, then (a) either there is a $a_l\geq 3$ with $t+s-1\leq m <2k$ or (b) all $a_l=2$ with $t+s-1<m=2k$.

First, we see the contribution of (a). By the moment assumption that the moment ${\mathbb{E}}|\varepsilon_{ij}|^a$ is bounded by $M^{a-2}$ for $a\geq 2$ and $a_1+\cdots +a_m=4k$, we conclude that the expectation $$\begin{aligned}
{\mathbb{E}}[\varepsilon_{j_1\,i_1}\varepsilon_{j_1\,i_2}\varepsilon_{j_2\,s+i_2}\varepsilon_{j_2\,s+i_3}\varepsilon_{j_3\,i_3}\varepsilon_{j_3\,i_4}\varepsilon_{j_4\,s+i_4}\varepsilon_{j_4\,s+i_5}\cdots \varepsilon_{j_{2k}\,s+i_{2k}}\varepsilon_{j_{2k}\,s+i_1}]\end{aligned}$$ in has magnitude at most $M^{4k-2m}$. Then we have bounded by $$\begin{aligned}
\label{b1}
&\quad \frac{1}{pT^{2k}}\cdot\sum_{t=1}^kT^tp^sM^{4k-2m}\#\{\text{isomorphism class in}~ Q_3\}\nonumber\\
&=O\left(\sum_{t=1}^kp^{t+s-k-m/2-1}\cdot \delta^{4k-2m}\right)~,\end{aligned}$$ where the equality is due to the fact 