\documentclass{aptpub}
\authornames{F.~CARON {\it et al}.} 

\shorttitle{Supplementary material: On sparsity, power-law and clustering properties of graphex processes}

\newenvironment{sketchproof}[1][Sketch of the proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\newcommand{\Nor}{N}   %%%{\mathord{\mathcal{N}}}% oppure {\textrm{N}} Normal
\newcommand{\Gam}{\mathord{\mathcal{G}}}% Gamma
\newcommand{\Unif}{\mathord{\mathcal{U}\mathit{nif}}}% Uniform
\renewcommand{\NG}{\mathord{\mathcal{NG}}}% Normal-Gamma
\newcommand{\Multinom}{\mathord{\mathcal{M}\mathit{ult}}}% Multinomial
\newcommand{\Wi}{\mathord{\mathcal{W}}}% Wishart
\newcommand{\Dir}{\mathord{\mathcal{D}\mathit{ir}}}% Dirichlet
\newcommand{\Stud}{\mathord{\mathcal{T}}}% Student-t
\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Nedges}{N_{\alpha}^{(e)}}
\newcommand{\Nnodes}{N_{\alpha}}
\newcommand{\Nnodesj}{N_{\alpha,j}}
\def\R{\mathbb R}
\usepackage{bbm}
\usepackage{xr-hyper}
\externaldocument{a18505}
\usepackage{graphicx,subfigure}
\usepackage{amsmath}
\allowdisplaybreaks
\graphicspath{ {./figures/} }
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{fit}
\usetikzlibrary{calc}
\usetikzlibrary{positioning}

\numberwithin{equation}{section}  

\begin{document}\recd{11 October 2021}{28 October 2022}

\title{Supplementary material: On sparsity, power-law and clustering properties of graphex processes} 

\authoronetwo[University of Oxford]{Fran\c{c}ois Caron} 
\authorthree[London School of Economics and Political Science]{Francesca Panero} 
\authoronefour[University of Oxford]{Judith Rousseau}

\addressone{Department of Statistics, 24-29 St Giles', Oxford OX1 3LB, United Kingdom} % Your postal address goes here.
\emailtwo{caron@stats.ox.ac.uk} %Authors email goes here.
\addressthree{Department of Statistics, 69 Aldwych, London WC2B 4RR,  United Kingdom. Email: f.panero@lse.ac.uk}
\emailfour{judith.rousseau@stats.ox.ac.uk}

The supplementary material is organised as follows. Section \ref{sec:proofsecond} contains proofs of asymptotic bounds on the variances of the number of nodes, number of nodes of a given degree, and number of triangles of nodes with a given degree, as well as the proof of a secondary proposition for the local clustering coefficient. Section \ref{sec:proofsecondCLT} contains proofs of secondary propositions for the central limit theorem. 
For the sake of simplicity, all Sections, Equations, Lemmas, etc., in the Supplementary material here are denoted with prefixes S and T, to differentiate them from the Sections, Equations, Lemmas, etc., of the main text \cite{CaronRousseau2017:main}. 

\appendix

\section{Proofs of secondary propositions for the variances and clustering coefficients}
\label{sec:proofsecond}

\subsection{Proof of Proposition \ref{prop:variancenbnodes} on $\var(N_\alpha)$}%
\label{sec:proofvarnnodes}


An application of the Slivnyak-Mecke and Campbell theorems gives
\begin{align*}
\var(N_{\alpha})    =&E(N_{\alpha})+2\alpha^{2}\int_{0}^{\infty}%
\mu(x)(1-W(x,x))e^{-\alpha\mu(x)}dx\\
&  +\alpha^{2}\int_{\mathbb{R}_{+}^{2}}(e^{\alpha\nu(x,y)}%
-1+W(x,y))(1-W(x,x))(1-W(y,y))\\
& \qquad\quad\times (1-W(x,y))e^{-\alpha\mu(x)-\alpha\mu(y)}dxdy.
\end{align*}


Using the inequality $e^x-1\leq xe^{x}$,
\begin{align*}
\var(N_{\alpha})
&\leq E(N_{\alpha}) +  2\alpha^2 \int_{\mathbb{R}_{+}}\mu(x)e^{-\alpha\mu(x)}dx  \\
&\quad+\alpha^{2}\int_{\mathbb{R}_{+}^{2}} e^{-\alpha\mu(x)-\alpha\mu(y)}\left\{ e^{\alpha \nu(x,y)} -1 +W(x,y) \right\}dxdy\\
& \leq E(N_{\alpha}) +  2\alpha^2 \int_{\mathbb{R}_{+}}\mu(x)e^{-\alpha\mu(x)}dx \\
&\quad +\alpha^{2}\int_{\mathbb{R}_{+}^{2}} e^{-\alpha\mu(x)-\alpha\mu(y)}\left\{ \alpha \nu(x,y)e^{\alpha \nu(x,y)} +W(x,y) \right\}dxdy\\
&=  E(N_{\alpha}) +  2\alpha^2 \int_{\mathbb{R}_{+}}\mu(x)e^{-\alpha\mu(x)}dx  \\
&\quad+\alpha^{2}\int_{\mathbb{R}_{+}^{2}} e^{-\alpha\mu(x)-\alpha\mu(y)} W(x,y)dxdy\\
&\quad+\sum_{k=1}^\infty \alpha^{2+k}\int_{\mathbb{R}_{+}^{2}}  \nu(x,y)^ke^{-\alpha\mu(x)-\alpha\mu(y)} dxdy.
\end{align*}


Now, using Lemmas \ref{lemma:abelianvariations} and~\ref{lemma:boundintnu}
\begin{align*}
\int_{\mathbb{R}_{+}^{2}}W(x,y) e^{-\alpha\mu(x)-\alpha\mu(y)}dxdy \leq  \int_{\mathbb R_+} \mu(x) e^{-\alpha\mu(x) } dx&=O\left ( \alpha^{\sigma-1}\ell_\sigma(\alpha)\right).\\
\int_{\mathbb{R}_{+}^{2}}  \nu(x,y)   e^{-\alpha\mu(x)-\alpha\mu(y)+\alpha\nu(x,y)} dxdy&=O\left (\alpha^{2\sigma-2a}\ell_\sigma^2(\alpha)\right).
\end{align*}

It follows that
$\var(N_{\alpha})  =O(\alpha^{3+2\sigma -2a}\ell_\sigma(\alpha)^2).
$



\label{sec:proofvariance}
Assume Assumption \ref{assumpt:1} and \ref{assumpt:2} are satisfied, with $a=1$. From the first part of Proposition \ref{prop:variancenbnodes}, we have the upper bound
$\var(N_{\alpha})=O\left(\alpha^{1+2\sigma}\ell_\sigma^{2}(\alpha)\right ).
$\\
We now derive a lower bound. If $\sigma=0$, $\var(N_{\alpha})\geq E(N_{\alpha})\gtrsim \alpha\ell_0(\alpha)$, hence $\var(N_{\alpha})\asymp \alpha\ell(\alpha)$. Consider now the case $\sigma>0$. We have
\begin{align*}
\var(N_{\alpha})\geq\alpha^{2}\int_{\mathbb{R}_{+}^{2}}&(e^{\alpha\nu
(x,y)}-1)(1-W(x,x))(1-W(y,y))(1-W(x,y))\\
&\times e^{-\alpha\mu(x)-\alpha\mu(y)}dxdy
\end{align*}
and using the inequality $e^{x}-1\geq x$ and Assumption \ref{assumpt:4}
\begin{align*}
\var(N_{\alpha})  &  \geq\alpha^{3}\int_{\mathbb{R}_{+}^{2}}\nu
(x,y)(1-W(x,x))(1-W(y,y))(1-W(x,y))e^{-\alpha\mu(x)-\alpha\mu(y)}dxdy\\
&  \geq C_{0}\alpha^{3}\int_{x_{0}}^{\infty}\int_{x_{0}}^{\infty}\mu
(x)\mu(y)(1-W(x,x))(1-W(y,y))(1-W(x,y))\\
&\qquad\qquad\qquad\times e^{-\alpha\mu(x)-\alpha\mu(y)}dxdy
\end{align*}
Using Lemmas~\ref{lemma:abelianvariations} and \ref{lemma:abelianvariationsv2}, we have
\begin{align*}
&  \int_{x_{0}}^{\infty}\int_{x_{0}}^{\infty}\mu(x)\mu
(y)(1-W(x,x))(1-W(y,y))(1-W(x,y))e^{-\alpha\mu(x)-\alpha\mu(y)}dxdy\\
&  \sim\int_{0}^{\infty}\int_{0}^{\infty}\mu(x)\mu(y)e^{-\alpha
\mu(x)-\alpha\mu(y)}dxdy=\left (\int_{\mathbb{R}_{+}}\mu(x)e^{-\alpha\mu(x)}dx\right )^2\sim\alpha^{2\sigma-2}\ell_\sigma^{2}(\alpha).
\end{align*}

It follows that, for $\sigma>0$, $\var(N_{\alpha})\gtrsim\alpha^{1+2\sigma}\ell_\sigma^{2}(\alpha).$
Combining this with the upper bound gives, for all $\sigma\in[0,1]$
$
\var(N_{\alpha})\asymp\alpha^{1+2\sigma}\ell_\sigma^{2}(\alpha).
$




\subsection{Proof of proposition \ref{prop:variancenbnodesj} on $\var(N_{\alpha,j})$}
\label{sec:proofvarnnodesj}



We have,
\begin{align*}
& E(N_{\alpha,j}^{2}\mid M)-E(N_{\alpha,j}\mid M)\nonumber\\
&=  \sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}\mathbb{P}\left\{  \sum
_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{1}k}=j\text{ and } \sum_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{2}%
,k}=j\mid M\right\}. \label{eq:bigexpectation}\\
%&=\sum_{j_1=0}^j \sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}\\
%&\times\mathbb{P}\left\{  \sum
%_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{1}k}=j\text{ and } \sum_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{2}%
%,k}=j\text{ and } \sum
%_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{1}k}Z_{i_{2}k}=j-j_1 \mid M\right\}\\
&=\sum_{b\in\{0,1\}^3} \sum_{j_1=0}^j \sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}\\
&\quad\times\mathbb{P}\left\{  \sum
_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{1}k}=j\text{ and } \sum_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{2}%
,k}=j\text{ and } \sum
_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{1}k}Z_{i_{2}k}=j-j_1\right .\\
&\quad\quad\quad\quad \left .\text{ and }Z_{i_1i_1}=b_{11},Z_{i_1i_2}=b_{12},Z_{i_2i_2}=b_{22} \mid M\right \}
\end{align*}
where $b=(b_{11},b_{12},b_{22})\in\{0,1\}^3$. Let $A_1,A_2,A_{12}$ be disjoint subsets of $\mathbb N\backslash \{i_1,i_2\}$ such that $|A_{12}|+b_{12}=j-j_1$, $|A_1| + |A_{1,2}|+b_{11}+b_{12} = |A_2| + |A_{1,2}|+b_{22}+b_{12} = j$ respectively corresponding to the indices of nodes only connected to node $i_1$, only to node $i_2$, or to both nodes $(i_1,i_2)$. Let $A=\{i_1,i_2\}\cup A_1\cup A_2\cup A_{12}$. We have
\begin{align*}
&\mathbb{P}\left\{  \sum
_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{1}k}=j, \sum_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{2}%
,k}=j, \sum
_{k}\mathbbm{1}_{\theta_{k}\leq\alpha}Z_{i_{1}k}Z_{i_{2}k}=j-j_1,\right.\\
& \left.\qquad  (Z_{i_1i_1},Z_{i_1i_2},Z_{i_2i_2})=b \mid M\right \}\\
&=\sum_{A_1,A_2,A_{12}}\frac{ \mathbbm{1}_{\theta_{i_1} \leq\alpha}\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}}{(j-j_1-b_{12})!(j_1-b_{11})!(j_1-b_{22})! }   W(\vartheta_{i_{1}}, \vartheta_{i_1})^{b_{11}}W(\vartheta_{i_{2}}, \vartheta_{i_2})^{b_{22}} W(\vartheta_{i_{1}}, \vartheta_{i_2})^{b_{12}}  \\
& \quad\times\{ 1 - W(\vartheta_{i_{1}}, \vartheta_{i_1})\}^{1-b_{11}}\{ 1 - W(\vartheta_{i_{2}}, \vartheta_{i_2})\}^{1-b_{22}} \{ 1 - W(\vartheta_{i_{1}}, \vartheta_{i_2})\}^{1-b_{12}}  \\
& \quad\times\left [ \prod_{k \in A_1} \mathbbm{1}_{\theta_k \leq \alpha}
W(\vartheta_{i_{1}},\vartheta_{i_{k}})\{1 - W(\vartheta_{i_{2}},\vartheta_{i_{k}})\} \right ] \left [ \prod_{k \in A_2} \mathbbm{1}_{\theta_k \leq \alpha}
\{1 - W(\vartheta_{i_{1}},\vartheta_{i_{k}})\}W(\vartheta_{i_{2}},\vartheta_{i_{k}}) \right ] \\
& \quad \left [ \prod_{k \in A_{12}} \mathbbm{1}_{\theta_k \leq \alpha}
W(\vartheta_{i_{1}},\vartheta_{i_{k}})W(\vartheta_{i_{2}},\vartheta_{i_{k}}) \right ] \exp\left[ - \sum_{k \in \mathbb N\backslash A} \{g_{\alpha,\vartheta_{i_1}}(\theta_k, \vartheta_k)+g_{\alpha,\vartheta_{i_2}}(\theta_k, \vartheta_k)\}\right]
\end{align*}
Using the extended Slivnyak-Mecke theorem,
 \begin{align*}
& E(N_{\alpha,j}^{2})-E(N_{\alpha,j})\\
&=\sum_{b\in\{0,1\}^3} \sum_{j_1=0}^j \frac{\alpha^{2+j+j_1-b_{11}-b_{12}-b_{22}}}{(j-j_1-b_{12})!(j_1-b_{11})!(j_1-b_{22})! }\mathbbm{1}_{j_1\geq b_{11}}\mathbbm{1}_{j_1\geq b_{22}}\mathbbm{1}_{j_1\leq{j-b_{12}}}\\
&\qquad \times\int_{\R_+^2} \{\mu(x)-\nu(x,y)\}^{j_1-b_{11}}\{\mu(y)-\nu(x,y)\}^{j_1-b_{22}} \nu(x,y)^{j-j_1-b_{12}}\\
 & \qquad  \qquad   \times e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x,y) } W(x,x)^{b_{11}}W(y,y)^{b_{22}}W(x,y)^{b_{12}}   \{1 - W(x,x)\}^{1-b_{11}}\\
  & \qquad  \qquad   \times\{1-W(y,y)\}^{1-b_{22}}\{1 - W(x,y)\}^{1-b_{12}}dxdy\stepcounter{equation}\\
 &\leq \sum_{b\in\{0,1\}^3} \sum_{j_1=0}^j \frac{\alpha^{2+j+j_1-b_{11}-b_{12}-b_{22}}}{(j-j_1-b_{12})!(j_1-b_{11})!(j_1-b_{22})! }\mathbbm{1}_{j_1\geq b_{11}}\mathbbm{1}_{j_1\geq b_{22}}\mathbbm{1}_{j_1\leq{j-b_{12}}}\\
 &\qquad\times\int_{\R_+^2} \mu(x)^{j_1-b_{11}}\mu(y)^{j_1-b_{22}} \nu(x,y)^{j-j_1-b_{12}}e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x,y) }  \\
 & \qquad  \qquad   \times W(x,x)^{b_{11}}W(y,y)^{b_{22}}W(x,y)^{b_{12}}\{1 - W(x,x)\}^{1-b_{11}}\{1-W(y,y)\}^{1-b_{22}}\\
& \qquad  \qquad   \times\{1 - W(x,y)\}^{1-b_{12}}dxdy
\end{align*}
We will need the following lemma.

\begin{lem}\label{lemma:Ir}
Let  $r\geq 1$,  $j_1, j_2 \geq 0$. Define
 \begin{equation*}
 \begin{split}
  I_r& := \int_{\R^2} [\alpha\mu(x)]^{j_1}[\alpha\mu(y)]^{j_2} (\alpha \nu(x,y))^{r} e^{ - \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x,y)}dxdy.
\end{split}
\end{equation*}
Under Assumptions 1 and 2, we have for all $r\geq 1$
$$I_r  =O( \alpha^{r-2ar+2\sigma }  \ell_\sigma^2(\alpha)).$$
\end{lem}

\begin{proof}
We have, using Assumption 2, that
 \begin{equation*}
 \begin{split}
  I_r& \leq  \alpha^{r}  \int_{\R_+^2} [\alpha\mu(x)]^{j_1}[\alpha\mu(y)]^{j_2} \nu(x,y)^{r} e^{ - \alpha \left\{ \mu(x) + \mu(y) \right\}/2 }dxdy\\
    &\leq C_1^r \alpha^{r-2ar}  \int_{\mathbb R_+} (\alpha \mu(x) )^{j_1+ar}e^{-\alpha \mu(x) /2} dx \int_{\mathbb R_+} (\alpha \mu(x) )^{j_2+ar}e^{-\alpha \mu(x) /2} dx +o(\alpha^{-p}).
\end{split}
\end{equation*}
for any $p>0$. Assumption 1 and Lemmas \ref{lemma:abelianvariations} ($\sigma=1$) and  \ref{lemma:abelianvariationsv3} ($\sigma\in[0,1)$) imply that for all $r\geq 1$
$$I_r  =O( \alpha^{r-2ar+2\sigma }  \ell_\sigma^2(\alpha)).$$
\end{proof}


It follows
\begin{align*}
& E(N_{\alpha,j}^{2})-E(N_{\alpha,j})\lesssim \sum_{b\in\{0,1\}^3} \frac{\alpha^{2+2j-b_{11}-b_{22}-2b_{12}}}{(j-b_{12}-b_{11})!(j-b_{12}-b_{22})! }\mathbbm{1}_{j\geq b_{11}+b_{12}}\mathbbm{1}_{j\geq b_{22}+b_{12}}\\
 &\times\int_{\R_+^2} \mu(x)^{j-b_{12}-b_{11}}\mu(y)^{j-b_{12}-b_{22}} e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x,y) }  W(x,x)^{b_{11}}W(y,y)^{b_{22}}W(x,y)^{b_{12}}\\
 &  \times\{1 - W(x,x)\}^{1-b_{11}}\{1-W(y,y)\}^{1-b_{22}}\{1 - W(x,y)\}^{1-b_{12}}dxdy+O\{ \alpha^{2+2\sigma+1-2a}\ell_\sigma^2(\alpha)\}.
\end{align*}
Let $V_{0}$ and $V_1$ respectively denote the sum of terms such that $b_{12}=0$ and $b_{12}=1$ in the above sum. Using the inequality $e^{x}\leq 1+xe^x$,
\bgroup
\allowdisplaybreaks
\begin{align*}
V_{0}&= \sum_{b_{11},b_{22}\in\{0,1\}^2}  \frac{\alpha^{2+2j-b_{11}-b_{22}}}{(j-b_{11})!(j-b_{22})! } \int_{\R_+^2} \mu(x)^{j-b_{11}}\mu(y)^{j-b_{12}} e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x,y) }\\
& \qquad   \times W(x,x)^{b_{11}}W(y,y)^{b_{22}}\{1 - W(x,x)\}^{1-b_{11}}\{1-W(y,y)\}^{1-b_{22}}\{1 - W(x,y)\}dxdy\\
&= \sum_{b_{11},b_{22}}  \frac{\alpha^{2+2j-b_{11}-b_{22}}}{(j-b_{11})!(j-b_{22})! } \int_{\R_+^2} \mu(x)^{j-b_{11}}\mu(y)^{j-b_{12}} e^{- \alpha \mu(x) -\alpha \mu(y)}\\
& \qquad  \times W(x,x)^{b_{11}}W(y,y)^{b_{22}}\{1 - W(x,x)\}^{1-b_{11}}\{1-W(y,y)\}^{1-b_{22}}dxdy\\
&\quad+O \left\{\sum_{b_{11},b_{22}}  \frac{\alpha^{3+2j-b_{11}-b_{22}}}{(j-b_{11})!(j-b_{22})! } \int_{\R_+^2} \mu(x)^{j-b_{11}}\mu(y)^{j-b_{12}}\nu(x,y) \right.\\
&\quad\qquad\qquad\left. \times e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x,y) }dxdy \right\} \\
&=\sum_{b_{11},b_{22}}  \left \{ \frac{\alpha^{1+j-b_{11}}}{(j-b_{11})!}\int_{\mathbb R_+}\mu(x)^{j-b_{11}}W(x,x)^{b_{11}}\{1 - W(x,x)\}^{1-b_{11}}e^{-\mu(x)} dx \right \}  \\
&\quad\quad\quad\times \left \{ \frac{\alpha^{1+j-b_{22}}}{(j-b_{22})!}\int_{\mathbb R_+}\mu(y)^{j-b_{22}}W(y,y)^{b_{22}}\{1 - W(y,y)\}^{1-b_{22}}e^{-\mu(y)} dy \right \}\\
&\quad+O\{\alpha^{2+2\sigma+1-2a}\ell_\sigma^2(\alpha)\}= E(N_{\alpha,j})^2  +O\{\alpha^{2+2\sigma+1-2a}\ell_\sigma^2(\alpha)\}
\end{align*}
\egroup
Similarly,
\begin{align*}
V_{1}&\leq  \sum_{b_{11},b_{22}}  \frac{\alpha^{2j-b_{11}-b_{22}}\mathbbm{1}_{j\geq 1+b_{11}}\mathbbm{1}_{j\geq 1+b_{22}}}{(j-1-b_{11})!(j-1-b_{22})! } \int_{\R_+^2} \mu(x)^{j-1-b_{11}}\mu(y)^{j-1-b_{12}}\\
& \qquad\qquad e^{\alpha \nu(x,y)- \alpha \mu(x) -\alpha \mu(y) }W(x,y)dxdy\\
&\leq \sum_{b_{11},b_{22}}  \frac{\alpha^{2j-b_{11}-b_{22}}\mathbbm{1}_{j\geq 1+b_{11}}\mathbbm{1}_{j\geq 1+b_{22}}}{(j-1-b_{11})!(j-1-b_{22})! } \int_{\R_+^2} \mu(x)^{j-1-b_{11}}\mu(y)^{j-1-b_{12}} \\
& \qquad\qquad e^{- \alpha \mu(x) -\alpha \mu(y) }W(x,y)dxdy+O\{\alpha^{2+2\sigma+1-2a}\ell_\sigma^2(\alpha)\}
\end{align*}

For $j_1\geq 1$ and $j_2\geq 1$, using Cauchy-Schwarz and Lemma \ref{lemma:abelianvariationsv3},
\begin{align*}
&\int W(x,y)\mu(x)^{j_1} \mu(y)^{j_2} e^{-\alpha \mu(x) -\alpha \mu(y)} dxdy\\
&\leq
\int_{\mathbb R_+}\mu(x)^{j_1}  e^{-\alpha \mu(x)} \left\{ \int W(x,y)  \mu(y)^{2j_2}e^{ -2\alpha \mu(y)} dy\right\}^{1/2} \mu(x)^{1/2} dx \\
& \leq \left\{\int\mu(x)^{j_1+1/2}  e^{-\alpha \mu(x)} dx\right \}  \left\{\int  \mu(y)^{2j_2}e^{ -2\alpha \mu(y)} dy\right\}^{1/2} \\
&=O\left \{\alpha^{3\sigma/2-j_1-j_2-1/2}\ell^{3/2}_\sigma(\alpha)\right \}
% &\leq \alpha^{3\sigma/2 - j_1- j_2-1/2} \ell(\alpha)^{3/2} L_0(\alpha)
\end{align*}
and for $j_1\geq 0$
\begin{align*}
&\int_{\R_+^2} \mu(x)^{j_1} e^{- \alpha \mu(x) -\alpha \mu(y) }W(x,y)dxdy\leq \int_{\R_+^2} \mu(x)^{j_1} e^{- \alpha \mu(x) }W(x,y)dxdy\\
&=\int_{\R_+} \mu(x)^{j_1+1} e^{- \alpha \mu(x) }dx=O\left \{\alpha^{\sigma-j_1-1}\ell_\sigma(\alpha)\right \}
\end{align*}
It follows that $V_1=O\{\alpha^{2+3\sigma/2-1/2}\ell^{3/2}_\sigma(\alpha)\}+O\{\alpha^{1+\sigma}\ell_\sigma^2(\alpha)\}+O\{\alpha^{2+2\sigma+1-2a}\ell_\sigma^2(\alpha)\}$. Combining the upper bounds on $V_0$ and $V_1$, we obtain
$
\var(N_{\alpha,j})=O(\alpha^{3-2a+2\sigma}\ell^2_\sigma(\alpha))
$
and this terminates the proof. In the case $\sigma=0$ and $a=1$, one can use Lemma \ref{lemma:abelianvariations} instead of  Lemma \ref{lemma:abelianvariationsv3} and replace big $O$ by little $o$ in the above bounds, together with the fact that $E(N_{\alpha,j})=o(\alpha\ell(\alpha))$ and $\ell(t)=O(\ell^2(t))$ if $\sigma=0$.





\subsection{Proof of Proposition \ref{prop:variancelocalclust}}

\label{sec:proofvariancelocalclust}


We first prove the first equality. The proof is similar to that of Proposition \ref{prop:variancenbnodesj}, given in Section~\ref{sec:proofvarnnodesj}.
For any $j\geq 2$,
\begin{align*}
2R_{\alpha,j}&=2\sum_i T_{\alpha i}\mathbbm{1}_{D_{\alpha i}=j}\mathbbm{1}_{\theta_i\leq \alpha}=\sum_{i\neq k\neq l} Z_{ik}Z_{il}Z_{kl}\mathbbm{1}_{D_{\alpha i}=j}\mathbbm{1}_{\theta_i\leq \alpha}.
\end{align*}
Let $S_{\alpha j}:=4R_{\alpha j}^2$. We have
\begin{align}\label{Big:R2}
S_{\alpha j}&=\left (\sum_{i\neq k\neq l} Z_{ik}Z_{il}Z_{kl}\mathbbm{1}_{D_{\alpha i}=j}\mathbbm{1}_{\theta_i\leq \alpha}\right )^2\\
%&=\sum_{i_1\neq k_1\neq l_1}\sum_{i_2\neq k_2\neq l_2} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_2}Z_{k_2l_2}  \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{D_{\alpha i_2}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}  \mathbbm{1}_{\theta_{i_2}\leq \alpha}\notag\\
&=\sum_{i_1\neq k_1\neq l_1\neq i_2\neq k_2\neq l_2} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_2}Z_{k_2l_2}  \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{D_{\alpha i_2}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}  \mathbbm{1}_{\theta_{i_2}\leq \alpha}\notag\\
&\quad+2\sum_{i_1\neq k_1\neq l_1\neq i_2\neq k_2} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_1}Z_{k_2l_1}  \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{D_{\alpha i_2}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}  \mathbbm{1}_{\theta_{i_2}\leq \alpha}\notag\\
&\quad+2\sum_{i_1\neq k_1\neq l_1\neq i_2} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_1}Z_{i_2l_1}  \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{D_{\alpha i_2}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}  \mathbbm{1}_{\theta_{i_2}\leq \alpha}\stepcounter{equation}\notag\\
&\quad+\sum_{i_1\neq k_1\neq l_1\neq k_2\neq l_2} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_1k_2}Z_{i_1l_2}Z_{k_2l_2}  \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}\notag\\
&\quad+2\sum_{i_1\neq k_1\neq l_1\neq k_2} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_1k_2}Z_{k_2l_1}  \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}\notag\\
&\quad+2\sum_{i_1\neq k_1\neq l_1} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1} \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}\notag
\end{align}

Note that some of the terms above are equal to 0 if $j\leq 4$. First note that for any $j_0\leq j$:
\begin{equation}
\label{eq:temp2}
\sum_{i\neq k_1\neq \ldots\neq k_{j_0}} \left ( \prod_{l=1}^{j_0} Z_{il}\right )\mathbbm{1}_{D_{\alpha i}=j}\mathbbm{1}_{\theta_i\leq \alpha}\leq \binom{j}{j_0} N_{\alpha j}
\end{equation}

Hence the last three terms of the right-handside of \eqref{Big:R2} are upper bounded by  $C_j N_{\alpha,j}$, for some constant $C_j$ that does not depend on $\alpha$. Consider now
\begin{align*}
S_{\alpha,j,1}&=\sum_{i_1\neq k_1\neq l_1\neq i_2\neq k_2\neq l_2} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_2}Z_{k_2l_2}  \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{D_{\alpha i_2}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}  \mathbbm{1}_{\theta_{i_2}\leq \alpha}\\
&=\sum_{j_1=2}^j S_{\alpha,j,1,j_1}
\end{align*}
where, for $j_1=2,\ldots,j$
\begin{align*}
S_{\alpha,j,1,j_1}&= \sum_{\substack{i_1\neq k_1\neq l_1\\\neq i_2\neq k_2\neq l_2}} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_2}Z_{k_2l_2}\mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{D_{\alpha i_2}=j}\\
&\qquad\times\mathbbm{1}_{\sum_k Z_{i_1k}Z_{i_2k}\mathbbm{1}_{\theta_k\leq \alpha}=j-j_1  }\mathbbm{1}_{\theta_{i_1}\leq \alpha}  \mathbbm{1}_{\theta_{i_2}\leq \alpha}\\
&=\sum_{b\in\{0,1\}^3} \sum_{\substack{i_1\neq k_1\neq l_1\\\neq i_2\neq k_2\neq l_2}} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_2}Z_{k_2l_2}\mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{D_{\alpha i_2}=j}\\
&\qquad\times\mathbbm{1}_{\sum_k Z_{i_1k}Z_{i_2k}\mathbbm{1}_{\theta_k\leq \alpha}=j-j_1  }\mathbbm{1}_{Z_{i_1 i_1}=b_{11}}\mathbbm{1}_{Z_{i_1 i_2}=b_{12}}\mathbbm{1}_{Z_{i_2 i_2}=b_{22}}\mathbbm{1}_{\theta_{i_1}\leq \alpha}  \mathbbm{1}_{\theta_{i_2}\leq \alpha}
\end{align*}
where we introduce $b=(b_{11},b_{12},b_{22})\in\{0,1\}^3$ as in Section~\ref{sec:proofvarnnodesj}. Using the extended Slivnyak-Mecke theorem, for $j_1=2,\ldots,j$,
 \begin{align}
& E(S_{\alpha,j,1,j_1})\\
&=\sum_{b\in\{0,1\}^3} \frac{\alpha^{2+j+j_1-b_{11}-b_{12}-b_{22}}}{(j-j_1-b_{12})!(j_1-b_{11}-2)!(j_1-b_{22}-2)! }\mathbbm{1}_{j_1\leq{j-b_{12}}}\mathbbm{1}_{j_1\geq b_{11}}\mathbbm{1}_{j_1\geq b_{22}}\nonumber\\
&\quad \times\int_{\R_+^6} \{\mu(x_1)-\nu(x_1,x_2)\}^{j_1-2-b_{11}}\{\mu(x_2)-\nu(x_1,x_2)\}^{j_1-2-b_{22}} \nu(x_1,x_2)^{j-j_1-b_{12}} \nonumber\\
 &\quad   \times e^{- \alpha \mu(x_1) -\alpha \mu(x_2) +\alpha \nu(x_1,x_2) }W(x_1,y_1)W(x_1,z_1)W(y_1,z_1)W(x_2,y_2)W(x_2,z_2)W(y_2,z_2)\nonumber\\
 &  \quad   \times W(x,x)^{b_{11}}W(y,y)^{b_{22}}W(x,y)^{b_{12}}\nonumber\\
 & \quad   \times\{1 - W(x,x)\}^{1-b_{11}}\{1-W(y,y)\}^{1-b_{22}}\{1 - W(x,y)\}^{1-b_{12}}dx_1dy_1dz_1dx_2dy_2dz_2\nonumber\\
 &\leq \sum_{b\in\{0,1\}^3} \frac{\alpha^{2+j+j_1-b_{11}-b_{12}-b_{22}}}{(j-j_1-b_{12})!(j_1-b_{11}-2)!(j_1-b_{22}-2)! }\mathbbm{1}_{j_1\leq{j-b_{12}}}\mathbbm{1}_{j_1\geq b_{11}}\mathbbm{1}_{j_1\geq b_{22}}\nonumber\\
 &\quad\times\int_{\R_+^6} \mu(x_1)^{j_1-2-b_{11}}\mu(x_2)^{j_1-2-b_{22}} \nu(x_1,x_2)^{j-j_1-b_{12}}e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x_1,x_2) }  \nonumber\\
 &  \quad   \times W(x_1,y_1)W(x_1,z_1)W(y_1,z_1)W(x_2,y_2)W(x_2,z_2)W(y_2,z_2)\nonumber\\
 & \quad     \times W(x_1,x_1)^{b_{11}}W(x_2,x_2)^{b_{22}}W(x_1,x_2)^{b_{12}}\nonumber\\
 & \quad   \times\{1 - W(x_1,x_1)\}^{1-b_{11}}\{1-W(x_2,x_2)\}^{1-b_{22}}\{1 - W(x_1,x_2)\}^{1-b_{12}}dx_1dy_1dz_1dx_2dy_2dz_2\label{eq:temp1}
\end{align}
For $b_{12}\neq 0$ or $j\neq j_1$, we can bound the terms in the above sum by
\begin{align}
 &\frac{\alpha^{2+j+j_1-b_{11}-b_{12}-b_{22}}}{(j-j_1-b_{12})!(j_1-b_{11}-2)!(j_1-b_{22}-2)! }\mathbbm{1}_{j_1\leq{j-b_{12}}}\mathbbm{1}_{j_1\geq b_{11}}\mathbbm{1}_{j_1\geq b_{22}} \nonumber\\
 &\qquad\times\int_{\R_+^4} \mu(x_1)^{j_1-b_{11}}\mu(x_2)^{j_1-b_{22}} \nu(x_1,x_2)^{j-j_1-b_{12}}e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x_1,x_2) } \nonumber \\
 & \qquad  \qquad   \times W(x_1,x_1)^{b_{11}}W(x_2,x_2)^{b_{22}}W(x_1,x_2)^{b_{12}} \nonumber\\
 & \qquad  \qquad   \times\{1 - W(x_1,x_1)\}^{1-b_{11}}\{1-W(x_2,x_2)\}^{1-b_{22}}\{1 - W(x_1,x_2)\}^{1-b_{12}}dx_1dx_2 \nonumber\\
 &=O(\alpha^{3+2\sigma-2a}\ell_\sigma(\alpha)^2)\label{eq:temp1b}
\end{align}
using the intermediate results of the proof in Section \ref{sec:proofvarnnodesj}.

Consider now the sum of terms such that $b_{12}=0$ and $j=j_1$ in \eqref{eq:temp1}. Using the inequality $e^x\leq 1+xe^x$, this sum is upper bounded by
\begin{align*}
 &\sum_{b_{11},b_{12}} \frac{\alpha^{2+2j-b_{11}-b_{22}}}{(j-b_{11}-2)!(j-b_{22}-2)! }\int_{\R_+^6} \mu(x_1)^{j-2-b_{11}}\mu(x_2)^{j-2-b_{22}} e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x_1,x_2) }  \\
 &\qquad     \times W(x_1,y_1)W(x_1,z_1)W(y_1,z_1)W(x_2,y_2)W(x_2,z_2)W(y_2,z_2)\\
 & \qquad     \times W(x_1,x_1)^{b_{11}}W(x_2,x_2)^{b_{22}}\{1 - W(x_1,x_1)\}^{1-b_{11}}\{1-W(x_2,x_2)\}^{1-b_{22}}dx_1dy_1dz_1dx_2dy_2dz_2\\
 &\leq  \sum_{b_{11},b_{12}} \frac{\alpha^{2+2j-b_{11}-b_{22}}}{(j-b_{11}-2)!(j-b_{22}-2)! }\int_{\R_+^6} \mu(x_1)^{j-2-b_{11}}\mu(x_2)^{j-2-b_{22}} e^{- \alpha \mu(x) -\alpha \mu(y) }  \\
 & \qquad   \times W(x_1,y_1)W(x_1,z_1)W(y_1,z_1)W(x_2,y_2)W(x_2,z_2)W(y_2,z_2)\\
 &  \qquad   \times W(x_1,x_1)^{b_{11}}W(x_2,x_2)^{b_{22}}\{1 - W(x_1,x_1)\}^{1-b_{11}}\{1-W(x_2,x_2)\}^{1-b_{22}}dx_1dy_1dz_1dx_2dy_2dz_2\\
 &\quad+\sum_{b_{11},b_{12}} \frac{\alpha^{3+2j-b_{11}-b_{22}}}{(j-b_{11}-2)!(j-b_{22}-2)! }\int_{\R_+^6} \mu(x_1)^{j-2-b_{11}}\mu(x_2)^{j-2-b_{22}} \nu(x,y)e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha\nu(x,y)}  \\
 &\qquad    \times W(x_1,y_1)W(x_1,z_1)W(y_1,z_1)W(x_2,y_2)W(x_2,z_2)W(y_2,z_2)\\
 & \qquad   \times W(x_1,x_1)^{b_{11}}W(x_2,x_2)^{b_{22}}  \times\{1 - W(x_1,x_1)\}^{1-b_{11}}\{1-W(x_2,x_2)\}^{1-b_{22}}dx_1dy_1dz_1dx_2dy_2dz_2\\
 &\leq 4E(R_{\alpha,j})^2\\
 &\quad+\sum_{b_{11},b_{12}} \frac{\alpha^{3+2j-b_{11}-b_{22}}}{(j-b_{11}-2)!(j-b_{22}-2)! }\int_{\R_+^2} \mu(x_1)^{j-b_{11}}\mu(x_2)^{j-b_{22}} \nu(x,y)e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha\nu(x,y)}  \\
 & \qquad   \times W(x_1,x_1)^{b_{11}}W(x_2,x_2)^{b_{22}}  \times\{1 - W(x_1,x_1)\}^{1-b_{11}}\{1-W(x_2,x_2)\}^{1-b_{22}}dx_1dx_2\\
 &=4E(R_{\alpha,j})^2+O(\alpha^{3+2\sigma-2a}\ell_\sigma(\alpha)^2)
\end{align*}
using Lemma \ref{lemma:Ir} in Section \ref{sec:proofvarnnodesj}. It follows that
\begin{align}
E(S_{\alpha,j,1})=E(4R_{\alpha,j})^2 +O(\alpha^{3+2\sigma-2a}\ell_\sigma(\alpha)^2)\label{eq:temp3}
\end{align}

Consider now
\begin{align*}
S_{\alpha,j,2}=\sum_{i_1\neq k_1\neq l_1\neq i_2\neq k_2} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_1}Z_{k_2l_1}  \mathbbm{1}_{D_{\alpha i_1}=j}\mathbbm{1}_{D_{\alpha i_2}=j}\mathbbm{1}_{\theta_{i_1}\leq \alpha}  \mathbbm{1}_{\theta_{i_2}\leq \alpha}
\end{align*}
We have similarly
\begin{align}
E(S_{\alpha,j,2})
%&\leq \sum_{j_1=1}^{j-1} \sum_{b\in\{0,1\}^3} \frac{\alpha^{2+j+j_1-b_{11}-b_{12}-b_{22}}}{(j-j_1-b_{12}-1)!(j_1-b_{11}-1)!(j_1-b_{22}-1)! }\mathbbm{1}_{j_1\leq{j-b_{12}}}\mathbbm{1}_{j_1\geq b_{11}}\mathbbm{1}_{j_1\geq b_{22}}\nonumber\\
% &\qquad\times\int_{\R_+^6} \mu(x_1)^{j_1-1-b_{11}}\mu(x_2)^{j_1-1-b_{22}} \nu(x_1,x_2)^{j-j_1-b_{12}-1}e^{- \alpha \mu(x_1) -\alpha \mu(x_2) +\alpha \nu(x_1,x_2) }  \nonumber\\
% &\qquad  \qquad   \times W(x_1,y_1)W(x_1,z_1)W(y_1,z_1)W(x_2,y_2)W(x_2,z_1)W(y_2,z_1)\nonumber\\
% & \qquad  \qquad   \times W(x_1,x_1)^{b_{11}}W(x_2,x_2)^{b_{22}}W(x_1,x_2)^{b_{12}}\nonumber\\
% & \qquad  \qquad   \times\{1 - W(x_1,x_1)\}^{1-b_{11}}\{1-W(x_2,x_2)\}^{1-b_{22}}\{1 - W(x_1,x_2)\}^{1-b_{12}}dx_1dy_1dz_1dx_2dy_2\nonumber\\
%&\leq \sum_{j_1=1}^{j-1} \sum_{b\in\{0,1\}^3} \frac{\alpha^{2+j+j_1-b_{11}-b_{12}-b_{22}}}{(j-j_1-b_{12}-1)!(j_1-b_{11}-1)!(j_1-b_{22}-1)! }\mathbbm{1}_{j_1\leq{j-b_{12}}}\mathbbm{1}_{j_1\geq b_{11}}\mathbbm{1}_{j_1\geq b_{22}}\nonumber\\
% &\qquad\times\int_{\R_+^6} \mu(x_1)^{j_1-b_{11}}\mu(x_2)^{j_1-b_{22}} \nu(x_1,x_2)^{j-j_1-b_{12}}e^{- \alpha \mu(x) -\alpha \mu(y) +\alpha \nu(x_1,x_2) } \nonumber \\
% & \qquad  \qquad   \times W(x_1,x_1)^{b_{11}}W(x_2,x_2)^{b_{22}}W(x_1,x_2)^{b_{12}}\nonumber\\
% & \qquad  \qquad   \times\{1 - W(x_1,x_1)\}^{1-b_{11}}\{1-W(x_2,x_2)\}^{1-b_{22}}\{1 - W(x_1,x_2)\}^{1-b_{12}}dx_1dy_1dz_1dx_2dy_2\nonumber\\
 &=O(\alpha^{3+2\sigma-2a}\ell_\sigma(\alpha)^2)\label{eq:temp4}
 \end{align}
 using Lemma \ref{lemma:Ir}.
Similarly, using Lemma \ref{lemma:Ir},
$
E(S_{\alpha,j,3})=O(\alpha^{3+2\sigma-2a}\ell_\sigma(\alpha)^2).
$
\\
Combining the above bound with \eqref{eq:temp3} and  \eqref{eq:temp4}, we obtain
$
\var(R_{\alpha,j})= O(\alpha^{3+2\sigma-2a}\allowbreak\ell_\sigma(\alpha)^2).
$
We now consider the second bound in Proposition~\ref{prop:variancelocalclust}. Consider an increasing sequence  $\alpha_n \rightarrow \infty$ such that $\alpha_{n+1} - \alpha_n = o(\alpha_n)$ as $n\to\infty$. Let $I_{\alpha_n} = \{ i, \, \theta_i \leq \alpha_n\}$ and $I_n^c = I_{\alpha_{n+1}}\backslash I_{\alpha_{n}}$.

For any $j\geq 1$, let
\begin{align*}
\widetilde R_{nj}^{(1)}&:=\sum_{i\in I_{\alpha_{n}} }T_{\alpha_{n+1} i}\mathbbm{1}_{D_{\alpha_n i}=j} \mathbbm{1}_{ \sum_{i'\in I_n^c} Z_{ii'}=1}.
\end{align*}


We have, similarly to Equation~\eqref{Big:R2}
\begin{equation*}
\begin{split}
(\widetilde R_{nj}^{(1)})^2 & =
\sum_{i_1, i_2}^{I_{\alpha_n}}\sum_{k_1\neq l_1\neq i_1}^{I_{\alpha_{n+1}}}\sum_{i_2\neq k_2\neq l_2}^{I_{\alpha_{n+1}}} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_2}Z_{k_2l_2}  \\
&\qquad\qquad\qquad\times\mathbbm{1}_{D_{\alpha_n i_1}=j}\mathbbm{1}_{D_{\alpha_n i_2}=j}\mathbbm{1}_{\sum_{i_1^{'}\in I_n^c}Z_{i_1^{'}i_1} =1}\mathbbm{1}_{\sum_{i_2^{'}\in I_n^c}Z_{i_2^{'}i_2} =1}.
\end{split}
\end{equation*}
Hence using the same decomposition as \eqref{Big:R2}  together with  the fact that $\mathbbm{1}_{\sum_{i'\in I_n^c}Z_{i'i} =1} \leq 1$, we derive the same bounds as \eqref{eq:temp2}, \eqref{eq:temp1b} and \eqref{eq:temp4} so that
\begin{equation*}
E( (\widetilde R_{nj}^{(1)})^2 ) \lesssim \alpha_n^{\sigma +1}\ell_\sigma(\alpha_n) + \alpha_n^{3 +2\sigma -2a}\ell_\sigma(\alpha_n)^2 + E( \widetilde S_{\alpha_n,j,1,j}  )
\end{equation*}
where, writing $b = (b_{11}, b_{22})$,
\begin{equation*}
\begin{split}
 \widetilde S_{\alpha_n,j,1,j}
&=\sum_{b\in\{0,1\}^2} \sum_{\substack{i_1\neq i_2}}^{\in I_{\alpha_{n}}}  \sum_{\substack{ k_1\neq l_1\\ \neq k_2\neq l_2}}^{\in I_{\alpha_{n+1} } \setminus\{ i_1,i_2\}} Z_{i_1k_1}Z_{i_1l_1}Z_{k_1l_1}Z_{i_2k_2}Z_{i_2l_2}Z_{k_2l_2}\mathbbm{1}_{D_{\alpha_n i_1}=j}\mathbbm{1}_{D_{\alpha_n i_2}=j}\\
&\quad\times\mathbbm{1}_{\sum_{k\in I_{\alpha_{n+1}}} Z_{i_1k}Z_{i_2k}=0  }\mathbbm{1}_{Z_{i_1 i_1}=b_{11}}\mathbbm{1}_{Z_{i_2 i_2}=b_{22}} \mathbbm{1}_{\sum_{i'\in I_n^c}Z_{i'i_1} =1} \mathbbm{1}_{\sum_{i'\in I_n^c}Z_{i'i_2} =1},
\end{split}
\end{equation*}
so that
$ E(\widetilde S_{\alpha_n,j,1,j})
  = E(\widetilde R_{nj}^{(1)})^2$.
We thus obtain that
$ \var(\widetilde R_{nj}^{(1)}) = O ( \alpha_n^{3+2\sigma-2a} \ell_{\sigma}(\alpha_n)^2) .$




\subsection{Proof of Lemma~\ref{lemma:asymptT}}
\label{sec:app:asymptT}

Let  $I_n^c = I_{\alpha_{n+1}}\backslash I_{\alpha_{n}}=\{i\mid \theta_i\in(\alpha_n,\alpha_{n+1}]\}$. First note that $\widetilde R_{nj}=\sum_{r=1}^j\widetilde R_{nr}^{(1)}+\widetilde R_{nr}^{(2)}+\widetilde R_{nr}^{(3)}$ where
\begin{align*}
\widetilde R_{nr}^{(1)}&=\sum_{i\in I_{\alpha_{n}} }T_{\alpha_{n+1} i}\mathbbm{1}_{D_{\alpha_n i}=r} \mathbbm{1}_{ \sum_{i'\in I_n^c} Z_{ii'}=1},\\
\widetilde R_{nr}^{(2)}&=\sum_{i\in I_{\alpha_{n+1}} }T_{\alpha_{n+1} i}\mathbbm{1}_{D_{\alpha_n i}=r} \mathbbm{1}_{ \sum_{i'\in I_n^c} Z_{ii'}\geq 2},\\
\widetilde R_{nr}^{(3)}&=\sum_{i\in  I_n^c}T_{\alpha_{n+1} i}\mathbbm{1}_{D_{\alpha_n i}=r} \mathbbm{1}_{ \sum_{i'\in I_n^c} Z_{ii'}=1}.
\end{align*}

For any $r\leq j$ %\fc{why is it $\geq 1$ in the third line below?}
\begin{equation*}
\begin{split}
E\left(\left.  \widetilde R_{nr}^{(2)} \right| M\right) \leq &\sum_{i\in I_{\alpha_{n+1}}}\sum_{l\neq k}^{\in I_{\alpha_{n}},\neq i} W(\vartheta_{i},\vartheta_{l})W(\vartheta_{i},\vartheta_{k})W(\vartheta_{l},\vartheta_{k})J_n(i, r-2) \mathbb{P}\left( \sum_{i'\in I_n^c}Z_{ii'}\geq 2|M\right) \\
& +2 \sum_{i  \in I_{\alpha_{n+1}}}\sum_{ l \neq i }^{\in I_{\alpha_{n}}} \sum_{ k\neq i}^{\in I_n^c} W(\vartheta_{i},\vartheta_{l})W(\vartheta_{i},\vartheta_{k})W(\vartheta_{l},\vartheta_{k})J_n(i, r-1)  \mathbb{P}\left( \sum_{i'\in I_n^c}Z_{ii'}\geq 1|M\right) \\
& + \sum_{i \in I_{\alpha_{n+1}}} \sum_{ l\neq k}^{\in I_n^c} W(\vartheta_{i},\vartheta_{l})W(\vartheta_{i},\vartheta_{k})W(\vartheta_{l},\vartheta_{k})J_n(i, r)
\end{split}
\end{equation*}
where, recalling the definition of $g_{\alpha,x}$ in Equation~\eqref{eq:defg},
$$J_n (i, r) = \sum_{i_{1}\neq i_{2}\ldots\neq i_{r}\neq l\neq k}^{\in  I_{\alpha_n}} \left[
\prod_{s=1}^{r}W(\vartheta_{i},\vartheta_{i_s})\right]  e^{-\sum_{s\neq
l,k,i_{1},\ldots,i_{r}}^{I_{\alpha_n}}g_{\alpha, \vartheta_{i}}(\theta_{s},\vartheta_{s})}.$$
Note that
\begin{align*}
 \mathbb{P}\left( \sum_{i'\in I_n^c}Z_{ii'}\geq 2|M\right) &= 1 - e^{-\sum_{s\in I_n^c}g_{\alpha_{n+1}-\alpha_n,\vartheta_{i}}(\theta_{s},\vartheta_{s})}- \sum_{i' \in I^c_n}W(\vartheta_i, \vartheta_{i'}) e^{-\sum_{s\neq i'}^{I_n^c}g_{\alpha_{n+1}-\alpha_n,\vartheta_{i}}(\theta_{s},\vartheta_{s})}. \end{align*}
Using the Slivnyak-Mecke theorem, the inequality $1-e^{-y}-ye^{-y}\leq y^2$ for $y\geq 0$, the condition \eqref{cond:localcluster} and Lemma \ref{lemma:abelianvariations2}, we obtain
\begin{align*}
E \left(  \widetilde R_{nr}^{(2)} \right) &\lesssim \alpha_{n+1}\alpha_n^{r}(\alpha_{n+1} - \alpha_n)^2 \int L_2(x) \mu(x)^{r+2}e^{-\alpha_n \mu(x)} dx\lesssim  \alpha_{n+1}\alpha_n^{\sigma-2}(\alpha_{n+1} - \alpha_n)^2\ell(\alpha_n) ,
\end{align*}
where $L_2(x)$ converges to $b\geq 0$ at infinity. Noting that
$
(\alpha_{n+1}-\alpha_n)/\alpha_n=O(1/n),
$
 we obtain
$
E \left(  \widetilde R_{nr}^{(2)} \right)  \lesssim  \alpha_n^{\sigma+1} \ell(\alpha_n) / n^2.
$
 This implies that
$\sum_n E \left(   \widetilde R_{nr}^{(2)} \right)/ (\alpha_n^{\sigma+1} \ell(\alpha_n) ) <+\infty$
so that, by Markov inequality and Borel-Cantelli lemma,
$\widetilde R_{nr}^{(2)}=o( \alpha_n^{\sigma+1} \ell(\alpha_n))$
almost surely as $n$ tends to infinity. \\We now study
$$  \widetilde R_{nr}^{(3)} := \sum_{i\in I_n^c }T_{\alpha_{n+1} i}\mathbbm{1}_{D_{\alpha_n i}=r} \mathbbm{1}_{ \sum_{i'\in I_n^c}Z_{ii'}=1}. $$
Similarly to before
\begin{equation*}
\begin{split}
E \left( \widetilde R_{nr}^{(3)} \mid M \right) & \leq\sum_{i\in I_n^c}  \sum_{ l\neq k}^{\in I_{\alpha_{n}}} W(\vartheta_{i},\vartheta_{l})W(\vartheta_{i},\vartheta_{k})W(\vartheta_{l},\vartheta_{k})J_n(i, r-2) \mathbb{P}\left( \sum_{i'\in I_n^c}Z_{ii'}=1|M\right) \\
& \quad+2\sum_{l \in I_n^c}^{l\neq i}  \sum_{ k \in I_{\alpha_{n}}} W(\vartheta_{i},\vartheta_{l})W(\vartheta_{i},\vartheta_{k})W(\vartheta_{l},\vartheta_{k})J_n(i, r-1)
\end{split}
\end{equation*}
 so that
 \begin{equation*}
\begin{split}
E\left( \widetilde R_{nr}^{(3)} \right)
& \lesssim (\alpha_{n+1} - \alpha_n)^2 \alpha_n^{r} \int L_3(x) \mu(x)^{r+1} e^{-\alpha_{n} \mu(x) } dx \lesssim   \frac{\alpha_n^{\sigma+1} \ell(\alpha_n)  }{ n^2 },
\end{split}
\end{equation*}
where $L_3(x)$ converges to $b$ and $\widetilde R_{nr}^{(3)}  = o(\alpha_n^{\sigma+1} \ell(\alpha_n)  )$ almost surely as $n$ tends to infinity.
Finally, we have
 \begin{equation*}
\begin{split}
E \left(\widetilde {R}_{nr}^{(1)} |M  \right) & \lesssim   \sum_{i\neq l\neq k}^{\in I_{\alpha_{n}}} W(\vartheta_{i},\vartheta_{l})W(\vartheta_{i},\vartheta_{k})W(\vartheta_{l},\vartheta_{k})J_n(i, r-2)\sum_{i' \in I_n^c} W(\vartheta_i, \vartheta_{i'}) \\
& \quad+2\sum_{i \neq l }^{\in I_{\alpha_{n}}} \sum_{ k \in I_n^c } W(\vartheta_{i},\vartheta_{l})W(\vartheta_{i},\vartheta_{k})W(\vartheta_{l},\vartheta_{k})J_n(i, r-1)
\end{split}
\end{equation*}
which implies that
 \begin{equation*}
\begin{split}
E \left(\frac{ \widetilde {R}^{(1)}_{nr}}{ \alpha_n^{\sigma+1} \ell_\sigma(\alpha_n)}  \right) & \quad \lesssim   \frac{ \alpha_n^{r+1}(\alpha_{n+1}-\alpha_n)}{ \alpha_n^{\sigma+1} \ell_\sigma(\alpha_n)} \int_{\mathbb R_+^{4}} L_1(x)\mu(x)^{ r+1}e^{-\alpha_n \mu(x)} dx \\
 & \quad = O\left(\frac{ \alpha_{n+1}-\alpha_n}{ \alpha_n}  \right) = o(1).
\end{split}
\end{equation*}
where $L_1(x)$ converges to $b$. Moreover, from Proposition \ref{prop:variancelocalclust} %{\color{red} FP: I don't get how this follows. Shouldn't it be a little o (from prop A.5 + to get the convergence for Borel Cantelli)? Also, shouldn't we have a $(\alpha_{n+1}-\alpha_n)^2$ in the statement of the proposition, to achieve the $1/n^2$?}
 $$ \var \left(\frac{ \widetilde {R}_{n r}^{(1)}}{ \alpha_n^{\sigma+1} \ell_\sigma(\alpha_n)}  \right) = O\left(  \alpha_n^{ 1-2a}  \right)
 $$
 so that, $\widetilde {R}_{nr}^{(1)}=o(\alpha_n^{1+\sigma}\ell(\alpha_n))$ almost surely. It finally follows that, for any $j\geq 1$,   $\widetilde R_{nj} = o(\alpha_n^{1+\sigma}\ell(\alpha_n))$ almost surely as $n$ tends to infinity.


\section{Proof of secondary propositions for the Central Limit Theorem}
\label{sec:proofsecondCLT}


\subsection{Proof of Proposition \ref{prop:normpart1}}%

Let
\[
Z_\alpha:=N_{\alpha}-E(N_{\alpha}\mid M)=\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha
}(\mathbbm{1}_{D_{\alpha,i}\geq1}-(1-e^{-M(g_{\alpha,\vartheta_{i}})}))
\]


where we recall that $g_{\alpha,x}(\theta,\vartheta)    =-\log(1-W(x,\vartheta))\mathbbm{1}_{\theta\leq
\alpha}$ and
\begin{equation*}\\
e^{-M(g_{\alpha,\vartheta_{i}})}    =e^{-\sum_{j}-\log(1-W(\vartheta
_{i},\vartheta_{j}))\mathbbm{1}_{\theta_{j}\leq\alpha}}
  =\prod_{j}(1-W(\vartheta_{i},\vartheta_{j}))^{\mathbbm{1}_{\theta_{j}\leq\alpha}}%
\end{equation*}
We have $E(Z_{\alpha}\mid M)=0$ hence $\var(Z_\alpha)=E(Z_\alpha^2)$. Note that
\begin{align*}
Z_{\alpha}^{2}  &  =Z_{\alpha}+\sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}%
\leq\alpha}(\mathbbm{1}_{D_{\alpha,i_{1}}\geq1}-(1-e^{-M(g_{\alpha,\vartheta_{i_{1}}}%
)}))\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}(\mathbbm{1}_{D_{\alpha,i_{2}}\geq1}-(1-e^{-M(g_{\alpha
,\vartheta_{i_{2}}})}))\\
&  =Z_{\alpha}+\sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}%
\mathbbm{1}_{D_{\alpha,i_{1}}\geq1}\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}\mathbbm{1}_{D_{\alpha,i_{2}}\geq
1} -\sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}\mathbbm{1}_{D_{\alpha,i_{1}}%
\geq1}\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}(1-e^{-M(g_{\alpha,\vartheta_{i_{2}}})})\\
&  -\sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}(1-e^{-M(g_{\alpha
,\vartheta_{i_{1}}})}))\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}\mathbbm{1}_{D_{\alpha,i_{2}}\geq1}+\sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}(1-e^{-M(g_{\alpha
,\vartheta_{i_{1}}})})\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}(1-e^{-M(g_{\alpha
,\vartheta_{i_{2}}})})
\end{align*}
We have
\begin{align*}
E\left(  Z_{\alpha}^{2}\mid M\right)
%&  =\sum_{i_{1}\neq i_{2}%
%}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}\mathbbm{1}_{\theta_{i_{2}}\leq\alpha}\left\{
%1-e^{-M(g_{\alpha,\vartheta_{i_{1}}})}-e^{-M(g_{\alpha,\vartheta_{i_{2}}}%
%)}+e^{-M(g_{\alpha,\vartheta_{i_{1}}})-M(g_{\alpha,\vartheta_{i_{2}}%
%})+g_{\alpha,\vartheta_{i_{1}}}(\theta_{i_{2}},\vartheta_{i_{2}})}\right\} \\
%&  -\sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}\mathbbm{1}_{\theta_{i_{2}}%
%\leq\alpha}(1-e^{-M(g_{\alpha,\vartheta_{i_{1}}})})(1-e^{-M(g_{\alpha
%,\vartheta_{i_{2}}})})\\
&  =\sum_{i_{1}\neq i_{2}}\mathbbm{1}_{\theta_{i_{1}}\leq\alpha}\mathbbm{1}_{\theta_{i_{2}}%
\leq\alpha}e^{-M(g_{\alpha,\vartheta_{i_{1}}})-M(g_{\alpha,\vartheta_{i_{2}}%
})}(e^{g_{\alpha,\vartheta_{i_{1}}}(\theta_{i_{2}},\vartheta_{i_{2}})}-1)
\end{align*}
Applying the extended Slivnyak-Mecke theorem%
\begin{align*}
E\left(  Z_{\alpha}^{2}\right)   &  =\alpha^{2}\int_{\mathbb{R}%
_{+}^{2}}(1-W(x,x))(1-W(y,y))(1-W(x,y))e^{-\alpha\mu(x)-\alpha\mu(y)+\alpha
\nu(x,y)}(1-(1-W(x,y)))dxdy\\
&  \leq\alpha^{2}\int_{\mathbb{R}_{+}^{2}}W(x,y)e^{-\alpha\mu(x)-\alpha
\mu(y)+\alpha\nu(x,y)}dxdy.
\end{align*}
Using Cauchy-Schwarz inequality, $\nu(x,y)\leq\sqrt{\mu(x)\mu(y)}\leq
\frac{1}{2}(\mu(x)+\mu(y))$, and Lemma \ref{lemma:abelianvariations}, we obtain
\begin{align*}
E\left(  Z_{\alpha}^{2}\right)
%&  \leq\alpha^{2}\int_{\mathbb{R}%
%_{+}^{2}}W(x,y)e^{-\alpha/2\mu(x)-\alpha/2\mu(y)}dxdy\\
%&  \leq\alpha^{2}\int_{\mathbb{R}_{+}^{2}}W(x,y)e^{-\alpha/2\mu(x)}dxdy\\
&  =\alpha^{2}\int_{0}^{\infty}\mu(x)e^{-\alpha/2\mu(x)}dx\asymp
\alpha^{1+\sigma}\ell(\alpha)
=\left \{
\begin{array}{ll}
  O(\alpha^{1+\sigma}\ell_\sigma(\alpha)) & \sigma\in[0,1)\\
  o(\alpha\ell(\alpha)) & \sigma=0
\end{array}\right. .
\end{align*}




It follows from Markov's inequality that, in probability
$$
Z_{\alpha}=\left \{
\begin{array}{ll}
  O(\alpha^{1/2+\sigma/2}\ell^{1/2}_\sigma(\alpha)) & \sigma\in[0,1)\\
  o(\alpha^{1/2}\ell^{1/2}(\alpha)) & \sigma=0
\end{array}\right. .
$$

\subsection{Proof of Proposition \ref{prop:normpart2}}

Define
$
M(h_{\alpha})=\sum_{i}\widetilde{Z}_{i}%
$
where
$
\widetilde{Z}_{i}=h_{\alpha}(\theta_{i},\vartheta_{i})=\mathbbm{1}_{\theta_{i}\leq
\alpha}\left[  1-(1-W(\vartheta_{i},\vartheta_{i}))e^{-\alpha\mu(\vartheta_{i}%
)}\right]  .
$
Using Campbell's formula%
\begin{align*}
E\left(  \sum_{i}\widetilde{Z}_{i}\right)   &  =\alpha\int%
_{0}^{\infty}(1-(1-W(x,x))e^{-\alpha\mu(x)})dx=E(N_{\alpha})\\
\var\left(  \sum_{i}\widetilde{Z}_{i}\right)   &  =\alpha\int_{0}^{\infty
}\left[  1-(1-W(x,x))e^{-\alpha\mu(x)}\right]  ^{2}dx\leq E(N_{\alpha
})
\end{align*}
Noting that $E(N_{\alpha
})\sim\alpha^{1+\sigma}\Gamma(1-\sigma)\ell(\alpha)$, it follows from Chebyshev's inequality that, in probability,
\begin{align*}
\sum_{i}\widetilde{Z}_{i}-E(N_{\alpha})  &  =O\left(
\sqrt{\var\left(  \sum_{i}\widetilde{Z}_{i}\right)  }\right) =O\left(  \alpha^{1/2+\sigma/2}\ell_\sigma^{1/2}(\alpha)\right)
\end{align*}
If $\mu$ has an unbounded support then, under Assumption \ref{assumpt:1}, either $\sigma>0$ or $\sigma=0$ and $\ell(t)\to\infty$. In both cases, in probability,
$\sum_{i}\widetilde{Z}_{i}-E(N_{\alpha}) =o\left(  \alpha^{1/2+\sigma}\ell_\sigma(\alpha)\right).$





\subsection{Proof of Proposition \ref{prop:normpart3}}
\label{sec:proofCLTsparse}

Let
\begin{align*}
f_{\alpha}(M)
&  =\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}\left[  (1-W(\vartheta_{i},\vartheta
_{i}))e^{-\alpha\mu(\vartheta_{i})}-e^{-M(g_{\alpha,\vartheta_{i}})}\right]
\end{align*}


The idea is to use Theorem 1.1 from \cite{Last2016}. To do so, define
\begin{equation}
F_\alpha=\frac{f_{\alpha}(M)}{\sqrt{v_{\alpha}}}
\end{equation} where $v_{\alpha}= \var(f_{\alpha}(M))\sim \var(N_{\alpha} )\asymp\alpha^{1+2\sigma}\ell_\sigma^{2}(\alpha)$. Note that $E(F_\alpha)=0$ and $\var(F_\alpha)=1$. Consider the difference operator $D_{z}F_\alpha$ defined by%
\[
D_{z}F_\alpha=\frac{1}{\sqrt{v_\alpha}}( f_\alpha(M+\delta_{z})-f_\alpha(M))
\]
Also%
\begin{align*}
D_{z_1,z_2}^{2}F_\alpha  &  =D_{z_2}(D_{z_1}F_\alpha)=D_{z_2}\left(\frac{1}{\sqrt{v_\alpha}} (f_\alpha(M+\delta_{z_1})-f_\alpha(M))\right )\\
&  =\frac{1}{\sqrt{v_\alpha}}\left( f_\alpha(M+\delta_{z_1}+\delta_{z_2})- f_\alpha(M+\delta_{z_1})- f_\alpha(M+\delta_{z_2}
)+ f_\alpha(M)\right ).
\end{align*}
Define
\begin{align*}
\gamma_{\alpha,1}  &  :=2\left(  \int_{\mathbb{R}_{+}^{6}}\sqrt{\mathbb{E}(D_{z_{1}%
}F_\alpha)^{2}(D_{z_{2}}F_\alpha)^{2}}\sqrt{\mathbb{E}(D_{z_{1},z_{3}}^{2}F_\alpha)^{2}%
(D_{z_{2},z_{3}}^{2}F_\alpha)^{2}}dz_{1}dz_{2}dz_{3}\right)  ^{1/2}\\
\gamma_{\alpha,2}  &  :=\left(  \int_{\mathbb{R}_{+}^{6}}\mathbb{E}\left [(D_{z_{1},z_{3}%
}^{2}F_\alpha)^{2}(D_{z_{2},z_{3}}^{2}F_\alpha)^{2}\right ]dz_{1}dz_{2}dz_{3}\right)
^{1/2}\\
\gamma_{\alpha,3}  &  :=\int_{\mathbb{R}_{+}^{2}}\mathbb{E}|D_{z}F_\alpha|^{3}dz
\end{align*}

We state a corollary of Theorem 1.1 from \cite{Last2016}.
\begin{cor}{\cite[Theorem 1.1]{Last2016}}
If $\gamma_{\alpha,1},\gamma_{\alpha,2},\gamma_{\alpha,3}\rightarrow0,$ then
$$F_\alpha=\frac{f_{\alpha}(M)}{\sqrt{v_{\alpha}}}\to\mathcal N(0,1).$$
\end{cor}
The rest of the proof aims to show that $\gamma_{\alpha,1},\gamma_{\alpha,2},\gamma_{\alpha,3}\rightarrow0$. The proof is rather lengthy and therefore split in different subsections. We first state a few notations and lemmas that will be useful in the following.

\subsubsection{Definitions and lemmas}

The following lemma, obtained with H\"older's inequality, will be used multiple times.
\begin{lem}\label{lemma:mu}
For any $d\geq 1$ and any $z_1,\ldots,z_d>0$,
\begin{align*}
E\left ( \prod_{k=1}^d e^{-M(g_{\alpha,z_k})}  \right )&\leq e^{-\frac{\alpha}{d}\sum_{k=1}^d \mu(z_k)}.
\end{align*}
\end{lem}
\begin{proof}
Using H\"older's inequality, for any $d\geq 1$
\begin{align*}
E\left ( \prod_{k=1}^d e^{-M(g_{\alpha,z_k})}  \right )&\leq \prod_{k=1}^d E\left (  e^{-d M(g_{\alpha,z_k})}  \right )^{1/d}= \prod_{k=1}^d e^{-\frac{\alpha}{d}\int_0^\infty (1-[1-W(z_k, y)]^d) dy}\\
&\leq \prod_{k=1}^d e^{-\frac{\alpha}{d}\int_0^\infty (1-[1-W(z_k, y)]) dy}=e^{-\frac{\alpha}{d}\sum_{k=1}^d \mu(z_k)}
\end{align*}
\end{proof}


For $i,j\geq 0$, let
\begin{align}
H_{i,j}(x_1,x_2)&=\int_{\mathbb R_+^2}W(x_1,y)^iW(x_2,y)^je^{-\frac{\alpha}{4}\mu(y)}dy , \quad H_{i}(x)  = H_{i,0}(x,x). \label{eq:H1}
\end{align}
The following lemma compiles various useful bounds.
\begin{lem}\label{lemma:bound} Assume Assumptions 1 and 5. Then
\begin{itemize}
\item For all $j\geq 1$ and all $x_1,\ldots,x_{j-1}>0$, $y_1>0$ and $p_1,\ldots,p_{j}\geq 1$, $\alpha>0$
\begin{align*}
&\int_0^\infty \left (\int_0^\infty W(y_1,x_j) ^{p_j}\left [\prod_{k=1}^{j-1} W(y_1,x_k)^{p_k}\right ]e^{-\alpha\mu(y_1) } dy_1 \right )^{1/2}W(y_2,x_j)dx_j\\
%&\leq  \left (\int W(y_1,x_j) ^{p_j}\left [\prod_{k=1}^{j-1} W(y_1,x_k)^{p_k}\right ]e^{-\alpha\mu(y_1) } dy_1 %dx_j \int  W(y_2,x_j)^2dx_j\right )^{1/2}\\
&\leq L(y_2)\mu(y_2) \left (\int L(y_1)^{p_j}\mu(y_1)^{p_j} \left [\prod_{k=1}^{j-1} W(y_1,x_k)^{p_k}\right ]e^{-\alpha\mu(y_1) } dy_1 \right )^{1/2}
\end{align*}
\item For any $x_2>0$,
\begin{align}
\int H_{1,1}(x_1,x_2)^2d x_1
&\leq \int L(y_1)L(y_2)\mu(y_1)\mu(y_2)W(x_2,y_1)W(x_2,y_2)e^{-\frac{\alpha}{4}(\mu(y_1)+\mu(y_2))}dy_1dy_2\label{eq:boundH2square}
\end{align}
\item For any $q=1,2,\ldots$ \begin{align}
\int_0^\infty H_1(x)^{q}dx&=\int \prod_{i=1}^q \int W(x,y_i) e^{-\frac{\alpha}{4} \mu(y_i)}dy_i dx
&=O(\alpha^{q\sigma-q}\ell_\sigma(\alpha)^q)\label{eq:boundintH1}
\end{align}
\item   For any $q\geq 1$, $p\geq 1$,
\begin{align}
\int (\alpha H_1(x_1))^{q/2}L(x_1)\mu(x_1)^{p/2} e^{-\frac{\alpha}{4}\mu(x_1)}dx_1
&=O(\alpha^{(q+1)\sigma/2 -p/2 }\ell_\sigma(\alpha)^{(q+1)/2})\label{eq:boundH1a}
\end{align}
\item If $q\leq 3$,
\begin{align}
\int L(x_1)\mu(x_1)^p W(x_1,x_2)^q e^{-\frac{\alpha}{4}\mu(x_1)}dx_1&\leq \mu(x_2)^q L(x_2)^q\left (\int L(x_1)^2\mu(x_1)^{2p}  e^{-\frac{\alpha}{2}\mu(x_1)}dx_1\right)^2 .\label{eq:boundH1c}
\end{align}
\end{itemize}
\end{lem}

\begin{proof}
The first inequality comes from H\"older's inequality, together with Assumptions \ref{assumpt:5} and \ref{assumpt:1}.
Also \eqref{eq:boundH2square} is a consequence of
$$ \int H_{1,1}(x_1,x_2)^2d x_1 =\int \nu(y_1,y_2)W(x_2,y_1)W(x_2,y_2)e^{-\frac{\alpha}{4}(\mu(y_1)+\mu(y_2))}dy_1dy_2$$

Under Assumption \ref{assumpt:5},
For any $q=1,2,\ldots$, using Assumption \ref{assumpt:1}, \ref{assumpt:5}, and Lemma \ref{lemma:abelianvariations2},
\begin{align*}
\int_0^\infty H_1(x)^{q}dx&=\int \prod_{i=1}^q \int W(x,y_i) e^{-\frac{\alpha}{4} \mu(y_i)}dy_i dx\nonumber \\
&\leq \prod_{i=1}^q \int L(y_i)\mu(y_i) e^{-\frac{\alpha}{4} \mu(y_i)}dy_i\nonumber=O(\alpha^{q\sigma-q}\ell_\sigma(\alpha)^q).
\end{align*}
Using H\"older's inequality,
\begin{align*}
\int (\alpha H_1(x_1))^{q/2}L(x_1)\mu(x_1)^{p/2} e^{-\frac{\alpha}{4}\mu(x_1)}dx_1&\leq \left (\int (\alpha H_1(x_1))^{q}dx_1 \int L(x_1)^2\mu(x_1)^{p} e^{-\frac{\alpha}{2}\mu(x_1)}dx_1    \right )^{1/2}\nonumber\\
&=O(\alpha^{(q+1)\sigma/2 -p/2 }\ell_\sigma(\alpha)^{(q+1)/2})
\end{align*}
which proves \eqref{eq:boundH1a}. Finally
recall that from Lemma \ref{lemma:abelianvariations2} that for any $p\geq 1$,
\begin{align*}
\int L(x_1)\mu(x_1)^pe^{-\frac{\alpha}{4}\mu(x_1)}dx_1&=O(\alpha^{\sigma -p }\ell_\sigma(\alpha)).
\end{align*}
so that using H\"older and Assumption \ref{assumpt:5}, for any $q\leq 3$
\begin{align*}
\int L(x_1)\mu(x_1)^p W(x_1,x_2)^q e^{-\frac{\alpha}{4}\mu(x_1)}dx_1&\leq \mu(x_2)^q L(x_2)^q\left (\int L(x_1)^2\mu(x_1)^{2p}  e^{-\frac{\alpha}{2}\mu(x_1)}dx_1\right)^2
\end{align*}

\end{proof}

\subsubsection{General bounds}

Let
$z=(t,x)$. Recall that $g_{\alpha,x}(\theta,\vartheta)=-\log(1-W(x,\vartheta))\mathbbm{1}_{\theta%
\leq\alpha}$.%
\begin{align*}
\sqrt{v_{\alpha}}\times D_{z}F_\alpha
%&  =\mathbbm{1}_{t\leq\alpha}\left[  (1-W(x,x))e^{-\alpha\mu(x)}-e^{-M(g_{\alpha,x})-g_{\alpha,x}(z)}\right] \\
%&  +\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}\left[  (1-W(\vartheta_{i},\vartheta
%_{i}))e^{-\alpha\mu(\vartheta_{i})}-e^{-M(g_{\alpha,\vartheta_{i}})-g_{\alpha
%,\vartheta_{i}}(z)}\right] \\
%&  -\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}\left[  (1-W(\vartheta_{i},\vartheta
%_{i}))e^{-\alpha\mu(\vartheta_{i})}-e^{-M(g_{\alpha,\vartheta_{i}})}\right] \\
%&  =\mathbbm{1}_{t\leq\alpha}\left[  (1-W(x,x))e^{-\alpha\mu(x)}-e^{-M(g_{\alpha,x})-g_{\alpha,x}(z)}\right] \\
%&  +\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}\left[  e^{-M(g_{\alpha,\vartheta_{i}}%
%)}(1-e^{-g_{\alpha,\vartheta_{i}}(z)})\right] \\
 &  =\mathbbm{1}_{t\leq\alpha}(1-W(x,x))\left[  e^{-\alpha\mu(x)}-e^{-M(g_{\alpha,x})}\right]   +\mathbbm{1}_{t\leq\alpha}\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}W(\vartheta_{i},x)e^{-M(g_{\alpha ,\vartheta_{i}})}.%
\end{align*}
We have
\begin{align}
\sqrt{v_\alpha}\left\vert D_{z}F_\alpha\right\vert
& \leq \mathbbm{1}_{t\leq\alpha}\left( \left|e^{-\alpha\mu(x)}-e^{-M(g_{\alpha,x})}\right| +\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}W(\vartheta_{i},x)e^{-M(g_{\alpha,\vartheta_{i}})}  \right)\label{eq:Dz}
\end{align}

Similarly,
\begin{align*}
\sqrt{v_\alpha}D^2_{z_1,z_2}(F_{\alpha})
%&  =\mathbbm{1}_{t_1\leq\alpha}\left[  (1-W(x_1,x_1))e^{-\mu(x_1)}-e^{-M(g_{\alpha,x_1})-g_{\alpha,x_1}(z_1)-g_{\alpha,x_1}(z_2)}\right] \\
%&+ \mathbbm{1}_{t_2\leq\alpha}\left[  (1-W(x_2,x_2))e^{-\mu(x_2)}-e^{-M(g_{\alpha,x_2})-g_{\alpha,x_2}(z_1)-g_{\alpha,x_2}(z_2)}\right] \\
%&  +\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}\left[  (1-W(\vartheta_{i},\vartheta
%_{i}))e^{-\mu(\vartheta_{i})}-e^{-M(g_{\alpha,\vartheta_{i}})-g_{\alpha
%,\vartheta_{i}}(z_1)-g_{\alpha
%,\vartheta_{i}}(z_2)}\right] \\
%&-\mathbbm{1}_{t_1\leq\alpha}\left[  (1-W(x_1,x_1))e^{-\mu(x_1)}-e^{-M(g_{\alpha,x_1})-g_{\alpha,x_1}(z_1)}\right] \\
%&  -\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}\left[  (1-W(\vartheta_{i},\vartheta
%_{i}))e^{-\mu(\vartheta_{i})}-e^{-M(g_{\alpha,\vartheta_{i}})-g_{\alpha
%,\vartheta_{i}}(z_1)}\right] \\
%&-\mathbbm{1}_{t_2\leq\alpha}\left[  (1-W(x_2,x_2))e^{-\mu(x_2)}-e^{-M(g_{\alpha,x_2})-g_{\alpha,x_2}(z_2)}\right] \\
%&  -\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}\left[  (1-W(\vartheta_{i},\vartheta
%_{i}))e^{-\mu(\vartheta_{i})}-e^{-M(g_{\alpha,\vartheta_{i}})-g_{\alpha
%,\vartheta_{i}}(z_2)}\right] \\
%&  +\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}\left[  (1-W(\vartheta_{i},\vartheta
%_{i}))e^{-\mu(\vartheta_{i})}-e^{-M(g_{\alpha,\vartheta_{i}})}\right] \\
&  =\mathbbm{1}_{t_1, t_2\leq\alpha} W(x_1,x_2)\left[  (1-W(x_1,x_1)) e^{-M(g_{\alpha,x_1})}
 + (1-W(x_2,x_2))e^{-M(g_{\alpha,x_2})}\right] \\
&  -\mathbbm{1}_{t_1, t_2\leq\alpha}\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha} W(\vartheta_i,x_1)W(\vartheta_i,x_2) e^{-M(g_{\alpha,\vartheta_{i}}) }
\end{align*}

Note that the above is equal to 0 if $t_1>\alpha$ or $t_2>\alpha$. For $t_1,t_2\leq \alpha$
\begin{align*}
|\sqrt{v_{\alpha}}\times D^2_{z_1,z_2}F_\alpha|^2&\leq 2
W(x_1,x_2)^2(e^{-M(g_{\alpha,x_1})}+e^{-M(g_{\alpha,x_2})})^2 \\
&  +2\left (\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha} W(\vartheta_i,x_1)W(\vartheta_i,x_2) e^{-M(g_{\alpha,\vartheta_{i}})}\right )^2\\
&\leq 4 W(x_1,x_2)^2(e^{-M(g_{\alpha,x_1})}+e^{-M(g_{\alpha,x_2})})+2\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha} W(\vartheta_i,x_1)^2 W(\vartheta_i,x_2)^2 e^{-2M(g_{\alpha,\vartheta_{i}})}\\
&+2\sum_{i\neq j}\mathbbm{1}_{\theta_{i}\leq\alpha}\mathbbm{1}_{\theta_{j}\leq\alpha} W(\vartheta_i,x_1)W(\vartheta_i,x_2)W(\vartheta_j,x_1)W(\vartheta_j,x_2) e^{-M(g_{\alpha,\vartheta_{i}})-M(g_{\alpha,\vartheta_{j}})}
\end{align*}
\begin{align*}
&v_\alpha^2(D_{z_{1},z_{3}%
}^{2}F_\alpha)^{2}(D_{z_{2},z_{3}}^{2}F_\alpha)^{2}\nonumber\\
&\leq 16 W(x_1,x_3)^2W(x_2,x_3)^2(e^{-M(g_{\alpha,x_1})}+e^{-M(g_{\alpha,x_3})})(e^{-M(g_{\alpha,x_2})}+e^{-M(g_{\alpha,x_3})})\nonumber\\
&\quad+8W(x_1,x_3)^2(e^{-M(g_{\alpha,x_1})}+e^{-M(g_{\alpha,x_3})})\left (\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha} W(\vartheta_i,x_2)W(\vartheta_i,x_3) e^{-M(g_{\alpha,\vartheta_{i}})}\right )^2\nonumber\\
&\quad+8W(x_2,x_3)^2(e^{-M(g_{\alpha,x_2})}+e^{-M(g_{\alpha,x_3})})\left (\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha} W(\vartheta_i,x_1)W(\vartheta_i,x_3) e^{-M(g_{\alpha,\vartheta_{i}})}\right )^2\nonumber\\
&\quad+4\sum_{i_1,i_2,i_3,i_4}\mathbbm{1}_{\theta_{i_1}\leq\alpha}\mathbbm{1}_{\theta_{i_2}\leq\alpha}\mathbbm{1}_{\theta_{i_3}\leq\alpha}\mathbbm{1}_{\theta_{i_4}\leq\alpha} W(\vartheta_{i_1},x_1)W(\vartheta_{i_1},x_3)W(\vartheta_{i_2},x_1)W(\vartheta_{i_2},x_3)\nonumber\\
&\qquad \quad\quad\times W(\vartheta_{i_3},x_2)W(\vartheta_{i_3},x_3)W(\vartheta_{i_4},x_2)W(\vartheta_{i_4},x_3)  e^{-\sum_{k=1}^4 M(g_{\alpha,\vartheta_{i_k}})}
\end{align*}

We obtain, using the inequality~\eqref{eq:boundH2square}
\begin{align}
&E\left (v_\alpha^2(D_{z_{1},z_{3}%
}^{2}F)^{2}(D_{z_{2},z_{3}}^{2}F)^{2}\right ) \nonumber\\
&\leq C\times\left ( W(x_1,x_3)^2W(x_2,x_3)^2(e^{-\alpha/2\mu(x_1))}+e^{-\alpha/2\mu(x_3)})(e^{-\alpha/2\mu(x_2)}+e^{-\alpha/2 \mu(x_3)})\right .\nonumber\\
&+(\alpha^2H_{1,1}(x_2,x_3)^2+\alpha H_{2,2}(x_2,x_3)) W(x_1,x_3)^2(e^{-\alpha/3 \mu(x_1)}+e^{-\alpha/3 \mu(x_3))})\nonumber\\
&+(\alpha^2H_{1,1}(x_1,x_3)^2+\alpha H_{2,2}(x_1,x_3)) W(x_2,x_3)^2(e^{-\alpha/3 \mu(x_2)}+e^{-\alpha/3 \mu(x_3))})\nonumber\\
&\left .+A_2(x_1,x_2,x_3)\right )\label{eq:longinequality}
\end{align}
for some constant $C>0$, where
\bgroup
\allowdisplaybreaks
\begin{align*}
&A_2(x_1,x_2,x_3)\\
&=E\left (\sum_{i_1,i_2,i_3,i_4}\prod_{\ell = 1}^4W(\vartheta_{i_\ell },x_3)\mathbbm{1}_{\theta_{i_\ell}\leq\alpha} W(\vartheta_{i_1},x_1)W(\vartheta_{i_2},x_1)
 W(\vartheta_{i_3},x_2)W(\vartheta_{i_4},x_2) e^{-\sum_{k=1}^4 M(g_{\alpha,\vartheta_{i_k}})}\right )\\
&=\alpha^4 \int \prod_{\ell = 1}^4W(y_\ell,x_3)  W(y_1,x_1)W(y_2,x_1)W(y_2,x_3)W(y_3,x_2)W(y_4,x_2) e^{-\alpha/4 (\sum_{i=1}^4\mu(y_i))}dy_{1:4} \\
&\quad+\alpha^3 \int  (W(y_1,x_1)^2W(y_1,x_3)^2W(y_2,x_2)W(y_2,x_3)W(y_3,x_2)W(y_3,x_3)\\
&\qquad \quad\quad +W(y_1,x_2)^2W(y_1,x_3)^2W(y_2,x_1)W(y_2,x_3)W(y_3,x_1)W(y_3,x_3)   ) e^{-\alpha/3 (\sum_{i=1}^3\mu(y_i))}dy_{1:3}\\
&\quad+\alpha^2 \int  (W(y_1,x_1)^2W(y_1,x_3)^2W(y_2,x_2)^2W(y_2,x_3)^2\\
&\qquad \quad\quad +W(y_1,x_2)W(y_1,x_1)W(y_1,x_3)^2W(y_2,x_1)W(y_2,x_2)W(y_2,x_3)^2   ) e^{-\alpha/2 (\sum_{i=1}^2\mu(y_i))}dy_{1:2}\\
&\quad\left .+\alpha \int  W(y_1,x_1)^2W(y_1,x_2)^2W(y_1,x_3)^4  e^{-\alpha \mu(y_1)}dy_1\right )
\end{align*}
\egroup


\subsubsection{Proof that $\gamma_{\alpha,3}\rightarrow 0$}
We show here that $\gamma_{\alpha,3}\rightarrow 0$, or equivalently $\int E \left\vert \sqrt{v_\alpha} D_{z}F\right\vert^3dz=o(\alpha^{3/2+3\sigma}\ell_\sigma^{3}(\alpha))$. From Equation~\eqref{eq:Dz} and using the inequality $(a+b)^3\leq 4(a^3+b^3)$ for any $a,b\geq 0$, a sufficient condition is
$$
\int_0^\infty  E \left [\left |e^{-\alpha\mu(x)}-e^{-M(g_{\alpha,x})}\right |^3 + \left (\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}W(\vartheta_{i},x)e^{-M(g_{\alpha,\vartheta_{i}})}\right )^3\right ]dx=o(\alpha^{1/2+3\sigma}\ell_\sigma^{3}(\alpha)).
$$

We have
\begin{align*}
\int_0^\infty \mathbb E \left [\left |e^{-\alpha\mu(x)}-e^{-M(g_{\alpha,x})}\right |^3 \right] dx
 &\leq 2
 \int E\left (\left (e^{-\alpha\mu(x)}-e^{-M(g_{\alpha,x})}\right )^2\right)dx \\
 %&=\int E(e^{-2M(g_{\alpha,x})}-e^{-2\alpha\mu(x)})dx\\
&\leq \int (1-e^{-2\alpha\mu(x)})dx=O(\alpha^\sigma\ell_\sigma(\alpha))\\
\end{align*}
Also  under Assumptions \ref{assumpt:1} and \ref{assumpt:5}, using Lemma \ref{lemma:mu}

\begin{align*}
&\int_0^\infty  E\left [ \left(\sum_{i}\mathbbm{1}_{\theta_{i}\leq\alpha}W(\vartheta_{i},x)e^{-M(g_{\alpha,\vartheta_{i}})}\right)^3 \right ]dx
  \leq
 \int_0^\infty  E \left[ \sum_{i_1,i_2,i_3}  \prod_{\ell=1}^3 \mathbbm{1}_{\theta_{i_\ell}\leq\alpha}W(\vartheta_{i_\ell},x)e^{-M(g_{\alpha,\vartheta_{i_\ell}}) } \right] dx \\
% &\leq  \alpha^3  \int   E[ \prod_{\ell=1}^3W(y_\ell,x)  e^{-M(g_{\alpha,y_\ell}) }] dx dy_{1:3} + 3\alpha^2  \int    W(y_1,x)^2W(y_2,x)  E[ e^{-M(g_{\alpha,y_1})-M(g_{\alpha,y_2}) }] dx dy_{1:2} \\
%& +\alpha  \int    W(y,x)^3  E[ e^{-M(g_{\alpha,y}) }] dx dy \\
& \leq \alpha^3 \left( \int L(y) \mu(y) e^{-\alpha \mu(y)/3} dy \right)^3 + 3 \alpha^2 \left( \int L(y)^2 \mu(y)^2 e^{-\alpha \mu(y)/3} dy \right)\left( \int L(y) \mu(y) e^{-\alpha \mu(y)/3} dy \right)\\
 & +\alpha \int L(y)^3 \mu(y)^3 e^{-\alpha \mu(y)/3} dy  = O( \alpha^{ 3\sigma} \ell_\sigma(\alpha)^3 ) .
\end{align*}

It follows that $\gamma_{\alpha,3}\to 0$ as $\alpha\to\infty$.

\subsubsection{Proof that $\gamma_{\alpha,2}\rightarrow 0$}
We now need to show that the integral of the right hand-side of Equation \eqref{eq:longinequality} with respect to $x_1,x_2,x_3$ is $o(\alpha^{-3}v_\alpha^2)=o(\alpha^{-1+4\sigma}\ell_\sigma^2(\alpha))$.
For the first term in the right hand-side of the inequality \eqref{eq:longinequality}, we have
\begin{align*}
&\int W(x_1,x_3)^2W(x_2,x_3)^2(e^{-\frac{\alpha}{2}\mu(x_1))}+e^{-\frac{\alpha}{2}\mu(x_3)})(e^{-\frac{\alpha}{2}\mu(x_2)}+e^{-\frac{\alpha}{2} \mu(x_3)})dx_1dx_2dx_3\\
&\leq 3\int W(x_1,x_3)W(x_2,x_3)(e^{-\frac{\alpha}{2}(\mu(x_1)+\mu(x_2))}+e^{-\frac{\alpha}{2}\mu(x_3)})dx_1dx_2dx_3\\
&\leq 3\int \nu(x_1,x_2)e^{-\frac{\alpha}{2}(\mu(x_1)+\mu(x_2))}dx_1dx_2+3\int \mu(x_3)^2 e^{-\frac{\alpha}{2}\mu(x_3)}dx_3 \\
&=O(\alpha^{2\sigma-2}\ell_\sigma^2(\alpha))+O(\alpha^{\sigma-2}\ell_\sigma(\alpha))\end{align*}

For the second line (and similarly for the third line) in the RHS of Equation \eqref{eq:longinequality}, we have, noting that $H_{2,2}(x_2,x_3)\leq H_{1,1}(x_2,x_3)$
\begin{align*}
&\int (\alpha^2H_{1,1}(x_2,x_3)^2+\alpha H_{1,1}(x_2,x_3)) W(x_1,x_3)^2(e^{-\alpha/3 \mu(x_1)}+e^{-\alpha/3 \mu(x_3))})dx_1dx_2dx_3\\
&\leq \alpha^2 \int W(x_1,x_3)^2 W(y_1,x_2)W(y_1,x_3)W(y_2,x_2)W(y_2,x_3)\\
&\times(e^{-\alpha/3(\mu(y_1)+\mu(y_2)+\mu(x_1))}+e^{-\alpha/3(\mu(y_1)+\mu(y_2)+\mu(x_3))})dx_1dx_2dx_3dy_1dy_2\\
&+\alpha \int W(x_1,x_3)^2 W(y_1,x_2)W(y_1,x_3)(e^{-\alpha/3(\mu(y_1)+\mu(x_1))}+e^{-\alpha/3(\mu(y_1)+\mu(x_3))})dx_1dx_2dx_3dy_1\\
&= \alpha^2 \int W(x_1,x_3)^2 \nu(y_1,y_2)W(y_1,x_3)W(y_2,x_3)\\
&\times(e^{-\alpha/3(\mu(y_1)+\mu(y_2)+\mu(x_1))}+e^{-\alpha/3(\mu(y_1)+\mu(y_2)+\mu(x_3))})dx_1dx_3dy_1dy_2\\
&+\alpha \int W(x_1,x_3)^2 \mu(y_1)W(y_1,x_3)(e^{-\alpha/3(\mu(y_1)+\mu(x_1))}+e^{-\alpha/3(\mu(y_1)+\mu(x_3))})dx_1dx_3dy_1\\
&\leq 2\alpha^2 \int  L(x_1)^2 L(y_1)L(y_2)\mu(x_1)^2\mu(y_1)\mu(y_2)e^{-\alpha/3(\mu(y_1)+\mu(y_2)+\mu(x_1))}dx_1dy_1dy_2\\
%&+\alpha^2 \int L(x_3)^2 \mu(x_3)^2 \mu(y_1)\mu(y_2)  e^{-\alpha/3(\mu(y_1)+\mu(y_2)+\mu(x_3))})dx_3dy_1dy_2\\
&+\alpha \int  \mu(y_1)^2 L(y_1)L(x_1)^2\mu(x_1)^2e^{-\alpha/3(\mu(y_1)+\mu(x_1))}dx_1dy_1+\alpha \int \mu(x_3)^2 L(x_3)^2 \mu(y_1)e^{-\alpha/3(\mu(y_1)+\mu(x_3))}dx_3dy_1\\
&=O(\alpha^{3\sigma-2}\ell_\sigma(\alpha)^3)
%&=\alpha^2 \int W(x_1,x_3)^2 (e^{-\alpha/3\mu(x_1)}+e^{-\alpha/3\mu(x_3)}) \left (\int  W(y,x_2)W(y,x_3)e^{-\alpha/3\mu(y)}dy\right )^2 dx_1dx_2dx_3\\
%&\alpha^3 \int  W(x_1,x_3)^2\left (\int W(y,x_2)W(y,x_3)e^{-\mu(y)}  dy\right )dx_1dx_2dx_3\\
%&\leq \alpha^3 \int  \left (\int W(y,x_2)\nu(x_1,y)e^{-\mu(y)}  dy\right )dx_1dx_2\\
%&\leq ...
\end{align*}
using Assumption \ref{assumpt:5} and Lemma \ref{lemma:abelianvariations2}. For the third term in the right-handside of Equation \eqref{eq:longinequality}, we obtain
\begin{align*}
&\int A_2(x_1,x_2,x_3)dx_1dx_2dx_3\\ &\leq \alpha^4 \left(\int  L(y)^2\mu(y)^2 e^{-\alpha/4\mu(y)}dy\right)^4 +\alpha^3 \int  L(y_1)^4\mu(y_1)^4 e^{-\alpha/3\mu(y_1)}dy_1 \left(\int  L(y)^2\mu(y)^2 e^{-\alpha/3\mu(y)}dy\right)^2\\
&\quad+\alpha^2\left(\int  L(y)^4\mu(y)^4 e^{-\alpha/2\mu(y)}dy\right)^2+\alpha \int  L(y)^8\mu(y)^8 e^{-\alpha\mu(y)}dy=O(\alpha^{4\sigma-4})\ell_\sigma^4(\alpha)
\end{align*}

It follows that $\gamma_{\alpha,2}\to 0$ as $\alpha\to\infty$.





\subsubsection{Proof that $\gamma_{\alpha,1}\rightarrow 0$}


For any $x>0$ and any unit-rate Poisson point measure $M$ on $\mathbb R_+^2$, denote
\begin{align}
r_\alpha(x,M)=e^{-\alpha\mu(x)}+e^{-M(g_{\alpha,x})}% \leq 2
\end{align}
%Note that
%\begin{align}
%E(r_\alpha(x,M))=2e^{-\alpha\mu(x)}
%\end{align}

For any $z_1=(t_1,x_1),z_2=(t_2,x_2)$, if $t_1>\alpha$ or $t_2>\alpha$,  then $\left\vert D_{z_1}(F_{\alpha})\right\vert^2\left\vert D_{z_2}(F_{\alpha})\right\vert^2=0$. Otherwise, if $t_1,t_2\leq\alpha$, we have from Equation~\eqref{eq:Dz},
\begin{align*}
&v_\alpha\left\vert D_{z_1}(F_{\alpha})\right\vert^2\left\vert D_{z_2}(F_{\alpha})\right\vert^2\\  &  \leq
\left (4r_\alpha(x_1,M)+2 \sum_{i,j}\mathbbm{1}_{\theta_{i}\leq\alpha}\mathbbm{1}_{\theta_{j}\leq\alpha}W(\vartheta_{i},x_1)W(\vartheta_{j},x_1)e^{-M(g_{\alpha,\vartheta_{i}})-M(g_{\alpha,\vartheta_{j}})}\right )
\\
&\quad \times\left (4r_\alpha(x_2,M)+2\sum_{i,j}\mathbbm{1}_{\theta_{i}\leq\alpha}\mathbbm{1}_{\theta_{j}\leq\alpha}W(\vartheta_{i},x_2)W(\vartheta_{j},x_2)e^{-M(g_{\alpha,\vartheta_{i}})-M(g_{\alpha,\vartheta_{j}})}\right )\\
&\leq 16r_\alpha(x_1,M)r_\alpha(x_2,M) \\
&+8\sum_{i,j}\mathbbm{1}_{\theta_{i},\theta_{j}\leq\alpha}\left \{ W(\vartheta_{i},x_1)W(\vartheta_{j},x_1)r_\alpha(x_2,M)+W(\vartheta_{i},x_2)W(\vartheta_{j},x_2)r_\alpha(x_1,M)\right) e^{-M(g_{\alpha,\vartheta_{i}})-M(g_{\alpha,\vartheta_{j}})}\\
%&\quad\quad \quad\quad\quad\quad\quad\quad\quad\left .+W(\vartheta_{i},x_1)W(\vartheta_{j},x_2)\right %\}\\
&+4 \sum_{i_1,i_2,i_3,i_4} W(\vartheta_{i_1},x_1)W(\vartheta_{i_2},x_1)W(\vartheta_{i_3},x_2)W(\vartheta_{i_4},x_2)\prod_{k=1}^4 \left ( \mathbbm{1}_{\theta_{i_k}\leq\alpha}e^{-M(g_{\alpha,\vartheta_{i_k}})} \right ).
\end{align*}
Note that, using Campbell theorem,  together with Lemma \ref{lemma:mu}
\begin{equation*}
\begin{split}
E(r_\alpha(x_1,M)r_\alpha(x_2,M)) \leq e^{-\alpha (\mu(x_1)+\mu(x_2))/2}
\end{split}
\end{equation*}
 It follows that, using the extended Slivnyak-Mecke theorem
\begin{align*}
&v_\alpha^2 E\left (\left\vert D_{z_1}(F_{\alpha})\right\vert^2\left\vert D_{z_2}(F_{\alpha})\right\vert^2\right)\\&\leq C\left ( e^{-\frac{\alpha}{2}(\mu(x_1)+\mu(x_2))}+\alpha\int_0^\infty (W(y,x_1)e^{-\frac{\alpha}{2}\mu(x_2)}+W(y,x_2)e^{-\frac{\alpha}{2}\mu(x_1)})e^{-\frac{\alpha}{2}\mu(y)}dy\right .\\
&+\alpha^2\int_{\mathbb R_+^2}\Big \{(W(y_1,x_1)W(y_2,x_1)e^{-\frac{\alpha}{3}\mu(x_2)}+W(y_1,x_2)W(y_2,x_2)e^{-\frac{\alpha}{3}\mu(x_1)}\Big \}e^{-\alpha\mu(y_1)/3-\alpha\mu(y_2)/3}dy_1dy_2\\
&+\alpha^3\int_{\mathbb R_+^3}(W(y_1,x_1)^2W(y_2,x_2)W(y_3,x_2)+W(y_1,x_2)^2W(y_2,x_1)W(y_3,x_1))e^{-\alpha\sum_{k=1}^3\mu(y_k)/3}dy_1dy_2dy_3\\
&\left .+\alpha^4 \int_{\mathbb R_+^4} W(y_1,x_1)W(y_2,x_1)W(y_3,x_2)W(y_4,x_2)e^{-\alpha\sum_{k=1}^4\mu(y_k)/4}dy_1dy_2dy_3dy_4 \right )\\
&\leq C\left ( e^{-\frac{\alpha}{3}(\mu(x_1)+\mu(x_2))}+\alpha(H_1(x_1)e^{-\frac{\alpha}{3}\mu(x_2)}+H_1(x_2)e^{-\frac{\alpha}{3}\mu(x_1)})+\alpha^2 (H_1(x_1)^2e^{-\frac{\alpha}{3}\mu(x_2)}+H_1(x_2)^2e^{-\frac{\alpha}{3}\mu(x_1)}) \right. \\
 & \left. \quad +\alpha^3(H_1(x_1)^2H_{2,0}(x_2)+H_{2,0}(x_1)H_1(x_2)^2) +\alpha^4H_1(x_1)^2H_1(x_2)^2\right )
\end{align*}
where $H_{i,j}$ are defined in Equation \eqref{eq:H1}; therefore, for any $t_1,t_2\leq\alpha$,  using the fact that $H_{2,0} \leq H_1$ and that $\sqrt{\sum_{i=1}^p a_i} \leq \sqrt{p} \sum_{i=1}^p\sqrt{a_i}$
\begin{align*}
v_\alpha\sqrt{\mathbb E\left\vert D_{z_1}F_\alpha\right\vert^2\left\vert D_{z_2}F_\alpha\right\vert^2}&\leq \sqrt{6C}\sum_{q_1=0}^2\sum_{q_2=0}^2  (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}
\end{align*}

Additionally, from Equation \eqref{eq:longinequality}, we have
\begin{align*}
&v_\alpha\int \sqrt{E\left ((D_{z_{1},z_{3}%
}^{2}F_\alpha)^{2}(D_{z_{2},z_{3}}^{2}F_\alpha)^{2}\right )}dx_3 \nonumber\\
&\leq C\times\Bigg ( \underbrace{\int W(x_1,x_3)W(x_2,x_3)(e^{-\frac{\alpha}{4}(\mu(x_1)+\mu(x_2))}+e^{-\alpha\mu(x_3)/4})dx_3}_{B_{\alpha,1}(x_1,x_2)}\Bigg .\nonumber\\
&\quad\quad+\underbrace{\int(\alpha H_{1,1}(x_2,x_3)+\sqrt{\alpha H_{2,2}(x_2,x_3)}) W(x_1,x_3)(e^{-\alpha/6 \mu(x_1)}+e^{-\alpha/6 \mu(x_3))})dx_3}_{B_{\alpha,2}(x_1,x_2)}\nonumber\\
&\qquad+\int(\alpha H_{1,1}(x_1,x_3)+\sqrt{\alpha H_{2,2}(x_1,x_3)}) W(x_2,x_3)(e^{-\alpha/6 \mu(x_2)}+e^{-\alpha/6 \mu(x_3))})dx_3\nonumber\\
&\qquad\Bigg .+\underbrace{\int\sqrt{A_2(x_1,x_2,x_3)}dx_3}_{B_{\alpha,3}(x_1,x_2)}\Bigg )%\label{eq:longinequality}
\end{align*}
for some constant $C$. To show that $\gamma_{\alpha,1}\to0$, we aim to show that, for any $q_1,q_2\in\{0,1,2\}$, and any $k=1,2,3$
\begin{align*}
I_{\alpha,k}(q_1,q_2)&:=\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}  B_{\alpha,k}(x_1,x_2)dx_1dx_2\\
&=o(\alpha^{4\sigma-1}\ell_\sigma(\alpha)^4)
\end{align*}
Consider first
\begin{align*}
I_{\alpha,1}(q_1,q_2)&=\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})} \\
&\quad\quad\times W(x_1,x_3)W(x_2,x_3)(e^{-\frac{\alpha}{4}(\mu(x_1)+\mu(x_2))}+e^{-\alpha\mu(x_3)/4})dx_1dx_2dx_3\\
&\leq \int (\alpha H_1(x_1))^{q_1/2}L(x_1)\mu(x_1)e^{-\frac{\alpha}{4}\mu(x_1)}dx_1
\int (\alpha H_1(x_2))^{q_2/2}L(x_2)\mu(x_2)e^{-\frac{\alpha}{4}\mu(x_2)}dx_2
\\
&\quad+\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2}W(x_1,x_3)W(x_2,x_3) e^{-\alpha\mu(x_3)/4}dx_1dx_2dx_3
\end{align*}


For $q_1,q_2\geq 1$, using H\"older's inequality and Assumptions \ref{assumpt:1} and \ref{assumpt:5},
\begin{align*}
&\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2}W(x_1,x_3)W(x_2,x_3) e^{-\alpha/4\mu(x_3)}dx_1dx_2dx_3\\
%&\leq \left (\int(\alpha H_1(x_1))^{q_1}W(x_2,x_3)^2 e^{-\alpha/4\mu(x_3)}dx_1dx_2dx_3 \right . \\
%& \quad\quad\times\left . \int(\alpha H_1(x_2))^{q_2}W(x_1,x_3)^2 e^{-\alpha\mu(x_3)/4}dx_1dx_2dx_3   \right )^{1/2}\\
&\leq \left (\int(\alpha H_1(x_1))^{q_1}dx_1 \int L(x_3)^2\mu(x_3)^2 e^{-\alpha\mu(x_3)/4}dx_3 \right . \\
& \quad\quad\times\left . \int(\alpha H_1(x_2))^{q_2}dx_2 \int L(x_3)^2\mu(x_3)^2 e^{-\alpha\mu(x_3)/4}dx_3   \right )^{1/2}\\
&=O(\alpha^{(q_1/2+q_2/2+1)\sigma -2 }\ell_\sigma(\alpha)^{q_1/2+q_2/2+1})
\end{align*}
Similarly,
\begin{align*}
\int (\alpha H_1(x_1))^{q_1/2}W(x_1,x_3)W(x_2,x_3) e^{-\alpha\mu(x_3)/4}dx_1dx_2dx_3&=O(\alpha^{(q_1/2+1/2)\sigma -2 }\ell_\sigma(\alpha)^{q_1/2+1/2})\\
\int W(x_1,x_3)W(x_2,x_3) e^{-\alpha\mu(x_3)/4}dx_1dx_2dx_3&=O(\alpha^{\sigma -2 }\ell_\sigma(\alpha))
\end{align*}
and it follows that for any $q_1,q_2\in\{0,1,2\}$,
$
I_{\alpha,1}(q_1,q_2)=O(\alpha^{3\sigma-2}\ell_\sigma(\alpha)^3)=o(\alpha^{-1+4\sigma}\ell_\sigma(\alpha)^2)\label{eq:I1}
$ as required. Consider now
\begin{align*}
I_{\alpha,2}(q_1,q_2)&=\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}\\
&\quad\quad\times(\alpha H_{1,1}(x_2,x_3)+\sqrt{\alpha H_{2,2}(x_2,x_3)}) W(x_1,x_3)(e^{-\alpha/6 \mu(x_1)}+e^{-\alpha/6 \mu(x_3))})dx_1dx_2dx_3.
\end{align*}
We have, using Lemma \ref{lemma:bound}
\begin{align*}
I_{\alpha,2,1}(q_1,q_2):=&\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}\\
&\quad\quad\times \alpha H_{1,1}(x_2,x_3) W(x_1,x_3)e^{-\alpha/6 \mu(x_1)}dx_1dx_2dx_3\\
&\leq \alpha\left(\int (\alpha H_1(x_1))^{q_1/2} e^{-\frac{\alpha}{6}\mu(x_1)\mathbbm{1}_{q_1=0}}L(x_1)\mu(x_1)e^{-\alpha/6 \mu(x_1)}dx_1 \right)\\
&\quad\times \left( \int (\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}\mu(x_2)\mathbbm{1}_{q_2=0}}W(x_2,y)L(y)\mu(y) e^{-\frac{\alpha}{4}\mu(y)}dx_2dy \right)
\end{align*}
If $q_1=0$,
$$\int e^{-\frac{\alpha}{3}\mu(x_1)}L(x_1)\mu(x_1)dx_1 =O(\alpha^{\sigma-1}\ell_\sigma(\alpha))$$
and if $q_1\geq 1$, using the bound \eqref{eq:boundH1a},
\begin{equation}
\int (\alpha H_1(x_1))^{q_1/2} L(x_1)\mu(x_1) e^{-\frac{\alpha}{6}\mu(x_1)}dx_1 =O(\alpha^{(q_1+1)\sigma/2-1}\ell_\alpha(\alpha)^{q_1/2+1/2})\label{eq:boundH1d}
\end{equation}
If $q_2=0$,
\begin{align*}
&\int  e^{-\frac{\alpha}{6}\mu(x_2)}W(x_2,y)L(y)\mu(y) e^{-\frac{\alpha}{4}\mu(y)}dx_2dy\leq \int  L(y)\mu(y)^2 e^{-\frac{\alpha}{4}\mu(y)}dy=O(\alpha^{\sigma-2}\ell_\sigma(\alpha))
\end{align*}
If $q_2\geq 1$, using H\"older's inequality, the bound \eqref{eq:boundintH1} and Assumptions \ref{assumpt:5} and \ref{assumpt:1},
\begin{align*}
&\int (\alpha H_1(x_2))^{q_2/2}W(x_2,y)L(y)\mu(y) e^{-\frac{\alpha}{4}\mu(y)}dx_2dy\\
&\leq \left(\int (\alpha H_1(x_2))^{q_2} \right )^{1/2} \int \left (\int W(x_2,y)^2dx_2 \right)^{1/2}L(y)\mu(y) e^{-\frac{\alpha}{4}\mu(y)}dy\\
&\leq \left(\int (\alpha H_1(x_2))^{q_2} \right )^{1/2} \int L(y)^2\mu(y)^2 e^{-\frac{\alpha}{4}\mu(y)}dy
=O(\alpha^{(q_2/2+1)\sigma-2}\ell_\sigma(\alpha^{q_2/2+1}))
\end{align*}
Hence $I_{\alpha,2,1}(q_1,q_2)=o(\alpha^{4\sigma-1} \ell_\sigma^2(\alpha))$. Consider now
\begin{align*}
I_{\alpha,2,2}(q_1,q_2)&:=\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}\\
&\quad\quad\times\alpha H_{1,1}(x_2,x_3) W(x_1,x_3)e^{-\alpha/6 \mu(x_3)}dx_1dx_2dx_3
\end{align*}
If $q_1=0$, we obtain the following bound, using the same computations as above,
\begin{align*}
I_{\alpha,2,2}(q_1,q_2)&\leq \alpha\int  e^{-\frac{\alpha}{6}\mu(x_1)}L(x_1)\mu(x_1)dx_1  \int (\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}\mu(x_2)\mathbbm{1}_{q_2=0}}W(x_2,y)L(y)\mu(y) e^{-\frac{\alpha}{4}\mu(y)}dx_2dy\\
&=O(\alpha^{4\sigma-2}\ell_\sigma(\alpha)^{4\sigma})
\end{align*}
If $q_1>0$, we have
\begin{align*}
I_{\alpha,2,2}(q_1,q_2)&\leq \int \left(\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\left(\int W(x_1,x_3)^2 dx_1\right )^{1/2} (\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}\mu(x_2)\mathbbm{1}_{q_2=0}}\\
&\quad\quad\times\alpha H_{1,1}(x_2,x_3) e^{-\alpha/6 \mu(x_3)}dx_2dx_3\\
&\leq \left(\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\int L(x_3)\mu(x_3)(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}\mu(x_2)\mathbbm{1}_{q_2=0}}\\
&\quad\quad\times\alpha H_{1,1}(x_2,x_3) e^{-\alpha/6 \mu(x_3)}dx_2dx_3\\
\end{align*}
If $q_2=0$, noting that $H_{1,1}(x_2,x_3)\leq L(x_2)\mu(x_2)L(x_3)\mu(x_3)$, we obtain
\begin{align*}
I_{\alpha,2,2}(q_1,q_2)&\leq \alpha\left(\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\int L(x_2)\mu(x_2)L(x_3)^2\mu(x_3)^2 e^{-\frac{\alpha}{6}\mu(x_2)}e^{-\alpha/6 \mu(x_3)}dx_2dx_3\\
&=O(\alpha^{(q_1/2+2)\sigma-2}\ell_\sigma(\alpha)^{q_1/2+2})
\end{align*}
If $q_2>0$, using H\"older's inequality and the bound \eqref{eq:boundH1c},
\begin{align*}
I_{\alpha,2,2}
%&(q_1,q_2)\leq\alpha  \left(\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\int L(x_3)\mu(x_3)(\alpha H_1(x_2))^{q_2/2}
%H_{1,1}(x_2,x_3) e^{-\alpha/6 \mu(x_3)}dx_2dx_3\\
%&= \alpha \left(\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\int L(x_3)\mu(x_3)(\alpha H_1(x_2))^{q_2/2}  W(x_2,y)W(x_3,y)e^{-\frac{\alpha}{4}[\mu(y) + 2\mu(x_3)/3]} dx_2dx_3dy\\
%&\leq  \alpha\left(\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\left(\int (\alpha H_1(x_2))^{q_2}dx_2\right )^{1/2}\\
%&\quad\quad\times \int L(y)\mu(y)L(x_3)\mu(x_3)W(x_3,y)e^{-\frac{\alpha}{4}[\mu(y) + 2\mu(x_3)/3]} dx_3dy\\
&\leq  \alpha\left(\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\left(\int (\alpha H_1(x_2))^{q_2}dx_2\right )^{1/2}\\
&\quad\quad\times \left (\int L(x_3)^2\mu(x_3)^2e^{-\alpha/3 \mu(x_3)}dx_3 \right )^{1/2} \int L(y)^2\mu(y)^2e^{-\frac{\alpha}{4}\mu(y)} dy=O(\alpha^{7\sigma/2-3}\ell(\alpha)^{7\sigma/2})
\end{align*}
Now, using H\"older and the fact that $H_{2,2} \leq H_{1,1}$,  together with  \eqref{eq:boundH1a},
\begin{align*}
I_{\alpha,2,3}(q_1,q_2)&:=\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}\\
&\quad\quad\times(\sqrt{\alpha H_{2,2}(x_2,x_3)}) W(x_1,x_3)e^{-\alpha/6 \mu(x_1)}dx_1dx_2dx_3\\
%&\leq\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}\\
%&\quad\quad\times(\sqrt{\alpha H_{1,1}(x_2,x_3)}) W(x_1,x_3)e^{-\alpha/6 \mu(x_1)}dx_1dx_2dx_3\\
&\leq \sqrt{\alpha}\int (\alpha H_1(x_1))^{q_1/2} L(x_1)\mu(x_1)e^{-\alpha \mu(x_1)/6}dx_1\\
&\quad\quad\times \int (\alpha H_1(x_2))^{q_2/2}\left (\int \mu(y)W(x_2,y)e^{-\frac{\alpha}{4}\mu(y)}dy \right)^{1/2}e^{-\frac{\alpha}{6}\mu(x_2)\mathbbm{1}_{q_2=0}} dx_2\\
%& = O(\alpha^{\sigma(q_1+1)/2-1/2 }\ell_\sigma(\alpha)^{(q_1+1)/2} ) \times
%\left (\int \mu(y)^2e^{-\frac{\alpha}{2}\mu(y)}dy \right)^{1/4} \\
% &\qquad \times \int (\alpha H_1(x_2))^{q_2/2}L(x_2)^{1/2}\mu(x_2)^{1/2}e^{-\frac{\alpha}{6}\mu(x_2)\mathbbm{1}_{q_2=0}} dx_2\\
&=O(\alpha^{(3+1/4)\sigma-3/2}\ell_\sigma(\alpha)^{3+1/4})
\end{align*}

Consider
\begin{align*}
I_{\alpha,2,4}(q_1,q_2)&:=\int (\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}\\
&\quad\quad\times\sqrt{\alpha H_{2,2}(x_2,x_3)} W(x_1,x_3)e^{-\alpha/6 \mu(x_3)}dx_1dx_2dx_3\\
\end{align*}
If $q_1=0$, then, using the above computations,
\begin{align*}
I_{\alpha,2,4}(q_1,q_2)&\leq \int (\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)+\mu(x_2)\mathbbm{1}_{q_2=0})}\\
&\quad\quad\times\sqrt{\alpha H_{1,1}(x_2,x_3)} W(x_1,x_3)dx_1dx_2dx_3\\
&=O(\alpha^{(3+1/4)\sigma-3/2}\ell_\sigma(\alpha)^{3+1/4})
\end{align*}
If $q_1>0$ and $q_2=0$, noting that $H_{2,2}(x_2,x_3)\leq L(x_2)^2\mu(x_2)^2L(x_3)^2\mu(x_3)^2$ and using H\"older's inequality and Assumptions \ref{assumpt:5} and \ref{assumpt:1},
\begin{align*}
I_{\alpha,2,4}(q_1,q_2)&\leq \sqrt{\alpha}\left (\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2} \int L(x_3)\mu(x_3) e^{-\frac{\alpha}{6}\mu(x_2)}\sqrt{ H_{2,2}(x_2,x_3)}e^{-\alpha/6 \mu(x_3)}dx_2dx_3\\
&\leq \sqrt{\alpha}\left (\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\int L(x_3)^{2}\mu(x_3)^{2}e^{-\alpha/6 \mu(x_3)}dx_3\int  e^{-\frac{\alpha}{6}\mu(x_2)}\mu(x_2)L(x_2)dx_2\\
&=O(\alpha^{3\sigma -3}\ell_\sigma(\alpha)^{3})
\end{align*}

If $q_1,q_2>0$, noting that $\int H_{1,1}(x_2,x_3)=\int \mu(y)^2e^{-\frac{\alpha}{4}\mu(y)}dy=O(\alpha^{\sigma-2}\ell_\sigma(\alpha))$,
\begin{align*}
&I_{\alpha,2,4}(q_1,q_2)\\
&\leq \sqrt{\alpha}\left (\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\times \int L(x_3)\mu(x_3) (\alpha H_1(x_2))^{q_2/2}\sqrt{ H_{1,1}(x_2,x_3)}e^{-\alpha/6 \mu(x_3)}dx_2dx_3\\
%&\leq \sqrt{\alpha}\left (\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\left (\int (\alpha H_1(x_2))^{q_2}dx_2\right )^{1/2}\times \int L(x_3)\mu(x_3) \left (\int H_{1,1}(x_2,x_3)dx_2 \right)^{1/2} e^{-\alpha/6 \mu(x_3)}dx_3\\
%&\leq \sqrt{\alpha}\left (\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\left (\int (\alpha H_1(x_2))^{q_2}dx_2\right )^{1/2}\times \left(\int L(x_3)^2\mu(x_3)^2e^{-\alpha/6 \mu(x_3)}dx_3\right)^{1/2} \left (\int H_{1,1}(x_2,x_3)e^{-\alpha/6 \mu(x_3)}dx_3dx_2 \right)^{1/2}\\
&\leq \sqrt{\alpha}\left (\int (\alpha H_1(x_1))^{q_1}dx_1\right )^{1/2}\left (\int (\alpha H_1(x_2))^{q_2}dx_2\right )^{1/2}\\
&\quad\quad\times \left(\int L(x_3)^2\mu(x_3)^2e^{-\alpha/6 \mu(x_3)}dx_3\right)^{1/2} \left (\int H_{1,1}(x_2,x_3)dx_3dx_2 \right)^{1/2}=O(\alpha^{3\sigma -3/2}\ell_\sigma(\alpha^{3\sigma}))
\end{align*}
It follows that
\begin{align}
I_{\alpha,2}(q_1,q_2)=o(\alpha^{-3}v_\alpha^2).\label{eq:I2}
\end{align}


Finally, consider
\begin{align*}
I_{\alpha,3}(q_1,q_2)=\int(\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}\sqrt{A_2(x_1,x_2,x_3)}dx_1dx_2dx_3
\end{align*}


where
\begin{align*}
&\sqrt{A_2(x_1,x_2,x_3)}\\&\leq \alpha^2 H_{1,1}(x_1,x_3)H_{1,1}(x_2,x_3) + \alpha^{3/2} \left ( H_{1,1}(x_2,x_3) \sqrt{ H_{2,2}(x_1,x_3) }  +H_{1,1}(x_1,x_3)\sqrt{ H_{2,2}(x_2,x_3) } \right)\\
&+\alpha \left (\sqrt{ H_{2,2}(x_1,x_3) }\sqrt{ H_{2,2}(x_2,x_3) } \right.\\
&\quad \quad\left . + \sqrt{W(y_1,x_2)W(y_1,x_1)W(y_1,x_3)^2W(y_2,x_1)W(y_2,x_2)W(y_2,x_3)^2   ) e^{-\alpha/2 (\sum_{i=1}^2\mu(y_i))}dy_{1:2}} \right)\\
&+\sqrt{\alpha} \left ( \int  W(y_1,x_1)^2W(y_1,x_2)^2W(y_1,x_3)^4  e^{-\alpha \mu(y_1)}dy_1\right )^{1/2}
\end{align*}
Using Assumption \ref{assumpt:5},
\begin{align*}
&\int  H_{1,1}(x_1,x_3))H_{1,1}(x_2,x_3)dx_3
%\\
%&= \int W(x_1,y_1)W(x_3,y_1) W(x_2,y_2)W(x_3,y_2)e^{-\frac{\alpha}{4}(\mu(y_1)+\mu(y_2)}dy_1dy_2dx_3\\
%&= \int W(x_1,y_1)W(x_2,y_2)\nu(y_1,y_2)e^{-\frac{\alpha}{4}(\mu(y_1)+\mu(y_2))}dy_1dy_2\\
&\leq \int W(x_1,y_1)L(y_1)\mu(y_1)e^{-\frac{\alpha}{4}\mu(y_1)}dy_1\int W(x_2,y_2)L(y_2)\mu(y_2)e^{-\frac{\alpha}{4}\mu(y_2)}dy_2
\end{align*}


Therefore, using the asymptotic bounds \eqref{eq:boundH1a} and Assumption \ref{assumpt:1},
\begin{align*}
&I_{\alpha,3,1}(q_1,q_2)\\
&:= \alpha^2\int(\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}  H_{1,1}(x_1,x_3)H_{1,1}(x_2,x_3)dx_1dx_2dx_3\\
&\leq \alpha^2\int(\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2}   H_{1,1}(x_1,x_3)H_{1,1}(x_2,x_3)dx_1dx_2dx_3\\
&\leq \alpha^2\int (\alpha H_1(x_1))^{q_1/2} L(y_1)\mu(y_1)W(x_1,y_1)e^{-\frac{\alpha}{4}\mu(y_1)}dy_1dx_1\\
&\quad\times\int (\alpha H_1(x_2))^{q_2/2} L(y_2)\mu(y_2)W(x_2,y_2) e^{-\frac{\alpha}{4}\mu(y_2)}dy_2dx_2\\
%&= \alpha^2\left (\int (\alpha H_1(x))^{q_1/2} L(y)\mu(y)W(x,y)e^{-\frac{\alpha}{4}\mu(y)}dydx\right )^2\\
%&=O(\alpha^{5\sigma -2}\ell(\alpha)^5)
\end{align*}
for any $q_1,q_2\leq 2$. If $q=0$,
\begin{align*}
\int (\alpha H_1(x))^{q/2} L(y)\mu(y)W(x,y)e^{-\frac{\alpha}{4}\mu(y)}dydx=\int  L(y)\mu(y)^2e^{-\frac{\alpha}{4}\mu(y)}dy=O(\alpha^{\sigma-2}\ell_\sigma(\alpha))
\end{align*}
If $q>0$, noting that, using H\"older's inequality and Assumption 3,
\begin{align*}
\int (\alpha H_1(x))^{q/2}W(x,y)dx     \leq L(y)\mu(y)\left (\int (\alpha H_1(x))^q dx\right )^{1/2}
\end{align*}
we have
\begin{align*}
\int (\alpha H_1(x))^{q/2} L(y)\mu(y)W(x,y)e^{-\frac{\alpha}{4}\mu(y)}dydx&\leq \left (\int (\alpha H_1(x))^q dx\right )^{1/2}\left( \int  L(y)^2\mu(y)^2e^{-\frac{\alpha}{4}\mu(y)}dy \right)^{1/2}\\
=O(\alpha^{(q/2+1)\sigma-2}\ell_\sigma(\alpha)^{q/2+1}).
\end{align*}
It follows that $I_{\alpha,3,1}(q_1,q_2)=O(\alpha^{4\sigma-2}\ell(\alpha)^{4\sigma})$ for any $q_1,q_2\in\{0,1,2\}$. Consider now
\begin{align*}
I_{\alpha,3,2}(q_1,q_2):=&  \alpha^{3/2} \int(\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2}e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})} H_{1,1}(x_2,x_3) \\
&\quad\times\left (\int  W(y_1,x_1)^2W(y_1,x_3)^2 e^{-\alpha/3 \mu(y_1)} dy_1\right )^{1/2}dx_1dx_2dx_3\\
&\leq  \alpha^{3/2}  \int(\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} H_{1,1}(x_2,x_3) \\
&\quad\times\left (\int  W(y_1,x_1)^2W(y_1,x_3)^2 e^{-\alpha/3 \mu(y_1)} dy_1\right )^{1/2}dx_1dx_2dx_3
\end{align*}

Using Lemma \ref{lemma:bound},
\begin{align*}
&\int H_{1,1}(x_2,x_3) \left (\int  W(y_1,x_1)^2W(y_1,x_3)^2 e^{-\alpha/3 \mu(y_1)} dy_1\right )^{1/2}dx_3\\
&\leq \int W(x_2,y_2)L(y_2)\mu(y_2)e^{-\frac{\alpha}{4}\mu(y_2)}dy_2 \times \left (\int L(y_1)^{2}\mu(y_1)^{2} W(y_1,x_1)^{2} e^{-\frac{\alpha}{3}\mu(y_1) } dy_1 \right )^{1/2}
\end{align*}
Therefore
\begin{align*}
I_{\alpha,3,2}(q_1,q_2)&\leq \alpha^{3/2}  \int  (\alpha H_1(x_2))^{q_1/2} W(x_2,y_2)L(y_2)\mu(y_2)e^{-\frac{\alpha}{4}\mu(y_2)}dy_2dx_2 \\
&\quad\times \int (\alpha H_1(x_1))^{q_2/2} \left (\int L(y_1)^{2}\mu(y_1)^{2} W(y_1,x_1)^{2} e^{-\frac{\alpha}{3}\mu(y_1) } dy_1 \right )^{1/2}dx_1
\end{align*}
For $q_1=0$,
$$\int  L(y_2)\mu(y_2)^2e^{-\frac{\alpha}{4}\mu(y_2)}dy_2=O(\alpha^{\sigma-2}\ell_\sigma(\alpha)),$$
while for $q_1\geq 1$,
\begin{align*}
&\int  (\alpha H_1(x_2))^{q_1/2} W(x_2,y_2)L(y_2)\mu(y_2)e^{-\frac{\alpha}{4}\mu(y_2)}dy_2dx_2\\
&\leq \left (\int (\alpha H_1(x_2))^{q_1} dx_2 \times \int W(x_2,y_2)^2L(y_2)^2\mu(y_2)^2e^{-\frac{\alpha}{4}\mu(y_2)}dy_2dx_2   \right )^{1/2}\\
&\leq \left (\int (\alpha H_1(x_2))^{q_1} dx_2 \times \int L(y_2)^4\mu(y_2)^4e^{-\frac{\alpha}{4}\mu(y_2)}dy_2dx_2   \right )^{1/2}=O(\alpha^{(q_1/2+1/2)\sigma-2}\ell_\sigma(\alpha)^{q_1+1/2}).
\end{align*}
Additionally, for $q_2=0$, using H\"older's inequality and Assumptions \ref{assumpt:1} and \ref{assumpt:5},
\begin{align*}
&\int \left (\int L(y_1)^{2}\mu(y_1)^{2} W(y_1,x_1)^{2} e^{-\frac{\alpha}{3}\mu(y_1) } dy_1 \right )^{1/2}dx_1\\
&\leq \int \left ( \int L(y_1)^{4}\mu(y_1)^{4}  e^{-\frac{2\alpha}{3}\mu(y_1) } dy_1 \int W(y_1,x_1)^{4} dy_1 \right )^{1/4}dx_1\\
&\leq \int L(x_1)\mu(x_1) dx_1 \left ( \int L(y_1)^{4}\mu(y_1)^{4}  e^{-\frac{2\alpha}{3}\mu(y_1) } dy_1 \right )^{1/4}=O(\alpha^{\sigma/4-1}\ell_\sigma(\alpha)^{1/4})
\end{align*}
while for $q_2\geq 1$
\begin{align*}
&\int (\alpha H_1(x_1))^{q_2/2} \left (\int L(y_1)^{2}\mu(y_1)^{2} W(y_1,x_1)^{2} e^{-\frac{\alpha}{3}\mu(y_1) } dy_1 \right )^{1/2}dx_1\\
&\leq \left ( \int(\alpha H_1(x_1))^{q_2}dx_1\int L(y_1)^{2}\mu(y_1)^{2} W(y_1,x_1)^{2} e^{-\frac{\alpha}{3}\mu(y_1) } dy_1dx_1    \right )^{1/2}\\
&\leq \left ( \int(\alpha H_1(x_1))^{q_2}dx_1\int L(y_1)^{4}\mu(y_1)^{4}e^{-\frac{\alpha}{3}\mu(y_1) } dy_1    \right )^{1/2}=O(\alpha^{(q_2+1/2)\sigma-2}\ell_\sigma(\alpha)^{q_2/2+1/2}).
\end{align*}
Hence, for any $q_1,q_2\leq 2$, $I_{\alpha,3,2}=O(\alpha^{3\sigma - 2}\ell_\sigma(\alpha)^3)=o(\alpha^{-3}v_\alpha^2).$ Consider now
\begin{align*}
I_{\alpha,3,3}(q_1,q_2)&:= \alpha\int(\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2}e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}   \\
&\quad\times \left [\left ( \int  W(y_1,x_1)^2W(y_1,x_3)^2W(y_2,x_2)^2W(y_2,x_3)^2e^{-\alpha/2 (\sum_{i=1}^2\mu(y_i))}dy_{1:2}\right )^{1/2}\right .\\
&\quad+\left .\left ( \int W(y_1,x_2)W(y_1,x_1)W(y_1,x_3)^2W(y_2,x_1)W(y_2,x_2)W(y_2,x_3)^2    e^{-\frac{\alpha}{2 (\sum_{i=1}^2\mu(y_i))}}dy_{1:2}\right )^{1/2}\right ]\\
&\quad\quad dx_1dx_2dx_3
\end{align*}
Using H\"older's inequality,
\begin{align*}
 &\int \left (\int  W(y_1,x_1)^2W(y_1,x_3)^2W(y_2,x_2)^2W(y_2,x_3)^2e^{-\alpha/2 (\sum_{i=1}^2\mu(y_i))}dy_{1:2}\right)^{1/2}dx_3\\
 &\leq \left (\int  W(y_1,x_1)^2W(y_1,x_3)^2e^{-\frac{\alpha}{2}\mu(y_1)}dy_{1}dx_3\right)^{1/2}\left (\int  W(y_2,x_2)^2W(y_2,x_3)^2e^{-\frac{\alpha}{2}\mu(y_2)}dy_{2}dx_3\right)^{1/2}\\
%&\leq  \left (\int  W(y_1,x_1)^4e^{-\alpha \mu(y_1)}dy_1\right )^{1/2}\left (\int W(y_1,x_3)^4dy_1\right )^{1/2}\\
%&\quad\times \left (\int  W(y_2,x_2)^4e^{-\alpha \mu(y_2)}dy_2\right )^{1/2}\left (\int W(y_2,x_3)^4dy_2\right )^{1/2}\\
%&\leq L(x_3)^4\mu(x_3)^4\left (\int  W(y_1,x_1)^4e^{-\alpha \mu(y_1)}dy_1\right )^{1/2}\left (\int  W(y_2,x_2)^4e^{-\alpha \mu(y_2)}dy_2\right )^{1/2}
\end{align*}
For $q=0$, using Assumption \ref{assumpt:5} and \ref{assumpt:1}
\begin{align*}
\int e^{-\frac{\alpha}{6}\mu(x)}&\left (\int H_{2,2}(x,x_3) dx_3\right)^{1/2}dx\leq \int e^{-\frac{\alpha}{6}\mu(x)}\left (\int  W(y,x)^2W(y,x_3)dydx_3\right)^{1/2}dx\\
%&\leq \left (\int  L(x_3)\mu(x_3) dx_3\right)^{1/2}\int L(x)\mu(x) e^{-\frac{\alpha}{6}\mu(x)}dx\\
&=O(\alpha^{\sigma-1}\ell_\sigma(\alpha))
\end{align*}
Additionally,
\begin{align*}
 & \left (\int  W(y_1,x_1)^2W(y_1,x_3)^2e^{-\frac{\alpha}{2}\mu(y_1)}dy_{1}dx_3\right)^{1/2}\left (\int  W(y_2,x_2)^2W(y_2,x_3)^2e^{-\frac{\alpha}{2}\mu(y_2)}dy_{2}dx_3\right)^{1/2}\\
 &\leq \left (\int  L(y_1)^2\mu(y_1)^2 W(y_1,x_1)^2 e^{-\frac{\alpha}{2}\mu(y_1)}dy_{1}\right)^{1/2}\left (\int  L(y_2)^2\mu(y_2)^2W(y_2,x_2)^2e^{-\frac{\alpha}{2}\mu(y_2)}dy_{2}\right)^{1/2}
%&\leq  \left (\int  W(y_1,x_1)^4e^{-\alpha \mu(y_1)}dy_1\right )^{1/2}\left (\int W(y_1,x_3)^4dy_1\right )^{1/2}\\
%&\times \left (\int  W(y_2,x_2)^4e^{-\alpha \mu(y_2)}dy_2\right )^{1/2}\left (\int W(y_2,x_3)^4dy_2\right )^{1/2}\\
%&\leq L(x_3)^4\mu(x_3)^4\left (\int  W(y_1,x_1)^4e^{-\alpha \mu(y_1)}dy_1\right )^{1/2}\left (\int  W(y_2,x_2)^4e^{-\alpha \mu(y_2)}dy_2\right )^{1/2}
\end{align*}
It follows that, for $q\geq 1$, using H\"older's inequality and Assumptions \ref{assumpt:5} and \ref{assumpt:1}
\begin{align*}
&\int (\alpha H_1(x))^{q/2}\left (\int  L(y)^2\mu(y)^2 W(y,x)^2 e^{-\frac{\alpha}{2}\mu(y)}dy\right)^{1/2}dx\\
&\leq \left ( \int (\alpha H_1(x))^{q}dx \right )^{1/2}\left(\int  L(y)^4\mu(y)^4 e^{-\frac{\alpha}{2}\mu(y)}dy \right)^{1/2}=O\left (\alpha^{(q/2+1/2)\sigma-2}\ell_\sigma(\alpha)^{(q/2+1/2)}\right )
\end{align*}
Similarly, for the second term of $I_{\alpha,3,3}(q_1,q_2)$
\bgroup
\allowdisplaybreaks
\begin{align*}
&\int\left ( \int W(y_1,x_2)W(y_1,x_1)W(y_1,x_3)^2W(y_2,x_1)W(y_2,x_2)W(y_2,x_3)^2    e^{-\alpha/2 (\sum_{i=1}^2\mu(y_i))}dy_{1:2}\right )^{1/2}dx_3\\
&\leq \left (\int  W(y_1,x_1)W(y_1,x_2)W(y_1,x_3)^2e^{-\frac{\alpha}{2}\mu(y_1)}dy_{1}dx_3\int  W(y_2,x_1)W(y_2,x_2)W(y_2,x_3)^2e^{-\frac{\alpha}{2}\mu(y_2)}dy_{2}dx_3\right)^{1/2}
\end{align*}
\egroup
and, using H\"older's inequality and \eqref{eq:boundH1a},
\bgroup
\allowdisplaybreaks
\begin{align*}
&\int(\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2}e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})}   \\
&\left (\int  W(y_1,x_1)W(y_1,x_2)W(y_1,x_3)^2e^{-\frac{\alpha}{2}\mu(y_1)}dy_{1}dx_3\right)^{1/2}\\
&\quad\times\left (\int  W(y_2,x_1)W(y_2,x_2)W(y_2,x_3)^2e^{-\frac{\alpha}{2}\mu(y_2)}dy_{2}dx_3\right)^{1/2}dx_1dx_2\\
&\leq \left (\int(\alpha H_1(x_1))^{q_1}e^{-\frac{\alpha}{6}\mu(x_1)\mathbbm{1}_{q_1=0}}W(y_1,x_1)\mu(y_1)^3L(y_1)^2e^{-\frac{\alpha}{2}\mu(y_1)}dy_{1}dx_1  \right )^{1/2}  \\
&\quad\times \left (\int(\alpha H_1(x_2))^{q_2}e^{-\frac{\alpha}{6}\mu(x_2)\mathbbm{1}_{q_2=0}}W(y_2,x_2)\mu(y_2)^3L(y_2)^2e^{-\frac{\alpha}{2}\mu(y_2)}dy_{2}dx_2  \right )^{1/2}
\end{align*}
\egroup
For $q=0$,
\begin{align*}
&\left (\int e^{-\frac{\alpha}{6}\mu(x)}W(y,x)\mu(y)^3L(y)^2e^{-\frac{\alpha}{2}\mu(y)}dydx  \right )^{1/2}
%\\
%&\leq \left (\int \mu(y)^4L(y)^2e^{-\frac{\alpha}{2}\mu(y)}dydx  \right )^{1/2}
=O(\alpha^{\sigma/2-2}\ell_\sigma(\alpha)^{1/2})
\end{align*}
while for $q\geq 1$,
\begin{align*}
&\left (\int (\alpha H_1(x))^qW(y,x)\mu(y)^3L(y)^2e^{-\frac{\alpha}{2}\mu(y)}dydx  \right )^{1/2}
%\\
%&\leq \left (\int (\alpha H_1(x))^qdx \right )^{1/2} \left ( \int \mu(y)^3L(y)^2e^{-\frac{\alpha}{2}\mu(y)}dydx  \right )^{1/2}\\
&=O(\alpha^{(q/2+1/2)\sigma-3/2}\ell_\sigma(\alpha)^{q/2+1/2})
\end{align*}
Combining the above results, we obtain, for any $q_1,q_2\in\{0,1,2\}$,   $I_{\alpha,3,3}(q_1,q_2)=o(\alpha^{-3}v_\alpha^2).$ Consider finally
\begin{align*}
I_{\alpha,3,4}(q_1,q_2):=&\int(\alpha H_1(x_1))^{q_1/2}(\alpha H_1(x_2))^{q_2/2} e^{-\frac{\alpha}{6}(\mu(x_1)\mathbbm{1}_{q_1=0}+\mu(x_2)\mathbbm{1}_{q_2=0})} \\
&\times\sqrt{\alpha} \left ( \int  W(y_1,x_1)^2W(y_1,x_2)^2W(y_1,x_3)^4  e^{-\alpha \mu(y_1)}dy_1\right )^{1/2} dx_1dx_2dx_3
\end{align*}
We have
\begin{align*}
&\int \left ( \int  W(y_1,x_1)^2W(y_1,x_2)^2W(y_1,x_3)^4  e^{-\alpha \mu(y_1)}dy_1\right )^{1/2} dx_3\\
%&\leq \int \left ( \int  W(y_1,x_1)^2W(y_1,x_2)^2W(y_1,x_3)^3  e^{-\alpha \mu(y_1)}dy_1\right )^{1/2} dx_3\\
%&\leq \int \left (\int  W(y_1,x_1)^6  e^{-\alpha \mu(y_1)}dy_1 \int  W(y_1,x_2)^6  e^{-\alpha \mu(y_1)}dy_1 \int  W(y_1,x_3)^{6}  dy_1  \right )^{1/6} dx_3\\
&\leq \left (\int L(x_3)\mu(x_3)dx_3\right )  \left (\int  W(y_1,x_1)^6  e^{-\alpha \mu(y_1)}dy_1\right)^{1/6} \left ( \int  W(y_1,x_2)^6  e^{-\alpha \mu(y_1)}dy_1  \right )^{1/6}
\end{align*}
and, for $q_1\geq1$, using H\"older's inequality,
\begin{align*}
&\int(\alpha H_1(x_1))^{q_1}  \left (\int  W(y_1,x_1)^6  e^{-\alpha \mu(y_1)}dy_1\right)^{1/6}dx_1=O(\alpha^{(q_1+1/6)\sigma-1}\ell_\sigma(\alpha)^{q_1+1/6}).
\end{align*}
For $q_1=0$,
\begin{align*}
\int  e^{-\frac{\alpha}{6}\mu(x_1)} \left (\int  W(y_1,x_1)^6  e^{-\alpha \mu(y_1)}dy_1\right)^{1/6}dx_1
%&\leq \int  e^{-\frac{\alpha}{6}\mu(x_1)} L(x_1)\mu(x_1)dx_1\\
&=O(\alpha^{\sigma-1}\ell_\sigma(\alpha))
\end{align*}
It follows that, for $q_1,q_2\leq 2$
\begin{align}
I_{\alpha,3,4}(q_1,q_2)=O(\alpha^{(4+1/3)\sigma-3/2 }\ell_\sigma(\alpha)^{4+1/3})=o(\alpha^{-3}v_\alpha^2)\label{eq:I34}\\
I_{\alpha,3}(q_1,q_2)=O(\alpha^{(4+1/3)\sigma-3/2 }\ell_\sigma(\alpha)^{4+1/3})=o(\alpha^{-3}v_\alpha^2)\label{eq:I3}
\end{align}
and, combining the bounds \eqref{eq:I1}, \eqref{eq:I2} and \eqref{eq:I3}, we obtain
$\gamma_{\alpha,1}\to0$.


\begin{thebibliography}{1}
\bibliographystyle{APT}
\footnotesize

\bibitem{CaronRousseau2017:main}
{\sc Caron, F., Panero, F. and Rousseau, J.} (2022).  
\newblock On sparsity, power-law and clustering properties of graphs based of graphex processes. 
\newblock {\em Advances in Applied Probability, to appear}.

\bibitem{Last2016}
{\sc Last, G., Peccati, G. and Schulte, M.} (2016).
\newblock Normal approximation on {P}oisson spaces: {M}ehler's formula, second
  order {P}oincar{\'e} inequalities and stabilization.
\newblock {\em Probability theory and related fields\/} {\bf 165,} 667--723.

\end{thebibliography}


\end{document}

