In words, $A_{kl}$ is an indicator for whether a change to the link $L_{i_rj_r}$ changes whether the link $jl$ is pairwise stable.

We then let $B_{j}(q;F)$ be an indicator for the event that there exists a proposal network $D^*\in \mathcal{D}^*(\mathbf{E_{1,r-1}},\mathbf{D_{E_{1,r-1}}})$ such that $D_{j_ri_r}^*=1$, $\sum_{k,l} A_{jl}=q$ and the conditional empirical distribution of $x_l,s^*_l,t^*_{jl}$ given $A_{jl}=1$ is equal to $F$. By Assumptions <ref> (ii) and <ref> (ii), $q$ is stochastically bounded. We then define $\mathcal{B}_{j,i_rj_r}$ as the sigma-field generated by $\{B_{j}(q;F):q=0,1,\dots,n-1,\hspace{0.3cm} F\textnormal{ is a c.d.f.}\}$. We define $\mathcal{B}_{k,i_rj_r}$ analogously.

Now conditional on $j$ and $k$ having the same relevant overlap with $\mathbf{E_{1,r}}$ for some permutation $\tau$,
\[R(\mathbf{E_1}^{\tau},\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X}^{\tau};j)
we have by the inductive hypothesis that for each $q$ and $F$, $B_{j}(q;F)$ and $B_{k}(q;F)$ have the same probability. We then distinguish all possible cases whether or not a change to $D_{i_rj_r}$ triggers a chain of adjustments through $j$ or $k$, or neither, corresponding to the events $\mathcal{B}_{j,i_rj_r}(q_j,F_j)$ and $\mathcal{B}_{k,i_rj_r}(q_k,F_k)$ for all combinations of $(q_j,F_j)$ and $q_k,F_k$. If we can show that conditional on a partition in terms of these events, the distribution of network statistics is the same for nodes $j$ and $k$, then conditional invariance given $x_i,x_j$ and equal relevant supports follows from the law of total probability.

Specifically we first consider the (potentially overlapping) events

\[C(q_1,q_2,F_1,F_2):=\{B_j(q_1,F_1)\cap B_k(q_2,F_2)\}\]

as the arguments $q_1,q_2$ and $F_1,F_2$ vary freely. We first consider the case $q_2=q_1$ and $F_2=F_1$: by definition of the event $C(q_1,q_1,F_1,F_1)$, a switch in $D_{i_1j_1}$ triggers $q_1$ chains of adjustments starting at node $j$. Any such chain may reach node $k$ after a change to $l$, and depending only on the signs of $U^*(x_l,x_k;s_l,s_k,t_{lk})+\varepsilon_{lk}-MC_l$ and $U^*(x_k,x_l;s_k,s_l,t_{lk})+\varepsilon_{kl}-MC_k$. Similarly, the chain reaches $j$ after that same change depending only on the signs of
$U^*(x_l,x_j;s_l,s_j,t_{lj})+\varepsilon_{lj}-MC_l$ and $U^*(x_j,x_l;s_j,s_l,t_{lj})+\varepsilon_{jl}-MC_j$. By inspection, conditional on $\mathcal{B}_{\mathbf{E_1},j}$ and $\mathcal{B}_{\mathbf{E_1}^{\tau},k}$ the distributions of $(\varepsilon_{lk},\varepsilon_{kl},MC_k,x_k)$ and $(\varepsilon_{lj},\varepsilon_{jl},MC_j,x_j)$ are the same, so that for $s_j=s_k$, $t_{lk}=t_{lj}$, and $t_{jl}=t_{kl}$, the probability that the chain proceeds to $j$ is equal to that of the chain proceeding to $k$. Finally, conditional on  $R(\mathbf{E_1}^{\tau},\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X}^{\tau};k)
=R(\mathbf{E_1},\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X};j)$, the resulting changes on the sets $\mathcal{S}_{j}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2,\mathbf{D}_{\mathbf{E}_2})$ and $\mathcal{S}_{\tau(j)}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2^{\tau},\mathbf{D}_{\mathbf{E}_2})$ are the same.

Hence, given $R(\mathbf{E_1}^{\tau},\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X}^{\tau};k)
=R(\mathbf{E}_1,\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X};j)$, the conditional probability that $\mathcal{S}_{j}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2,\mathbf{D}_{\mathbf{E}_2})$ is supported given $C(q_1,q_1;F_1,F_1)$ is equal to that of $\mathcal{S}_{k}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2^{\tau},\mathbf{D}_{\mathbf{E}_2})$ being supported given $C(q_1,q_1;F_1,F_1)$.

For the case $(q_1,F_1)\neq (q_2,F_2)$, we can follow an analogous line of argument to conclude that conditional on $R(\mathbf{E_1}^{\tau},\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X}^{\tau};k)
=R(\mathbf{E}_1,\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X};j)$, the conditional probability that $\mathcal{S}_{j}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2,\mathbf{D}_{\mathbf{E}_2})$ is supported given $C(q_1,q_2;F_1,F_2)$ is equal to that of $\mathcal{S}_{k}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2^{\tau},\mathbf{D}_{\mathbf{E}_2})$ being supported given $C(q_2,q_1;F_2,F_1)$. Since the events $C(q_2,q_1;F_2,F_1)$ and $C(q_2,q_1;F_2,F_1)$ have equal probability, it follows from the law of total probability that the conditional distributions are the same given the union $C(q_1,q_2;F_1,F_2)\cup C(q_2,q_1;F_2,F_1)$. A similar line of reasoning gives us the analogous conclusion conditional on any intersections of “symmetrized" events $C(q_1,q_2;F_1,F_2)\cup C(q_2,q_1;F_2,F_1),\dots,C(q_R,q_{R+1};F_R,F_{R+1})\cup C(q_{R+1},q_R;F_{R+1},F_R)$, allowing us to construct a partition of the event $R(\mathbf{E_1}^{\tau},\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X}^{\tau};k)
=R(\mathbf{E}_1,\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X};j)$ such that invariance with respect to $\tau$ holds conditional on each element in that partition. By the law of total probability, this completes the inductive step from $r-1$ to $r$.

This establishes the conclusion of the Lemma for a binary permutation $\tau$ such that for $j,k$, $\tau(j)=k$ and $\tau(k)=j$, but $\tau(i)=i$ for any $i\neq j,k$. Since an arbitrary permutation can be generated by a sequence of pairwise swaps of indices, this is sufficient to establish the first conclusion of the Lemma.

To establish the second part of the claim we can adapt the same argument to the level of an edge, requiring only notational adjustments. Since the argument is otherwise completely analogous to the case of node-level statistics, we do not give it here explicitly.

For future reference we now also state without proof an immediate generalization of the result for the leading case to the joint distribution for multiple nodes.

Suppose Assumptions <ref>-<ref> hold. Then for any $i_1,\dots,i_D=1,\dots,n$, $\mathbf{E}$, and $\mathbf{D}_{\mathbf{E}}$ we have

\begin{eqnarray}
\nonumber \left(\mathcal{S}_{i_1}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2,\mathbf{D}_{\mathbf{E}_2}),\right.&\dots,&\left. \mathcal{S}_{i_D}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2,\mathbf{D}_{\mathbf{E}_2})\right)\\
\nonumber&\stackrel{d}{=}&\left(\mathcal{S}_{\tau(i_1)}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2^{\tau},\mathbf{D}_{\mathbf{E}_2}),
\dots, \mathcal{S}_{\tau(i_D)}^*(\mathbf{E}_1,\mathbf{D}_{\mathbf{E}_1},\mathbf{E}_2^{\tau},\mathbf{D}_{\mathbf{E}_2})\right)
\end{eqnarray}

conditional on

\begin{eqnarray}\nonumber R(\mathbf{E_1}^{\tau},\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X}^{\tau};\tau(i_1))
\nonumber&\vdots&\\
\nonumber R(\mathbf{E_1}^{\tau},\mathbf{L}_{\mathbf{E}_1}(\mathbf{D}_0,\mathbf{D_{E_1}}),\mathbf{X}^{\tau};\tau(i_D))
\end{eqnarray}

$x_{\tau(i_1)}=x_{i_1},\dots,x_{\tau(i_D)}=x_{i_D}$, and $\mathcal{F}$. The analogous conclusion holds for edge-level statistics among $i_1,\dots,i_D$ as well.

It can be seen from the proof of the main result that this generalization of the leading case, where we only need to account for the rates at which any chain of non-robust edges links back to either node $i_1,\dots,i_D$ relative to its image under the permutation $\tau$.

§.§ Proof of Lemma <ref>

First note that a Poisson process with an intensity equal to the empirical distribution over $n$ values of the variable can equivalently be represented as a Poisson process on $\{1,\dots,n\}$ with uniform mean measure. In contrast, $\mathcal{N}_i$ is generated by drawing from $\{1,\dots,n\}\backslash\{i_1,\dots,i_D\}$ uniformly at random and without replacement  conditional on realizations $(x_j,s_j^*)$ for $j\notin\{i_1,\dots,i_D\}$. Hence the point measure associated with $\mathcal{N}_i$ has the same distribution as that associated with $\tilde{\mathcal{N}}_i$ conditional on $\tilde{\mathcal{N}}_i$ assigning zero weight to the values corresponding to $i_1,\dots,i_D$ and attaining no multiplicities greater than one for any $i\in\{1,\dots,n\}$.

Furthermore, if the neighborhoods $\mathcal{N}_{i_1},\dots,\mathcal{N}_{i_D}$ are disjoint, the relevant overlap for any  $j\in\mathcal{N}_{i_d}$ with $i_{d'}, d'\neq d$ corresponds to the empty graph. Hence in the absence of ties in $\mathcal{N}_{i_1},\dots,\mathcal{N}_{i_D}$ and $\tilde{\mathcal{N}}_{i_1},\dots,\tilde{\mathcal{N}}_{i_D}$, respectively, the relevant overlap of any node $j$ with $\{i_1,\dots,i_D\}$ can without loss of generality be taken to be $R_{i_dj}$ if $j\in\mathcal{N}_{i_d}$.

We therefore next verify that the probability of ties across $\mathcal{N}_{i_1},\dots,\mathcal{N}_{i_D}$ and $\{i_1,\dots,i_D\}$ is bounded by a sequence of the order $O(n^{-1})$: Denoting $\kappa_n:=n\mathbb{P}\left(D_{ij0}=D_{ji0}=1\right)$, Assumption <ref> immediately implies that $\kappa_n$ is bounded. We then let $q_{i_d}:=|\tilde{\mathcal{N}}_{i_d}|$ and $Q:=\sum_{d=1}^Dq_{i_d}$, we have that conditional on $Q$, the probability that the $q$th draw for $\mathcal{N}_{i_d}$ constitutes a tie is at most $\frac{D+Q-1}{n}$, so the conditional probability of a tie given $Q$ is bounded from above by

\[\bar{\mu}_n:=1 - \left(1-\frac{D+Q-1}{n}\right)^Q= \frac{D+Q-1}{n} + o\left(n^{-1}\right)\]

for any finite $Q$.

We can then use the law of iterated expectations with respect to $Q$ together with the fact that the Poisson distribution for $Q$ has exponential tails to conclude that the sampling representation approximates probability of the network event up to an error $\frac{2D(1+\kappa_n)}{n}$. Specifically, for a truncation argument we can choose $Q_n=O(\log n)$ such that $P(Q\geq Q_n)\leq\frac{2D(1+\kappa_n)}{n}$. We can then bound

\begin{eqnarray}
\nonumber \mathbb{E}\left[\left(1-\frac{Q}n\right)^Q\right] &=& \sum_{q=0}^{\infty}\left(1-\frac{q}n\right)^q\frac{(D\kappa_n)^qe^{-D\kappa_n}}{q!}\\
\nonumber &\geq&\sum_{q=0}^{Q_n}\left(1-\frac{q}n\right)\left(1-\frac{Q_n}n\right)^{q-1}\frac{(D\kappa_n)^qe^{-D\kappa_n}}{q!}\\
\nonumber &=&\left(1-\frac{\mathbb{E}[Q]}n\right)\left(1-P(Q\geq Q_n) - o_p(n^{-1})\right)\\
\nonumber &\geq&1-\frac{2D(1+\kappa_n)}{n} + o_p(n^{-1})
\end{eqnarray}

which is of the desired order of magnitude. Note in particular that this bound does not depend on the empirical distribution of realizations for $(x_1,s_1^*),\dots,(x_n,s_n^*)$ and therefore also holds conditional on $\mathcal{F}$. Furthermore, by Assumption <ref>, the size of $\mathcal{N}_{i_d}$ is also bounded for each $d=1,\dots,D$ so that by an analogous argument the probability of ties among $\mathcal{N}_{i_1},\dots,\mathcal{N}_{i_D}$ is also of the order $O(n^{-1})$ with the same bounding constants.

We next argue that conditional on the event $\mathbf{B}_{\mathbf{i}}$ of no ties in the sets $\mathcal{N}_{i_1},\dots,\mathcal{N}_{i_D},\{i_1,\dots,i_D\}$ and no ties in
$\tilde{\mathcal{N}}_{i_1},\dots,\tilde{\mathcal{N}}_{i_D},\{i_1,\dots,i_D\}$, the probability of the subnetwork event on $\{i_1,\dots,i_D\}$ on the $n$-agent network is the same as that generated by the sampling process. To this end, consider any node $j\in\mathcal{N}_i$ in the original network. It then follows from Lemma <ref> that conditional on $R_{j}$, $(x_j,s_j^*)$ has the same distribution as $(x_j,s_{\tilde{j}}^*(R_{j}),t_{i\tilde{j}}^*(R_{j}))$ for any node $\tilde{j}\in\mathcal{\tilde{N}}_i$. Moreover, Lemma <ref> also implies that conditional on $\mathcal{F}$ drawing from $(x_j,s_{\tilde{j}}^*(R_{j}),t_{i\tilde{j}}^*(R_{j}))_{j=1}^n$ with replacement results in the same conditional probabilities as drawing from the distribution $\hat{M}_n(x,s|R_j)$.

By Lemma <ref>, for all $i\in\{i_1,\dots,i_D\}$ and $j\in\mathcal{N}_{\mathbf{i}}$, $\varepsilon_{ij},MC_i$ are furthermore independent of $s_j^*(R)$ for each $R$ conditional on $\mathcal{F}$. In particular, proposals

\[D_{ji}^*(s,R_{\mathbf{i}j}):=\dum\left\{U^*(x_j,x_i;s_j^*(R_{\mathbf{i}j}))+\varepsilon_{ji}\geq MC_j\right\}\]

are fully determined by initial taste shocks, and potential values $s_{j}^*(R_{\mathbf{i}j})$ for each $s\in\mathcal{S}$ and therefore also independent of $\varepsilon_{ij},MC_i$.

Since the event $\mathbf{A_{E_i}}$ was assumed to be the union of disjoint elementary events regarding the subnetwork $\mathbf{\mathbf{L}_{i}}$, these steps establish that $\pi_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathbf{B_i},\mathcal{F})=\tilde{\pi}_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathbf{B_i},\mathcal{F})$ and that $\mathbb{P}(\mathbf{B_i}^c|\mathcal{F})=O(n^{-1})$, where $\mathbf{B_i}^c$ denotes the complement of $\mathbf{B}_{\mathbf{i}}$. By the law of total probability,

\[|\pi_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathcal{F})-\tilde{\pi}_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathcal{F})|\leq |\pi_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathbf{B_i}^c,\mathcal{F})-\tilde{\pi}_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathbf{B_i}^c,\mathcal{F})|
\mathbb{P}(\mathbf{B_i}^c|\mathcal{F})\]

By assumption, the event $\mathbf{A_{E_i}}$ is invariant to permutations of edges $(ij)\in\mathbf{E_i}$ for $j\notin\{i_1,\dots,i_D\}$, so that $\pi_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathbf{B_i}^c,\mathcal{F})=\pi_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathcal{F})$ and $\tilde{\pi}_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathbf{B_i}^c,\mathcal{F})=\tilde{\pi}_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathcal{F})$ so that $\frac{\pi_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathbf{B_i}^c,\mathcal{F})}{\tilde{\pi}_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathcal{F})}$ is also bounded. By the triangle inequality it therefore follows that

\[\frac{|\pi_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathcal{F})-\tilde{\pi}_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathcal{F})|}{\tilde{\pi}_n(\mathbf{A_{E_i}}|\mathbf{x}_{\mathbf{i}},\mathcal{F})}=O(n^{-1})\]

establishing the main claim. The generalization to finite collections of events is immediate

§ PROOFS FOR SECTION <REF>

Before establishing the rate of convergence to an extreme value distribution in Lemma <ref>, we first re-state the result from <cit.> establishing that the limits for conditional link acceptance probabilities essentially correspond to their analogs under the assumption of independent extreme-value type-I taste shifters:

Suppose that Assumptions <ref>-<ref> hold. Then as $n\rightarrow\infty$,

\begin{equation}\label{ccp_limit} \left|n^{r/2}\Phi(i,j_1,\dots,j_r)- \frac{r!\prod_{s=1}^{r}\exp\{U^*_{ij_s}\}}
{\left(1+n^{-1/2}\sum_{i=1}^nD_{ji}^*(s)\exp\{U^*_{ij}\}\right)^{r+1}}\right|\rightarrow 0\end{equation}

for any $r=0,1,2,\dots$.

This is Lemma 6.2 in <cit.>, and we include the proof below for easier reference.

§.§ Proof of Lemma <ref>
This result is a generalization of Lemma B.1 in <cit.>. We therefore refer to the proof of that result for some of the intermediate technical steps below.

Denote the event $\mathcal{B}_i:=\left\{\mathbf{X},\mathbf{s}_{-i}\in\mathcal{S}_{-i}^*(\mathbf{D_{E_i}}),
\mathbf{D_{E_{-i}}}\in\mathcal{D}_{\mathbf{{E_{-i}}}}^*(\mathbf{D_{E_i}})\right\}$. We also let $J=\lceil n^{1/2}\rceil$ as in Assumption <ref>, and $j_{r+1},\dots,j_n$ be an enumeration of the indices in $\{1,\dots,n\}\backslash\{j_1,\dots,j_r\}$. Then by independence of $\eta_{i1},\dots,\eta_{iN}$,

\begin{eqnarray}
\nonumber J^r\Phi(i,j_1,\dots,j_r)&=&J^rP(U_{ij_1}\geq MC_i,\dots,U_{ij_r}\geq MC_i,U_{ij_{r+1}}<MC_i,\dots,U_{ij_{J}}<MC_i|\mathcal{B}_i)\\
\nonumber&=&J^r\int\left(\prod_{q=1}^r P(U_{ij_q}\geq \sigma s|\mathcal{B}_i)\right)\left(\prod_{q=r+1}^nP(U_{ij_q}<\sigma s|\mathcal{B}_i)^{D_{j_qi}}\right)JG(s)^{J-1}g(s)ds\\
\nonumber&=&J^r\int \left(\prod_{q=1}^r(1-G(s-\sigma^{-1}U^*_{ij_q}))\right)\left(\prod_{q=r+1}^nG(s-\sigma^{-1}U^*_{ij_q})^{D_{j_qi}}\right)
\nonumber&=&\int \left(\prod_{q=1}^rJ(1-G(s-\sigma^{-1}U^*_{ij_q}))\right)J\frac{g(s)}{G(s)}\\
\label{app_ccp_eqn}&&\times\exp\left\{J\log G(s) + \frac1J\sum_{q=r+1}^nJ\log G(s-\sigma^{-1}U^*_{ij_q})D_{j_qi}\right\}ds
\end{eqnarray}

Now let $b_J:=G^{-1}\left(1-\frac1J\right)$ and $a_J=a(b_J)$, where $a(\cdot)$ is the auxiliary function in Assumption <ref> (ii). By Assumption <ref> (iii), $\sigma =\frac{1}{a(b_J)}$, so that a change of variables $s=a_Jt + b_J$ yields

\begin{eqnarray}\label{ccp_lem_change_vars}\nonumber J^r\Phi(i,j_1,\dots,j_r)&=&\int \left(\prod_{q=1}^rJ(1-G(b_J + a_J(t-U^*_{ij_q})))\right)J\frac{a_Jg(b_J + a_Jt)}{G(b_J + a_Jt)}\\
\nonumber &&\times\exp\left\{J\log G(b_J + a_Jt) + \frac1J\sum_{q=r+1}^nJ\log G(b_J + a_J(t-U^*_{ij_q}))D_{ji}\right\}dt\end{eqnarray}

By Assumption <ref> (ii), $J(1-G(b_J+a_Jt))\rightarrow e^{-t}$ and

\[Ja_Jg(b_J + a_Jt)=Ja(b_J)g(b_J+a(b_J)t)=a(b_J)\frac{1-G(b_J+a_Jt)}{a(b_J+a_Jt)(1-G(b_J))}\rightarrow e^{-t}\]

where the last step uses Lemma 1.3 in <cit.>. Also, following steps analogous to the proof of Lemma B.1 in <cit.>, we can take limits and obtain

\begin{eqnarray}
\nonumber \prod_{q=1}^rJ(1-G(b_J+a_J(t-U^*_{ij_q})))&\rightarrow&\exp\left\{-rt + \sum_{q=1}^rU^*_{ij_q}\right\}\\
\nonumber J\log G(b_J + a_J(t-U^*_{ij_q}))&\rightarrow&-e^{-t}\exp\{U^*_{ij_q}\}
\end{eqnarray}

Combining the different components, we can take the limit of the integrand in (<ref>),

\begin{eqnarray}\nonumber R_J(t)&:=&\left(\prod_{q=1}^rJ(1-G(b_J + a_J(t-U^*_{ij_q})))\right)J\frac{a_Jg(b_J + a_Jt)}{G(b_J + a_Jt)}\\
\nonumber&&\times\exp\left\{J\log G(b_J + a_Jt) + \frac1J\sum_{q=r+1}^nJ\log G(b_J + a_J(t-U^*_{ij_q}))D_{j_qi}\right\}\\
\label{logit_lem_appx}&=&\exp\left\{-e^{-t}\left(1+\frac1J\sum_{q=r+1}^{n}D_{j_qi}\exp\{U^*_{ij_q}\}\right)-(r+1)t + \sum_{q=1}^rU^*_{ij_q}\right\}

for all $t\in\mathbb{R}$. Using the same argument as in the proof of Lemma B.1 in <cit.>, pointwise convergence and boundedness of the integrand under Assumption <ref> imply convergence of the integral by dominated convergence, so that we obtain

\begin{eqnarray}\nonumber J^r\Phi(i,j_1,\dots,j_r|\mathbf{z}_i^*)&\rightarrow&\int_{-\infty}^{\infty}\exp\left\{-e^{-t}\left(1+\frac1J\sum_{q=r+1}^{n}
D_{j_qi}\exp\{U^*_{ij_q}\}\right)-(r+1)t + \sum_{q=1}^rU^*_{ij_q}\right\}dt\\
\nonumber&=&\int_{-\infty}^{0}\exp\left\{s\left(1+\frac1J\sum_{q=r+1}^{n}D_{j_qi}\exp\{U^*_{ij_q}\}\right)+ \sum_{q=1}^rU^*_{ij_q}\right\}s^{r}ds\\
\nonumber &=&\frac{r!\exp\{\sum_{q=1}^rU^*_{ij_q}\}}{\left(1+\frac1J\sum_{q=r+1}^nD_{j_qi}\exp\left\{U^*_{ij_q}\right\}\right)^{r+1}}
\end{eqnarray}

where the first step uses a change of variables $s=-e^{-t}$, and the last step can be obtained recursively via integration by parts. Furthermore, if $\frac{r}J\rightarrow0$, boundedness of the systematic parts from Assumption <ref> implies that
\[\left|\frac1J\sum_{j=1}^JD_{ji}\exp\left\{U^*_{ij}\right\}-\frac1J\sum_{q=r+1}^nD_{j_qi}\exp\left\{U^*_{ik_q}\right\}\right|\rightarrow0\]

so that

\[J^r\Phi(i,j_1,\dots,j_r)\rightarrow \frac{r!\prod_{q=0}^r\exp\{U^*_{ij_q}\}}{\left(1+\frac1J\sum_{j=1}^nD_{ji}\exp\left\{U^*_{ij}\right\}\right)^{r+1}}\]

which completes the proof

To establish the leading terms in the asymptotic distribution for generalized subgraph counts below, we also need to show that the limiting distribution conditional on $MC_i$ is strictly decreasing in $MC_i$: Defining

\[\Phi(i,j_1,\dots,j_r|\mu) = \mathbb{P}\left(\left.\mathbf{D_{E_i}}\in\mathcal{D}_{\mathbf{E_i}}^*(\mathbf{s}_{-i})\right|\mathbf{X},\mathbf{s}_{-i}\in\mathcal{S}_{-i}^*(\mathbf{D_{E_i}}),
\mathbf{D_{E_{-i}}}\in\mathcal{D}_{\mathbf{{E_{-i}}}}^*(\mathbf{D_{E_i}}),MC_i-b_J/a_J=\mu\right)\]

we have the following lemma:

Suppose the assumptions of Lemma <ref>hold. sLet $b_J:=G^{-1}\left(1-\frac1J\right)$ and $a_J=a(b_J)$ and suppose that $r\geq1$. Then for the conditional probability given $MC_i-b_J/a_J=\mu$, we have that
$n^{r/2}\Phi(i,j_1,\dots,j_r|\mu)$ converges to a function of $\mu$ that is strictly decreasing in $\mu$ for $\mu$ large enough.

Proof of Lemma <ref> As before we let $J=\lceil n^{1/2}\rceil$. Considering the conditional version of (<ref>),

\begin{eqnarray}
\nonumber J^r\Phi(i,j_1,\dots,j_r|\mu)&=&\left(\prod_{q=1}^rJ(1-G(\mu+b_J/a_J-\sigma^{-1}U^*_{ij_q}))\right)
\left(\prod_{q=r+1}^nG(\mu+b_J/a_J-\sigma^{-1}U^*_{ij_q})^{D_{j_qi}}\right)\\
\nonumber&=&\left(\prod_{q=1}^rJ(1-G(b_J-a_J(U^*_{ij_q}-\mu)))\right)\exp\left\{\frac1J\sum_{q=r+1}^nJ\log G\left(b_J
\end{eqnarray}

where the last equality follows from a change of variables $s=a_Jt + b_J$ with $a_J,b_J$ defined as in the proof of Lemma <ref> and Assumption <ref> regarding the asymptotic sequence for $\sigma$. Substituting in the component-wise limits derived in the proof of Lemma <ref>, we obtain

\begin{eqnarray}\nonumber J^r\Phi\left(i,j_1,\dots,j_r\left|\mu\right.\right)&\rightarrow&\exp\left\{\sum_{q=1}^r(U^*_{ij_q}-\mu)
\end{eqnarray}

By inspection, there is a sufficiently large (but finite) value of $\mu_0<\infty$ such that the function on the right-hand side is strictly decreasing in $\mu$ for all $\mu\geq\mu_0$

§.§ Proof of Lemma <ref>.

This result requires only a few modifications to the proof of Lemma <ref> to control the rate at which $J(1-G(b_J+a_Jv))$ converges to $e^{-v}$. For this proof, $b_J$ is again chosen according to

\[1-G(b_J)= \frac1J\]

and $a_J:=\frac{1-G(b_J)}{g(b_J)}$, so that in particular,

\[J(1-G(b_J+a_Jv)) = \frac{1-G(b_J+a_Jv)}{1-G(b_J)}\]

For the main results, we only require rates that are pointwise in $v$, where pointwise convergence rates are invariant under mappings that are continuously differentiable. The mapping $w\mapsto\log(w)$ is continously differentiable for $w\in(0,\infty)$, so it is sufficient to establish rates in levels or logs.

First consider the term

\[\left|\log (1-G(b_J+a_Jv))+\log J +v\right|\rightarrow 0\]

In the following, we let

\[h(u):=\log(1-G(u))\]

By the choice of $b_J$ and the fundamental theorem of calculus, we have

\begin{eqnarray}
\nonumber \log\left(J(1-G(b_J+a_Jv))\right)&=&\log(1-G(b_J+a_Jv) - \log(1-G(b_J))\\
\nonumber&=&\int_{0}^{a_Jv}h'(b_J+u)du
\end{eqnarray}

By the mean-value theorem, for any value of $u\in[0,a_Jv]$, there exists an intermediate value $\bar{u}(u)\in[0,u]$ such that

\[h'(u) = h'(b_J) + h''(b_J+\bar{u}(u))u\]

Furthermore, from the choice of $a_J$ we have

\[a_Jh'(b_J) = -a_J\frac{g(b_J)}{1-G(b_J)}=-1\]

\begin{eqnarray}
\nonumber \left|\log\left(J(1-G(b_J+a_Jv))\right)+v\right|&=& \left|h'(b_J)a_Jv + \int_{0}^{a_Jv}a_Jvh''(b_J+\bar{u}(u))udu +v\right|\\
\nonumber&=&\left|\int_{b_J}^{b_J+a_Jv}h''(b_J + \bar{u}(u))udu\right|\\
\nonumber&\leq&\int_{b_J}^{b_J+a_Jv}|h''(\bar{u}(u))||a_Jv|du\\
\nonumber&=&v\int_{b_J}^{b_J+a_Jv}\frac{|h''(\bar{u}(u))|}{|h'(b_J)|}du
\end{eqnarray}

Since $a_J/b_J\rightarrow0$, we can bound $|h''(\bar{u})|\leq 2|h''(b_J(1-\varepsilon))|$ asymptotically. From Assumption <ref> (ii), it then follows that the right-hand side of this expression is of the order $o\left(\exp\left\{-\frac12b_J\right\}\right)=o(J^{-1})$.

Finally we need to establish the rate of convergence for $J\log G(b_J+a_Jv)+e^{-v}$. The argument is analogous to the previous case - noting that we can rewrite the logarithm via a mean-value expansion around $G=1$,

\[J\log G(b_J+a_Jv)=J\log(1) + \frac{J}1(G(b_J+a_Jv)-1) + \frac{J}{\bar{G}}(G(b_J+a_Jv)-1)^2\]

so that by the same arguments as before,

\[J\log G(b_J+a_Jv)=-\exp\{-v\} + o(J^{-1})\]

Since all multiplicative terms of the integrand in (<ref>) converge to finite limits, the rate of convergence of the product is governed by the term that converges at the slowest rate, which is also $o(J^{-1})$. This establishes the point-wise convergence rate of the integrand, so that Lemma <ref> follows by dominated convergence

§.§ Proof of Lemma <ref>

We first establish a law of large numbers for the bound $\hat{\Psi}_{1n}$ in equation (<ref>), showing that
\[\left|\hat{\Psi}_{1n}(x,s|R_0)-\frac{\frac1n\sum_{i=1}^n\mathbb{E}_{\Eta}[\tilde{\psi}_{1ni}[\hat{M}_n]](s,R_0,x)}

for each $s,R_0,x$. We then establish that $\bar{\psi}_{1ni}[M,\Eta]:=\mathbb{E}_{\Eta}[\tilde{\psi}_{1ni}[M]]$, allowing us to express the fixed point condition as a known expression in terms of the aggregate state variables $M,\Eta$ and economic primitives. To simplify notation, we also let $L_{ij0}:=D_{ij0}D_{ji0}$.

For a point-wise law of large numbers for $\frac1{nq_n}\sum_{i=1}^n\psi_{1ni}^*(s,R_0,x)$ it is then sufficient to show that its variance goes to zero as $n$ grows. Now, by Lemma <ref>,

\[n\mathbb{P}(L_{ik0}=1)\leq \left(\frac{\exp\{\bar{U}\}}{1+\exp\{\bar{U}\}}\right)^2=:\bar{C}^2\]

We also let $q_n:=\mathbb{P}(R_{ik}=R_0|L_{ik0}=1)$, where by assumption $nq_n\rightarrow\infty$. We can then bound

\begin{eqnarray}\nonumber\var\left(\frac1{nq_n}\sum_{i=1}^n\sum_{k\neq i}\dum_{A_{k,i}(s,R_0,x)}\right)&=&
\frac{n(n-1)}{n^2q_n^2}\var\left(\dum_{A_{1,2}(s,R_0,x)}\right)\\
\nonumber&&+\frac{n(n-1)(n-2)}{n^2q_n^2}\cov(\dum_{A_{1,2}(s,R_0,x)},\dum_{A_{1,3}(s,R_0,x)})\\
\nonumber&&+\frac{n(n-1)(n-2)}{n^2q_n^2}\cov(\dum_{A_{1,3}(s,R_0,x)},\dum_{A_{2,3}(s,R_0,x)})\\
\nonumber&&+\frac{n(n-1)(n-2)(n-3)}{n^2q_n^2}\cov\left(\dum_{A_{1,2}(s,R_0,x)},\dum_{A_{3,4}(s,R_0,x)}\right)\\
\nonumber&\leq&\frac{\bar{C}^2}{nq_n}\var\left(\dum_{A_{1,2}(s,R_0,x)}\left|L_{210}=1,R_{21}=R_0\right.\right)\\
\nonumber&&+\frac{\bar{C}^4}{nq_n}\cov\left(\dum_{A_{1,2}(s,R_0,x)},\dum_{A_{1,3}(s,R_0,x)}\left|L_{210}=L_{310}=1,R_{21}=R_0\right.\right)\\
\nonumber&&+\frac{\bar{C}^4}{nq_n}
\cov\left(\dum_{A_{1,3}(s,R_0,x)},\dum_{A_{2,3}(s,R_0,x)}\left|L_{310}=L_{320}=1,R_{31}=R_0\right.\right)\\
\nonumber&&+\bar{C}^4\cov\left(\dum_{A_{1,2}(s,R_0,x)},\dum_{A_{3,4}(s,R_0,x)}\left|L_{210}=L_{430}=1,R_{21}
\end{eqnarray}

Since $nq_n\rightarrow\infty$, the variance therefore vanishes whenever

\[\cov\left(\dum_{A_{1,2}(s,R_0,x)},\dum_{A_{3,4}(s,R_0,x)}\left|L_{210}=L_{430}=1,R_{21}

\begin{eqnarray}
\nonumber&\equiv& \mathbb{P}\left(A_{1,2}(s,R_0,x)\cap A_{3,4}(s,R_0,x)\left|L_{210}=L_{430}=1,R_{21}
\nonumber&&-\mathbb{P}\left(A_{1,2}(s,R_0,x)\right)\mathbb{P}\left(A_{3,4}(s,R_0,x)\left|L_{210}=L_{430}=1,R_{21}
\nonumber&\rightarrow&0
\end{eqnarray}

To keep notation manageable, we will take probabilities and expectations for the remainder of this step to be conditional on $L_{210}=L_{430}=1$ and $R_{21}
=R_{43}=R_0$, noting also that the event $A_{3,4}(s,R_0,x)$ does not involve $L_{21}$ and $R_{21}$, and vice versa.

To bound $\cov\left(\dum_{A_{1,2}(s,R_0,x)},\dum_{A_{3,4}(s,R_0,x)}\right)$, we partition the event $A_{1,2}(s,R_0,x)\cap A_{3,4}(s,R_0,x)$ according to whether both events are supported by common non-zero edges in $\mathbf{E}_1$ and $\mathbf{E}_2$. That is, by the law of total probability we can write

\begin{eqnarray}
\nonumber \mathbb{P}\left(A_{1,2}(s,R_0,x)\cap A_{3,4}(s,R_0,x)\right)&=&
\mathbb{P}\left(A_{1,2}(s,R_0,x)\cap A_{3,4}(s,R_0,x)\left|\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right.\right)\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right)\\
\nonumber &&+\mathbb{P}\left(A_{1,2}(s,R_0,x)\cap A_{3,4}(s,R_0,x)\left|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right.\right)\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)
\end{eqnarray}

Now since the number of edges in $\mathbf{E}_1\cap\mathbf{E}_2$ is fixed, we have by Assumption <ref> that

\[\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right) = O(n^{-1})\]

Therefore we have

\[\left|\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)\right.-\left.\mathbb{P}\left(A_{1,2}(s,R_0,x)\right)\right|\hspace{8cm}\textnormal{ }\]

\begin{eqnarray}
\nonumber&=&\left|\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right)-\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)\right|
\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right)\\
\nonumber&\leq&\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right) = O(n^{-1})
%\nonumber\left|\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)-\mathbb{P}\left(A_{1,2}(s,R_0,x)\right)\right|&\leq&
%\left|\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right)
%\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right)\right|\\
%\left|\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)
%\left(1-\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right)\right)\right|\\
%\nonumber&\leq&2\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right)=O(n^{-1})
\end{eqnarray}

Furthermore, by Lemma <ref>,

\[\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\cap A_{3,4}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)
=\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)\mathbb{P}\left(\left.A_{3,4}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)\]

Hence, we can bound

\begin{eqnarray}
\nonumber \left|\cov\left(\dum_{A_{1,2}(s,R_0,x)},\dum_{A_{3,4}(s,R_0,x)}\right)\right|&\leq&
\left|\mathbb{P}\left(\left.A_{1,2}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)\mathbb{P}\left(\left.A_{3,4}(s,R_0,x)\right|\mathbf{L_{E_1\cap E_3}}=\mathbf{0}\right)\right.\\
\nonumber&&\left.-\mathbb{P}\left(A_{1,2}(s,R_0,x)\right)\mathbb{P}\left(A_{1,2}(s,R_0,x)\right)\right|+O(n^{-1})\\
\nonumber&\leq&4\mathbb{P}\left(\mathbf{L_{E_1\cap E_3}}\neq\mathbf{0}\right)+O(n^{-1})=O(n^{-1})
\end{eqnarray}

Hence, the variance of the sample mean of $\dum_{A_{1,2}(s,R_0,x)}$ is of the order $O(n^{-1})$ so that by Chebyshev's inequality $\frac1{n}\sum_{i=1}^n\sum_{k\neq i}\psi^*_{1ni}(s,R_0,x)$ converges in probability to its expectation given $\hat{M}_n$. Since Lemma <ref> also implies that

\[\mathbb{E}_{\hat{M}_n}[\psi_{1ni}^*(s,R_0,x)] = \mathbb{E}[\tilde{\psi}_{1ni}[\hat{M}_n]](s,R_0,x) + O\left(\frac1n\right),\]

\[\frac1{n^2q_n}\sum_{i=1}^n\sum_{k\neq i}\psi^*_{1ni}(R_0)-\frac1{n^2}\sum_{i=1}^n\sum_{k\neq i}\mathbb{E}[\tilde{\psi}_{1ni}[\hat{M}_n]](R_0)\stackrel{p}{\rightarrow}0\]

where $\frac1{n^2q_n}\sum_{i=1}^n\sum_{k\neq i}\mathbb{E}[\tilde{\psi}_{1ni}[\hat{M}_n]](R_0)$ is bounded away from zero. Convergence of $\hat{\Psi}_{1n}(s,x|R_0)$ then follows from the continuous mapping theorem.

For the second step, we can evaluate the expectation of $\tilde{\psi}_{1ni}[\hat{M}]$ under the sampling representation in Lemma <ref>. Specifically, by the law of total probability the probability that $s_i^*=s$ and $R_i^*=R_0$ are supported by a pairwise stable network, and $D_{ik0}=1$ for an arbitrary node $k\neq i$,

\begin{eqnarray}
\nonumber\mathbb{E}[\tilde{\psi}_{1ni}[M]]&=&\frac1{n}\sum_{k\neq i}^n\mathbb{P}\left(\mathcal{L}_{k,i,\{j_1,\dots,j_r\}}^*\cap
\mathcal{L}(s,R_0;k,i,\{j_1,\dots,j_r\})\neq\emptyset,x_i= x,\tilde{L}_{ik0}=1|M\right)\\
\nonumber&=&\frac1{n}\sum_{k\neq i}\sum_{r\geq0}\sum_{N_{i,\{j_1,\dots,j_r\}}}
\mathbb{P}\left(\mathcal{L}_{k,i,\{j_1,\dots,j_r\}}^*\cap
\mathcal{L}(s,R_0;k,i,\{j_1,\dots,j_r\})\neq\emptyset,x_i= x,\tilde{D}_{ik0}=1|
\tilde{\mathcal{N}}_i=N_{i,\{j_1,\dots,j_r\}}\right)\\
\nonumber&&\times\mathbb{P}(\tilde{\mathcal{N}}_i=N_{i,\{j_1,\dots,j_r\}}|M)\\
\label{ref_dist_expect}&=&\sum_{r\geq0}\sum_{N_{i,\{j_1,\dots,j_r\}}}\sum_{\mathbf{L}\in\{0,1\}^r} \bar{\psi}_{1ni,j_1\dots j_r}[M,\Eta](\mathbf{L},N_{i,\{j_1,\dots,j_r\}},x)\dum_{\mathbf{L}\in\mathcal{L}(s,R_0;i,N_{i,\{j_1,\dots,j_r\}})}
\end{eqnarray}

where the second equality uses the law of total probability and that by Lemma <ref>, $\tilde{\mathcal{N}}_i$ and $\hat{M}_n$ are independent. For the third equality, we used Lemma <ref> and (<ref>). Furthermore, by standard arguments, drawing from $\hat{M}_n$ independently with replacement differs from drawing from the underlying population without replacement, and leaving out $i$ by a factor that is at most of the magnitude $O\left(\frac1n\right)$ as $n$ grows.

Averaging over all nodes, we obtain

\begin{eqnarray}
\nonumber\bar{\Psi}_{1n}[M,\Eta](x,s|R_0)&:=&\frac{\frac1n\sum_{i=1}^n\mathbb{E}_{\Eta}\left[\tilde{\psi}_{1ni}[M](s,R_0,x)\right]}
\nonumber&=:&\frac{\frac1n\sum_{i=1}^n\bar{\psi}_{1ni}[M,\Eta](s,R_0,x)}{\frac1n\sum_{i=1}^n\bar{\psi}_{1ni}[M,\Eta](R_0)}
\end{eqnarray}

as claimed, where $\mathcal{L}_{i}^*$ is the set of subnetworks $\mathbf{L_{E_i}}^*$ that are supported by a pairwise stable subnetwork on $\mathbf{E}_i$.

Combining this with convergence of $\hat{\Psi}_{1n}$ to $\bar{\Psi}_{1n}[\hat{M}_n,H]$ we obtain the conclusion of the Lemma for a singleton argument $s,R_0$. The analogous conclusion for sets $\{(s_1,R_1),\dots,(s_K,R_K)\}$ follows in a completely analogous fashion

§ PROOFS FOR SECTION <REF>

The main result in this paper is a central limit theorem for moments of the form (<ref>) for pairwise stable networks among finitely many agents. The main argument proceeds along the following steps:

* For the $D$-fold array of contributions $\left(m_{i_1\dots i_D}\right)_{i_1\dots i_D}$,  we first develop an approximate sampling representation in terms of jointly exchangeable arrays (defined below).

* Using results by <cit.> and <cit.>, we show that these approximating arrays can be characterized by a function of (vector-valued) i.i.d. factors $\alpha_{i_1}^{(1)},\dots,\alpha_{i_1\dots i_D}^{(D)}$ that are indexed by subsets of $\{i_1,\dots,i_D\}$.

* This characterization allows us to decompose the D-adic average in (<ref>) of, generally dependent, contributions to the network moment, into a sum of multi-linear forms of random vectors whose entries are independent. We can then use results by <cit.> to establish CLT-type results to these multi-linear forms.

* Having established that the network moment is asymptotically normal, we can then characterize the first two moments of its asymptotic distribution in terms of model primitives.

This appendix establishes a central limit theorem in terms of high-level conditions, which will then be used to prove Theorem <ref>.

§.§ Coupling to Exchangeable Array

We start out by approximating the array $\left(Y_{i_1,\dots,i_{D+1}}\right)_{i_1\dots i_{D+1}}$ with elements

\[Y_{i_1,\dots,i_{D+1}}:=\left(m_{i_1\dots i_D}',\psi_{i_{D+1}}\right)_{i_1\dots i_{D+1}}\]

determining the joint distribution of the network moment $\hat{m}_n(\theta)$ and the fixed point conditions for the aggregate state $\eta=[M,\Eta]$, with another array $\left(\tilde{Y}_{i_1,\dots,i_{D+1}}\right)_{i_1\dots i_{D+1}}$ consisting of entries

\[\tilde{Y}_{i_1,\dots,i_{D+1}}[\hat{\eta}_n]:=\left(\tilde{m}_{i_1\dots i_D}',\tilde{\psi}_{i_{D+1}}\right)_{i_1\dots i_{D+1}}\]

constructed under the sampling representation in Lemma <ref>, where the reference distribution $\hat{M}_n$ is given by the first component of $\hat{\eta}_n = [\hat{M}_n,\hat{\Eta}_n]$.

Specifically, we let $\hat{M}_n$ be the reference distribution corresponding to the realization of $(Y_{\mathbf{i}})_{\mathbf{i}}$, and generate $(\tilde{Y}_{\mathbf{i}})_{\mathbf{i}}$ by drawing node attributes, network neighborhoods, and taste shocks for each node $i$ at random from $\hat{M}_n$, independently from the original array. We also denote the properly normalized sample averages with

\[\bar{Y}_n:=\left(\left[\binom{n}{D}p_n\right]^{-1}\sum_{\mathbf{i}}m_{\mathbf{i}},\frac1n\sum_{i=1}^n\psi_{i}\right)\]

\[\overline{\tilde{Y}}_n[\hat{\eta}_n]:=\left(\left[\binom{n}{D}p_n\right]^{-1}\sum_{\mathbf{i}}\tilde{m}_{\mathbf{i}},\frac1n\sum_{i=1}^n\tilde{\psi}_{i}\right)\]

respectively. We first note that the array $\tilde{Y}_{i_1\dots i_{D+1}}$ is jointly exchangeable:

A jointly exchangeable array is an infinite array $(Y_{i_1\dots i_D})_{i_1\dots i_D}$ such that for any integer $\tilde{N}<\infty$ and permutation $\tau:\{1,\dots,\tilde{N}\}\rightarrow\{1,\dots,\tilde{N}\}$, we have
\[\left(Y_{\tau(i_1)\dots\tau(i_D)}\right)_{i_1\dots i_D}\stackrel{d}{=}\left(Y_{i_1\dots i_D}\right)_{i_1\dots i_D},\]
where “$\stackrel{d}{=}$" denotes equality in distribution.

By construction of either component potential values for the network statistics were drawn independently from the reference distribution so that $\left(\tilde{Y}_{i_1,\dots,i_{D+1}}\right)_{i_1\dots i_{D+1}}$ is in fact jointly exchangeable. It remains to be shown that this array approximates its finite-network analog under the appropriate metric. Specifically, we establish the following:

Suppose Assumptions <ref>-<ref> hold. Then we have

\[\overline{\tilde{Y}}_n[\hat{\eta}_n]\stackrel{d}{=}\bar{Y}_n + O_p\left(\frac1n\right)\]

conditional on $\hat{\eta}_n=\hat{\Psi}_n[\hat{\eta}_n]=\hat{\tilde{\Psi}}_n[\hat{\eta}_n]$.

Proof: Note first that $\hat{M}_n$ is the conditional empirical distribution given relevant overlap of draws $(x_k,s_k^*)$ such that $D_{ik0}=D_{ki0}=1$ for some $i\neq k$. As an intermediate step for the approximation, we now consider an array $\left(\ddot{Y}_{\mathbf{i}}[M]\right)_{\mathbf{i}}$ for a fixed value of $M$ which results from permuting the pre-network at random, specifically for a permutation $\tau$ on $\{1,\dots,n\}$ that is generated uniformly at random we let $\ddot{D}_{\tau(i)k0}=\ddot{D}_{k\tau(i)0}=1$ iff $D_{ik0}=D_{ki0}=1$, while leaving $(\ddot{x}_k,\ddot{s}_k^*)=(x_k,s_k^*)$ unchanged. $\ddot{Y}_{\mathbf{i}}[M]$ is then defined in analogy $\tilde{Y}_{\mathbf{i}}[M]$ where network neighborhoods $\ddot{\mathcal{N}}_i$ are defined by the pre-network $\ddot{D}_{ik0}=\ddot{D}_{ki0}=1$. We also denote the polyadic average with

\[\overline{\ddot{Y}}_n:=\left(\left[\binom{n}{D}p_n\right]^{-1}\sum_{\mathbf{i}}\ddot{m}_{\mathbf{i}},\frac1n\sum_{i=1}^n\ddot{\psi}_{i}\right)

Since for the original array the joint distribution of $\varepsilon_{ik}$ and $MC_i$ is i.i.d. and therefore in particular invariant to permutations of this kind, and polyadic averages are also a symmetric function of node identifiers,  we have that
\[\overline{\ddot{Y}}_n[\hat{\eta}_n]\stackrel{d}{=}\bar{Y}_n\]

conditional on $\hat{\eta}_n\in\hat{\ddot{\Psi}}_n[\hat{\eta}_n]$ and $\hat{\eta}_n\in\hat{\Psi}_n[\hat{\eta}_n]$. The most important implication of this step is that for the approximating array the pairwise stability conditions only need to hold at the level of the aggregate fixed point condition $\hat{\eta}_n\in\hat{\Psi}_n[\hat{\eta}_n]$ rather than for each node.

To complete the argument, it remains to be shown that $\overline{\tilde{Y}}_n[\hat{\eta}_n]-\overline{\ddot{Y}}_n[\hat{\eta}_n]=O_P\left(\frac1n\right)$. Here the key difference between the two arrays is that the array $\left(\ddot{Y}_{\mathbf{i}}\right)_{\mathbf{i}}$ can be thought of drawing the pre-network around a node $k$ without replacement, whereas the array $\left(\ddot{Y}_{\mathbf{i}}\right)_{\mathbf{i}}$ is generated by drawing with replacement with probabilities equal to the corresponding frequencies for the former array.

We bound the approximation error via a coupling between the two arrays that is constructed as follows: suppose that for a given node $i$ we have $\ddot{D}_{ik_q0}\ddot{D}_{k_qi0}=1$ for $q=1,\dots,Q(i)$, and
$\ddot{D}_{ik_q0}\ddot{D}_{k_qi0}=0$ for all $k\notin\{k_1,\dots,k_{Q(i)}\}$. We then draw $l_1,\dots,l_{Q(i)}$ uniformly at random and with replacement from $1,\dots,Q(i)$, resulting in the network neighborhood $\tilde{\mathcal{N}}_i$ consisting of nodes $k_{l_1},\dots,k_{l_{Q(i)}}$. This construction $\tilde{\mathcal{N}}_i$ is equivalent to the sampling representation in Lemma <ref>.

In particular it allows for the neighborhood to contain replicates of the same node, whereas in the absence of such ties, $\tilde{\mathcal{N}}_i=\ddot{\mathcal{N}}_i$. Comparing a sequence of $Q<\infty$ draws, the probability of such a tie is bounded by $\frac{Q}n$. Now, a given entry of the array $\tilde{Y}_{\mathbf{i}}\neq \ddot{Y}_{\mathbf{i}}$ only if $\tilde{\mathcal{N}}_{i_d}\neq\ddot{\mathcal{N}}_{i_d}$ for at least one component $i_d$ of the multi-index $\mathbf{i}=(i_1,\dots,i_D)$. However since the pre-network degree $|\mathcal{N}_i|$ is stochastically bounded and $D$ is fixed, the probability for that event is also bounded by a nonstochastic sequence at the rate $\frac1n$. Since the event $\tilde{\mathcal{N}}_i\neq\ddot{\mathcal{N}}_i$ is furthermore independent across $i=1,\dots,n$, this establishes the desired conclusion

§.§ Aldous-Hoover Representation

By construction, $\tilde{Y}_{i_1\dots i_D}$ is determined by the draws for marginal costs, taste shocks, and local neighborhoods $\tilde{\mathcal{N}}_{i_1},\dots,\tilde{\mathcal{N}}_{i_D}$, which are generated as i.i.d. random draws from their respective distributions. In particular, $\tilde{Y}_{i_1\dots i_D}$ is jointly exchangeable in $i_1,\dots,i_D$.

From an Aldous-Hoover representation (Theorem 7.22 in <cit.>), we can represent such a jointly exchangeable $D$-fold array on a Borel space $\mathcal{B}$ as

\[\tilde{Y}_{i_1\dots i_D} = h\left(\alpha^{(0)},\alpha_{i_1}^{(1)},\dots,\alpha_{i_D}^{(1)},\alpha_{\{i_1,i_2\}}^{(2)},\dots,\alpha_{\{i_1,\dots, i_D\}}^{(D)}\right)\]

for some measurable function $h:[0,1]^{2^D}\rightarrow\mathcal{B}$, and $(\alpha^{(0)},\alpha_{i_1}^{(1)},\dots,\alpha_{i_D}^{(1)},\alpha_{i_1i_2}^{(2)},\dots,\alpha_{i_1\dots i_D}^{(D)})$ is a uniform random array whose $2^D$ components are random variables that are independently and uniformly distributed on the unit interval, and which are indexed by the subsets of $\{i_1,\dots,i_D\}$.

§.§ Decomposition for CLT

In what follows we therefore restrict our attention to the term

\[\tilde{Z}_n=\binom{n}{D}^{-1}\sum_{i_1<\dots<i_D}(\tilde{Y}_{i_1\dots i_D}-\mathbb{E}[Y_{i_1\dots i_D}|\alpha^{(0)}])\]

For a decomposition of $\tilde{Z}_n$, we now denote the conditional expectations of $Y_{i_1\dots i_D}$ with

\begin{eqnarray}
%\nonumber y^{(0)}&:=& \mathbb{E}[\tilde{Z}_n|\alpha^{(0)}] = \mathbb{E}[Y_{i_1\dots i_D}|\alpha^{(0)}]\\
\nonumber y_{j_1\dots j_D}^{(1)}&:=& \mathbb{E}[\tilde{Z}_n|\alpha^{(0)},\alpha_{j_1}^{(1)},\dots,\alpha_{j_D}^{(1)}]\\
\nonumber y_{j_1\dots j_D}^{(2)}&:=& \mathbb{E}[\tilde{Z}_n|\alpha^{(0)},\alpha_{j_1}^{(1)},\dots,\alpha_{j_2}^{(D)},\alpha_{j_1j_2}^{(2)},\dots,\alpha_{j_{D-1}j_D}^{(2)}]\\
\nonumber \vdots&&\vdots\\
\nonumber y_{j_1\dots j_D}^{(d)}&:=& \mathbb{E}[\tilde{Z}_n|\alpha^{(0)},\alpha_{j_1}^{(1)},\dots,\alpha_{j_{1}\dots j_d}^{(d)}\dots,\alpha_{j_{D-d}\dots j_D}^{(d)}]
\end{eqnarray}

for $d=1,\dots,D$, noting that $y_{j_1\dots j_D}^{(D)}=Y_{j_1\dots j_D}$. We then form recursive projection residuals

\begin{eqnarray}
%\nonumber \dot{y}^{(0)} = y^{(0)}\\
\nonumber \dot{y}_{j_1\dots j_D}^{(1)}&:=& y_{j_1\dots j_D}^{(1)}\\
\nonumber \dot{y}_{j_1\dots j_D}^{(2)}&:=&y_{j_1\dots j_D}^{(2)} - y_{j_1\dots j_D}^{(1)}\\
\nonumber \vdots&&\vdots\\
\nonumber \dot{y}_{j_1\dots j_D}^{(d)}&:=& y_{j_1\dots j_D}^{(d)} - y_{j_1\dots j_D}^{(d-1)}\\
\nonumber \vdots&&\vdots\\
\nonumber \dot{y}_{j_1\dots j_D}^{(D)}&:=& y_{j_1\dots j_D}^{(D)} - y_{j_1\dots j_D}^{(D-1)}
\end{eqnarray}

By construction, these projection residuals are uncorrelated within and across index sets.

Given these definitions, we can decompose the sum

\begin{equation}\label{app_z_tilde_decomp}\tilde{Z}_n  = \tilde{Z}_{1n} + \dots + \tilde{Z}_{Dn}\end{equation}

\begin{eqnarray}
\nonumber \tilde{Z}_{1n}&:=&\binom{n}{D}^{-1}\sum_{i_1<\dots<i_D}\dot{y}_{i_1\dots i_D}^{(1)}\\
\nonumber \vdots&&\vdots\\
\nonumber \tilde{Z}_{Dn}&:=&\binom{n}{D}^{-1}\sum_{i_1<\dots<i_D}\dot{y}_{i_1\dots i_D}^{(D)}
\end{eqnarray}

By construction, each terms $\tilde{Z}_{dn}$ is mean-independent of its precursors $\tilde{Z}_{1n},\dots,\tilde{Z}_{(d-1)n}$.

We can now analyze the asymptotic distribution of each term $\tilde{Z}_{1n},\tilde{Z}_{2n},\dots$, noting that our analysis is conditional on $\alpha^{(0)}$ so that $\tilde{Z}_{0n}$ is regarded as fixed. We can rewrite

\begin{eqnarray}
\nonumber \dot{y}_{i_1\dots i_D}^{(1)}&=&\sum_{r=1}^D\sum_{q\in Q_1(r,D)} e_{i_1\dots i_r;d_{q(1)}\dots d_{q(r)}}^{(1)}
\end{eqnarray}

where $Q_1(r,d)$ denotes the set of all strictly increasing mappings $\{1,\dots,r\}\rightarrow\{1,\dots,d\}$, and we define

\begin{eqnarray}
%\nonumber e^{(0)}&:=&\mathbb{E}[Y_{i_1\dots i_D}|\alpha^{(0)}]\\
\nonumber e_{j_1;d_1}^{(1)}&:=&\mathbb{E}[\dot{y}_{i_1\dots i_D}^{(1)}|\alpha^{(0)},\alpha_{i_{d_1}}^{(1)}=\alpha_{j_1}^{(1)}]\\
%\nonumber&=&\mathbb{E}[\dot{y}_{i_1\dots i_D}^{(1)}|\alpha^{(0)},\alpha_{i_{d_1}}^{(1)}=\alpha_{j_1}^{(1)}]\\
\nonumber e_{j_1,j_2;d_1,d_2}^{(1)}&:=&\mathbb{E}\left[\left.
\dot{y}_{i_1\dots i_D}^{(1)}\right|\alpha^{(0)},\alpha_{i_{d_1}}^{(1)}=\alpha_{j_1}^{(1)},\alpha_{i_{d_2}}^{(1)}=\alpha_{j_2}^{(1)}\right]\\
\nonumber&&-e_{j_1;d_1}^{(1)} - e_{j_2;d_2}^{(1)}\\
%Y_{i_1\dots i_D}\right|\alpha^{(0)},\alpha_{i_{d_1}}^{(1)}=\alpha_{j_1}^{(1)}\right]
%Y_{i_1\dots i_D}\right|\alpha^{(0)},\alpha_{i_{d_2}}^{(1)}=\alpha_{j_2}^{(1)}\right]+\mathbb{E}\left[\left.Y_{i_1\dots i_D}\right|\alpha^{(0)}\right]\\
\nonumber e_{j_1\dots j_r;d_1\dots d_r}^{(1)}&:=&\mathbb{E}\left[\left.
\dot{y}_{i_1\dots i_D}^{(1)}\right|\alpha^{(0)},\alpha_{i_{d_1}}^{(1)}=\alpha_{j_1}^{(1)},\dots,\alpha_{i_{d_r}}^{(1)}=\alpha_{j_r}^{(1)}\right]\\
\nonumber&&-\sum_{s=1}^{r-1}\sum_{q\in Q_1(s,r-1)}e_{j_1\dots j_s;d_{q(1)}\dots d_{q(s)}}^{(1)}
\end{eqnarray}

recursively. This expansion can be viewed as a generalized Hoeffding composition, whose individual components are mean-independent from their respective precursors, again by construction.

Now let $Q_2(r,d)$ denotes the set of all nondecreasing mappings $\mathbf{q}:\{1,\dots,r\}\rightarrow\{1,\dots,d\}^2$ such that the second component is larger than the first, and for each $s=1,\dots,r$, at least one component of $\mathbf{q}(s)$ is strictly greater than the corresponding component of $\mathbf{q}(s-1)$. We then write

\begin{eqnarray}
\nonumber \dot{y}_{i_1\dots i_D}^{(2)}&=&\sum_{r=2}^D\sum_{q\in Q_2(r,D)} e_{\mathbf{i}_1\dots \mathbf{i}_r;\mathbf{d}_{q(1)}\dots \mathbf{d}_{q(r)}}^{(2)}
\end{eqnarray}

where with some abuse of notation we let $\mathbf{i}_{q_1q_2}:=(i_{q_1}i_{q_2})$, $\mathbf{d}_{q_1q_1}:=(d_{q_1}d_{q_2})$, and we define

\begin{eqnarray}
\nonumber e_{\mathbf{j}_1;\mathbf{d}_1}^{(2)}&:=&\mathbb{E}\left[\left.
\dot{y}_{i_1\dots i_D}^{(2)}\right|\alpha^{(0)},\alpha_{i_1}^{(1)},\dots,\alpha_{i_D}^{(1)},\alpha_{i_{\mathbf{d}_1}}^{(2)}=\alpha_{\mathbf{j}_1}^{(2)}\right]\\
\nonumber e_{\mathbf{j}_1\dots\mathbf{j}_r;\mathbf{d}_1\dots\mathbf{d}_r}^{(1)}&:=&\mathbb{E}\left[\left.
\dot{y}_{i_1\dots i_D}^{(2)}\right|\alpha^{(0)},\alpha_{i_1}^{(1)},\dots,\alpha_{i_D}^{(1)},\alpha_{i_{\mathbf{d}_1}}^{(2)}=\alpha_{\mathbf{j}_1}^{(2)},
\dots,\alpha_{i_{\mathbf{d}_r}}^{(2)}=\alpha_{\mathbf{j}_r}^{(2)}\right]\\
\nonumber&&-\sum_{s=2}^{r-1}\sum_{q\in Q_2(s,r-1)}e_{\mathbf{j}_1,\dots \mathbf{j}_s;\mathbf{d}_{q(1)}\dots\mathbf{d}_{q(s)}}^{(2)}
\end{eqnarray}

More generally, for $k=1,\dots,D$, we let $Q_d(r,k)$ denote the set of all nondecreasing mappings $\mathbf{q}:\{1,\dots,r\}\rightarrow\{1,\dots,k\}^d$ such that for each $s=1,\dots,r$, at least one component of $\mathbf{q}(s)$ is strictly greater than the corresponding component of $\mathbf{q}(s-1)$, and each component $q_l(s)<q_{l+1}(s)$. We then write

\begin{eqnarray}
\label{y_dot_k_decomp} \dot{y}_{i_1\dots i_D}^{(k)}&=&\sum_{r=k}^D\sum_{q\in Q_k(r,D)} e_{\mathbf{i}_1\dots \mathbf{i}_r;\mathbf{d}_{q(1)}\dots \mathbf{d}_{q(r)}}^{(k)}
\end{eqnarray}

where $\mathbf{i}_{q_1\dots q_r}:=(i_{q_1}\dots i_{q_r})$, $\mathbf{d}_{q_1\dots q_r}:=(d_{q_1}\dots d_{q_r})$, and

\begin{eqnarray}
\nonumber e_{\mathbf{j}_1;\mathbf{d}_1}^{(k)}&:=&\mathbb{E}\left[\left.
\dot{y}_{i_1\dots i_D}^{(k)}\right|\alpha^{(0)},\alpha_{i_1}^{(1)},\dots,\alpha_{i_D}^{(1)},\dots,\alpha_{i_{D-k+1}\dots i_D}^{(k-1)},\alpha_{i_{\mathbf{d}_1}}^{(k)}=\alpha_{\mathbf{j}_1}^{(k)}\right]\\
\nonumber e_{\mathbf{j}_1\dots\mathbf{j}_r;\mathbf{d}_1\dots\mathbf{d}_r}^{(k)}&:=&\mathbb{E}\left[\left.
\dot{y}_{i_1\dots i_D}^{(k)}\right|\alpha^{(0)},\alpha_{i_1}^{(1)},\dots,\alpha_{i_D}^{(1)},,\dots,\alpha_{i_{D-k+1}\dots i_D}^{(k-1)},\alpha_{i_{\mathbf{d}_1}}^{(k)}=\alpha_{\mathbf{j}_1}^{(k)},
\dots,\alpha_{i_{\mathbf{d}_r}}^{(k)}=\alpha_{\mathbf{j}_r}^{(k)}\right]\\
\nonumber&&-\sum_{s=k}^{r-1}\sum_{q\in Q_k(s,r-1)}e_{\mathbf{j}_1\dots \mathbf{j}_s;\mathbf{d}_{q(1)}\dots\mathbf{d}_{q(s)}}^{(k)}
\end{eqnarray}

are again defined recursively.

§.§ Rate of Convergence

In the next subsections <ref>-<ref>, we present generic results for a scalar component of the multivariate exchangeable array $\tilde{\mathbf{Y}}{\mathbf{i}}$, which will subsequently be extended to the full array using the Cramér-Wold device.

For the scalar case, we can write

\[\sigma_{k,\mathbf{d}_1\dots\mathbf{d}_r}^2:=\var(e_{\mathbf{j}_1\dots\mathbf{j}_r;\mathbf{d}_1\dots\mathbf{d}_r}^{(k)})  \]

for any $k=1,\dots,D$, where each $\mathbf{d}_s\in\{1,\dots,D\}^k$ and $\mathbf{j}_s\in\{1,\dots,D\}^k$. Under sparse asymptotics, $\sigma_{k,\mathbf{d}_1\dots\mathbf{d}_r}^2$ will in general not be constant, but some components will go to zero at some rate as $n$ increases. We first establish the stochastic order of the terms the expansion (<ref>) and (<ref>) corresponding to projections of degree $r=1$.

Suppose that $\tilde{Y}_{i_1\dots i_D}$ is a jointly exchangeable array with scalar entries which are symmetric across dimensions $d=1,\dots,D$ and satisfy the conditions of Assumption <ref>. Then for any $k=1,\dots, D$ such that $\binom{n}{k}\sigma_{k,\mathbf{d}_1}^2\rightarrow\infty$, the terms

\[\sum_{1\leq i_1<\dots<i_k\leq n}\sum_{1\leq d_1<\dots<d_k\leq D}e_{i_1\dots i_k,d_1\dots d_k}^{(k)} = O_P(r_{n,k}^{-1/2})\]

\[r_{n,k}^{-1}:=\binom{n}{k}^{-1}\sigma_{k,\mathbf{d}_1}^2\]

and $\mathbf{d}_1:=(1,\dots,k)'$.

Proof: From the recursive construction of $e_{i_1\dots i_{k},d_1\dots d_{k}}^{(k)}$ and independence of $\{\alpha_{i_{d_1},\dots,i_{d_s}}^{(s)}:s=1,\dots,D\}$, we have that conditional on $\{\alpha_{i_{d_1},\dots,i_{d_s}}^{(s)}:s=1,\dots,k-1\}$, $e_{i_1\dots i_{k},d_1\dots d_{k}}^{(k)}$ are mean-independent, although not necessarily i.i.d.. Furthermore, $Y_{i_1\dots i_D}$ are bounded by Assumption <ref> (a), so that the conclusion follows from a martingale difference central limit theorem for triangular arrays, see e.g. the main theorem in <cit.>. The unconditional statement follows from the law of iterated expectations and the fact that $\{\alpha_{i_{d_1},\dots,i_{d_s}}^{(s)}:s=1,\dots,k-1\}$ are also i.i.d.

The next lemma establishes that under regularity conditions, the projections of degree $r=1$ do in fact provide the leading terms in that expansion, possibly for multiple values of $k=1,\dots,D$, whereas terms corresponding to $r>1$ are generally dominated.

Suppose that $\tilde{Y}_{i_1\dots i_D}$ is a jointly exchangeable array with scalar entries which are symmetric across dimensions $d=1,\dots,D$ and satisfy the conditions of Assumption <ref>. Furthermore assume that the component variances satisfy $\sigma_{k,\mathbf{d}_1}^2/\sigma_{k,\mathbf{d}_1\dots\mathbf{d}_r}^2\geq B$ for some lower bound $B>0$. Then for any $k=1,\dots, D$ and $r=2,\dots,k$, the terms

\[\sum_{1\leq i_1<\dots<i_D\leq n}\sum_{q\in Q_k(r,D)}e_{\mathbf{i}_{q_1}\dots \mathbf{i}_{q_r},\mathbf{d}_{q_1}\dots \mathbf{d}_{q_r}}^{(k)} = o_P(r_{n,k}^{-1/2})\]

Proof: We first notice that by Assumption <ref> and the triangle inequality all moments of $e_{i_1\dots i_{r},d_1\dots d_{r}}$ exist. Also by construction, $e_{i_1\dots i_{r},d_1\dots d_{r}}$ are mean-independent and therefore uncorrelated. Since there are $\binom{\binom{n}{k}}{r}$ of these terms, the variance of the $r$-fold mean equals $\left[\binom{\binom{n}{k}}{r}\sigma_{k,\mathbf{d}_1\dots\mathbf{d}_r}^2\right]^{-1}\leq \left[\binom{\binom{n}{k}}{r}B\sigma_{K,\mathbf{d}_1}^2\right]^{-1}$, which converges to zero at a rate faster than $r_{n,K}^{-1}$, so that the conclusion follows directly from Chebyshev's inequality

§.§ Generic LLN and CLT

Given the previous two lemmata, we can establish a Law of Large Numbers for the average $\tilde{Z}_n$.

Suppose that $\tilde{Y}_{i_1\dots i_D}$ is a jointly exchangeable array with scalar entries which are symmetric across dimensions $d=1,\dots,D$ and satisfy the conditions of Assumption <ref>. Then for any sequence $q_n$ such that for each $k=1,\dots,D$, $\sigma_{n,k}^2/p_n$ is bounded and $\binom{N}{D}p_n\rightarrow\infty$, we have

\[p_n^{-1}\tilde{Z}_{n}\stackrel{p}{\rightarrow}0\]

This lemma is an immediate consequence of Lemmas <ref> and <ref> and Chebyshev's Inequality. These lemmas furthermore identify the leading terms in the expansion for $\tilde{Z}_n$, allowing us to establish a generic central limit theorem. Let $r_n:=\min\{r_{n,1},\dots,r_{n,D}\}$ and define

\[V_n:=r_n\sum_{k=1}^D\binom{n}{k}\binom{D}{k}\sigma_{k,\mathbf{d}_1}^2\]

We then have the following:

Suppose that $\tilde{Y}_{i_1\dots i_D}$ is a jointly exchangeable array with scalar entries which are symmetric across dimensions $d=1,\dots,D$ and satisfy the conditions of Assumption <ref>. If furthermore $V_n\rightarrow V$ for some $\mathcal{F}$-measurable positive definite matrix $V$ almost surely, we have

\[r_n^{1/2}V^{-1/2}\tilde{Z}_{n}\stackrel{d}{\rightarrow}N(0,\mathbf{I}),\]

mixing with respect to $\mathcal{F}$.

Note that the asymptotic variance $V$ is defined only implicitly as a limit in order to allow for drifting sequences to accommodate sparse network asymptotics. We analyze this expression in terms of the primitives of the economic model further below.

Proof: As before, we decompose $\tilde{Z}_n$ into

\[\tilde{Z}_n = \tilde{Z}_{1n} + \tilde{Z}_{2n} + \dots +\tilde{Z}_{Dn}\]

with $\tilde{Z}_{kn}$ defined in (<ref>). It follows from Lemmas <ref> and <ref> that

\[\tilde{Z}_{kn} = \sum_{i_1<\dots<i_D}\sum_{q\in Q_k(D,k)} e_{i_1\dots i_k;d_{q_1}\dots d_{q_k}}^{(k)}+o_P(r_{nk}^{-1/2})\]

so that in the following we can focus on the leading term of each component $\tilde{Z}_{kn}$.

We now define the filtration

\[\mathcal{G}_i:=\sigma\left(\left\{\alpha^{(0)},\alpha_{i_1}^{(1)},\dots,\alpha_{i_1\dots i_D}^{(D)}:i_1\leq i\right\}\right)\]

By definition of $e_{\mathbf{i}_1\dots\mathbf{i}_r;\mathbf{d}_1\dots\mathbf{d}_r}^{(k)}$,

\[\mathbb{E}[e_{i_1\dots i_k;d_1\dots d_k}^{(k)}|\mathcal{G}_i]=\left\{\begin{array}{lcl}
e_{i_1\dots i_k;d_1\dots d_k}^{(k)}&\hspace{0.3cm}&\textnormal{if }i_1\leq i\\0&&\textnormal{otherwise}\end{array}\right.\]

Hence, $\tilde{Z}_{kn}$ is a martingale adapted to $\mathcal{G}_I$ with the martingale differences

\begin{eqnarray}
\nonumber X_{kni}&=&\mathbb{E}[Z_{kn}|\mathcal{G}_i] - \mathbb{E}[Z_{kn}|\mathcal{G}_{i-1}]\\
\nonumber &=&\sum_{i=i_1<i_2<\dots<i_D}\sum_{q\in Q_k(D,k)} e_{i_1\dots i_k;d_{q_1}\dots d_{q_k}}^{(k)}+o_P(r_{nk}^{-1/2}/n)
\end{eqnarray}

Since each component of $e_{i_1\dots i_k;d_{q_1}\dots d_{q_k}}^{(k)}$ is mean-independent of its predecessors $e_{i_1\dots i_{k-s};d_{q_1}\dots d_{q_{k-s}}}^{(k-s)}$, $s=1,\dots, k-1$, the first three conditional moments of $X_n:=X_{n1i}+ \dots + X_{nDi}$ given $\mathcal{G}_{j}$ are equal to the sum of the respective conditional moments of $X_{n1i},\dots,X_{nDi}$.

By construction, $\mathbb{E}[X_{ni}|\mathcal{G}_{i-1}]=0$, and

\[\var(r_n^{1/2}X_{ni}|\mathcal{G}_{i-1}) =r_n\sum_{k=1}^D\binom{n-1}{k-1}\binom{D-1}{k-1}\sigma_{k,\mathbf{d}_1}^2+o_P\left(\frac{r_n}{nr_{n,k}}\right)\]

which is bounded by the definition of $r_n$. Summing over $i=1,\dots,n$ and rearranging terms, we get

\[\var(r_n^{1/2}\tilde{Z}_n)=r_n\sum_{k=1}^D\binom{n}{k}\binom{D}{k}\sigma_{k,\mathbf{d}_1}^2+o_P\left(1\right)\]

Furthermore, the third absolute moments of the joint distribution of $r_n^{1/2}(X_{1ni},\dots,X_{Dni})$ are bounded, so that the third moments of $r_n^{1/2}X_{ni}$ are also bounded. This implies a Lindeberg condition as stated in (2.4) in <cit.>, which ensures that conditions (a) and (b) of Theorem (2.3)in <cit.> are satisfied. Condition (c) follows from similar considerations for the fourth moments and Chebyshev's inequality. Hence, the conclusion of this Lemma follows from Theorem (2.3) in <cit.>

Given these intermediate results, we now conclude with the proofs of the main results for Section <ref>.

§.§ Proof of Theorem <ref>
The asymptotic representation of the moment in terms of the aggregate state variables follows from Lemmas <ref> and <ref>, and Theorem <ref> ensures that the limiting model is well-defined. The existence a sequence of fixed points

\[\eta_n\in\Psi_0(\eta_n)\]

supporting a converging sequence of the moment follows immediately from Theorem 4.1 in <cit.>. Furthermore Lemma B.3 in <cit.> implies convergence of $\hat{\eta}_n\equiv \hat{\eta}_n(\alpha^{(0)})$ to the set of fixed points of $\Psi_0$ conditional on $\mathcal{F}$. In particular, given the conclusion of Lemma <ref>, a law of large numbers for the mapping $\hat{\Psi}_n$ follows from the Birkhoff LLN for exchangeable sequences together with the continuous mapping theorem and the Glivenko-Cantelli property for $\tilde{\psi}_i(\eta)$. The conclusion then follows from Lemmas <ref> and <ref> with $q_n:=p_n$, noting that $\sigma_{k,n}^2/p_n$ is asymptotically bounded and that $m_0(\theta;\eta)$ is continuous in $\eta$

§.§ Embedding into Asymptotic Sequence

For a central limit theorem with mixing we embed the approximating array $\left(\tilde{Y}_{\mathbf{i}}\right)_{\mathbf{i}}$ into an asymptotic sequence along which we then take limits. Since the fixed points for $\hat{\eta}_n$ and $\hat{\tilde{\eta}}_n$ are not necessarily unique but may be selected at random, this requires that the approximating sequence is defined on a common probability space such that $\hat{\tilde{\eta}}_n$ converges to a well-defined limit $\eta_0^*$ which may be stochastic. Specifically we have the following:

Suppose Assumptions <ref>-<ref>, <ref>, and <ref> hold. Then for any regular fixed point of the limiting mapping $\eta_0^*\in\Psi_0[\eta_0^*]$ we can construct a sequence of exchangeable arrays $\left(\tilde{Y}_{\mathbf{i}n}\right)_{\mathbf{i}}$, $n=1,2,\dots$ defined on a common probability space according to the sampling representation in Lemma <ref> such that $\hat{\tilde{\eta}}_n\in\overline{\tilde{\Psi}}_n[\hat{\tilde{\eta}}_n]$ and $\hat{\tilde{\eta}}_n\rightarrow\eta_0^*$ almost surely.

Proof: From the law of large numbers in Theorem <ref> it follows that for the set $\mathcal{H}_0^*:=\left\{\eta_0^*:\eta_0^* = \Psi_0[\eta_0^*]\right\}$ we have $d(\hat{\tilde{\eta}}_n,\mathcal{H}_0^*)\stackrel{p}{\rightarrow}0$. Hence we can choose $\eta_0^*$ as the element of $\mathcal{H}_0^*$ that is closest to $\hat{\eta}_n$. Recall that Assumption <ref> (iv) states that the fixed point $\eta_0^*$ must be a regular point of the mapping $\Psi_0[\eta]$. Local existence of a fixed point $\hat{\tilde{\eta}}_{n'}\in\overline{\tilde{\Psi}}_n[\hat{\tilde{\eta}}_{n'}]$ for each $n'$ and $\hat{\tilde{\eta}}_{n'}\stackrel{a.s.}{\rightarrow}\eta_0^*$ then follows from Theorem 3.1(b) in <cit.>

Since we are coupling realizations of $\left(\tilde{Y}_{\mathbf{i}}\right)_{\mathbf{i}}$ rather than the initial array $\left(Y_{\mathbf{i}}\right)_{\mathbf{i}}$, this does not require existence of a pairwise stable network supporting values $\hat{\eta}_n\rightarrow\eta_0^*$ for any given realization of payoffs, which would be more difficult to establish. For the remainder of the argument, we take the approximating sequence to be defined on a common probability space according to the coupling in Lemma <ref>.

Given the generic CLT in Lemma <ref> in the appendix, we now turn to our main results which will be stated in terms of primitives of the network formation model. We first note that the network moments have the specific structure

\[m_{\mathbf{i}}(\mathbf{L},\mathbf{X};\theta) = \dum\left\{\mathbf{L}_{|\mathbf{i}}\in\mathcal{A}_{\mathbf{i}}\right\}h_{\mathcal{A}_i}(\mathbf{x}_{\mathbf{i}};\theta)\]

For simple (i.e. singleton) events $\mathcal{A}_{\mathbf{i}}$, the corresponding indicator variable is the product of edge-specific indicators $L_{i_qi_r},(1-L_{i_qi_r})$ for $q,r\in\{1,\dots,D\}$. If the two-dimensional approximating array $\left(\tilde{L}_{ij}\right)$ is jointly exchangeable, it follows that it can be represented as a function

\[\tilde{L}_{i_1i_2} = g(\xi^{(0)},\xi_{i_1}^{(1)},
\xi_{i_2}^{(1)},\xi_{i_1i_2}^{(2)})\]

Since attributes $x_{i_1}$ are i.i.d. by assumption, it follows that we can represent

\begin{equation}\label{app_2nd_order_rep_struc}m_{\mathbf{i}}(\tilde{\mathbf{L}},\mathbf{X};\theta) = \tilde{m}(\xi^{(0)},\xi_{i_1}^{(1)},\dots,\xi_{i_D}^{(1)},\xi_{i_1i_2}^{(2)},\dots,\xi_{i_{D-1}i_D}^{(2)},x_{i_1},\dots,x_{i_D};\theta)
\end{equation}

as a function of i.i.d. random variables at the node or edge level alone, which does not depend on any shocks at the level of $d$-ads of any order higher than $d=2$.

For composite events, corresponding to the case where $\mathcal{A}_{\mathbf{i}}$ is a set of size greater than one, the same argument holds, noting that the indicator for the composite event is the maximum over the indicators for each of the simple events contained in $\mathcal{A}_{\mathbf{i}}$. Lemma <ref> in the appendix then determines convergence rates given the network formation model in Assumptions <ref>-<ref>.

Given the projection representation of the network moment introduced before, we next establish the convergence rate given the network formation model:

Assume the network formation model in Assumptions <ref>-<ref>, and <ref>. Furthermore suppose that Assumption <ref> holds and that the network event $\mathcal{A}_{i_1\dots i_D}$ implies that $L_{i_1i_s}=1$ for at least one $s\in\{2,\dots,D\}$. We then have that
$\sigma_1^2/p_n^2$ converges almost surely to a limit that is finite and bounded away from zero, and $\sigma_2^2/p_n^2 = O\left(1\right)$ so that $r_n = \left(\binom{n}{1}p_n^2\right)^{-1}$. Furthermore,

\[r_n\sigma_{k,\mathbf{d}_1}^2\rightarrow0\]

almost surely for all $k>2$.

Proof: Note first that by Lemma <ref>, the second moments converge almost surely to their $\mathcal{F}$-measurable limits. For the first claim, consider the conditional probability of $\mathcal{A}_i$ given $MC_i$,

\[\pi_{n1}(\mu):=\mathbb{P}\left(\left.\mathbf{L}_{|\mathbf{i}}\in\mathcal{A}_{\mathbf{i}}\frac{}{}\right|MC_{i_1}-\frac12\log n=\mu\right)\]
where by the law of iterated expectations, $\mathbb{E}[\pi_{n1}(MC_{i_1})]=p_n$. Since

\[e_{i_1;1}^1:=(\pi_{n1}(MC_{i_1}) - p_n)\mathbb{E}[h(x_{i_1},\dots,x_{i_D})|x_{i_1}]\]

it that it will be sufficient to show that $\var\left(\frac{\pi_{n1}(MC_{i_1})-p_n}{p_n}\right)$ remains bounded away from zero.

In order to evaluate $\pi_{n1}(MC_{i_1})$, we make use of the conditional independence result in Lemma <ref>. As before, we define $D_{ij}:=\dum\left\{U_{ij}(\mathbf{L},\mathbf{X})\geq MC_i\right\}$ as an indicator whether $i$ would agree to form a link to $j$ given that network $\mathbf{L}$, and let $D_{ij}^*:=\dum\left\{U_{ij}(\mathbf{L}^*,\mathbf{X})\geq MC_i\right\}$ denote the corresponding indicator given the pairwise stable network $\mathbf{L}^*$.

Now for every network $\bar{\mathbf{L}}_{|\mathbf{i}}\in\mathcal{A}_{\mathbf{i}}$, we let $\mathcal{A}_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}})$ denote the event that $D_{i_1i_s}=1$ for each $i_s\in\{i_2,\dots, i_D\}$ with $\bar{L}_{i_1i_s}=1$. Since it was assumed that $\mathcal{A}_{i_1\dots i_D}$ contains at least one non-zero link between $i_1$ and another node $i_s\in\{i_2,\dots, i_D\}$, there exists at least one such $i_s$. We also let $A_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}})\in\mathcal{A}_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}})$ denote the (singleton) event $D_{i_1i_s}=\bar{L}_{i_1i_s}$ for all $s=2,\dots,D$. Since under sparse asymptotics in Assumption <ref>, $\mathbb{P}(D_{ij}=1)$ goes to zero, $\frac{\mathbb{P}(A_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}}))}{\mathbb{P}(\mathcal{A}_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}}))}\rightarrow1$ so that we can without loss of generality restrict our attention to the singleton event $A_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}})$.

We then have by the law of iterated expectations that

\begin{eqnarray}
\nonumber \pi_{n1}(\mu)&:=&\mathbb{P}\left(\left.\mathbf{L}_{|\mathbf{i}}\in\mathcal{A}_{\mathbf{i}}\right|\mathcal{A}_{Di_1;i_2\dots i_D},MC_{i_1}-\frac12\log n=\mu\right)\mathbb{P}(\mathcal{A}_{Di_1;i_2\dots i_D}|MC_{i_1}-\frac12\log n=\mu)\\
\nonumber&=&\sum_{\bar{\mathbf{L}}_{|\mathbf{i}}\in\mathcal{A}_{\mathbf{i}}}
\mathbb{P}\left(\left.\mathbf{L}_{|\mathbf{i}}\right|A_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}}),MC_{i_1}-\frac12\log n=\mu\right)\mathbb{P}(\mathcal{A}_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}})|MC_{i_1}-\frac12\log n=\mu) + o(p_n)\\
\nonumber&=&\sum_{\bar{\mathbf{L}}_{|\mathbf{i}}\in\mathcal{A}_{\mathbf{i}}}
\mathbb{P}\left(\left.\mathbf{L}_{|\mathbf{i}}\right|A_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}})\right)\mathbb{P}(\mathcal{A}_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}})|MC_{i_1}-\frac12\log n=\mu) + o(p_n)
\end{eqnarray}

where the last step follows from Lemma <ref>.

Since $\eta_{i_1i_s}$ is continuously distributed with full support, $\mathbb{P}(\mathcal{A}_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}})|MC_{i_1}-\frac12\log n=\mu)\mathbb{P}(\mathcal{A}_{Di_1;i_2\dots i_D}(\bar{\mathbf{L}}_{|\mathbf{i}}))$ is strictly increasing in $\mu$ for any $n$, and by Lemma <ref>, its limit is also strictly decreasing in $\mu$. Since $MC_i$ is continuously distributed with full support on $\mathbb{R}$, it follows that $\var\left(\frac{\pi_{n1}(MC_{i_1})-p_n}{p_n}\right)$ converges to a strictly positive limit, establishing the first claim.

From analogous arguments, we can show that $\sigma_2^2 = O\left(p_n\right)$ and $\sigma_{k,\mathbf{d}_1\dots\mathbf{d}_r}^2=O\left(p_n\right)$ for every $k\geq2$ and $r=1,\dots,k$. By Lemma <ref>, this establishes the rate $r_n = \left(\binom{n}{D}p_n\right)^{-1}$. Finally (<ref>) implies that jointly exchangeable approximation to the network moment can be represented in terms of Aldous-Hoover factors of degree less than 3, establishing the last claim of this Lemma

We can now state the proofs for the remaining results in Section <ref>.

§.§ Proof of Lemma <ref>
Without loss of generality, we assume that the approximating sequence of arrays $\left(\tilde{Y}_{\mathbf{i}n}\right)_{\mathbf{i}}$ is defined on a common probability space using the construction in Lemma <ref>.

Using the notation in Lemma <ref>, we let $\mathcal{J}_i$ denote the set of nodes available to $i$ given the selected potential values of network statistics, $\mathbf{s_{E_{i,2}}}$,

\[\mathcal{J}_i^*\equiv :=\mathcal{J}_i^*(\mathbf{D_{E_i}}):=\{j: D_{ji}^*=1\}\subset\{1,\dots,n\}\]

where $\mathbf{D_{E_i}}\in\mathcal{D}_{\mathbf{E_i}}^*(\mathbf{s_{E_{i,2}}})$. We also let

\[J^*:=|\mathcal{J}_i^*|\]

denote the size of the link opportunity set $\mathcal{J}_i^*$. By Assumption <ref>, $J^*/\sqrt{n}=O_p(1)$ with finite variance.

The expectation of $\hat{m}_n$ and $\hat{\Psi}_n$ will be determined by conditional link acceptance probabilities of the form

\[\Phi(i,j_1,\dots,j_r|\mathbf{z}_i^*) = P\left(U_{ij_1},\dots,U_{ij_r}\geq MC_i >U_{ij'}\textnormal{ all other }j'\in W_i(\mathbf{L}^*)|\mathbf{z}_i^*\right)\]

For a given node set $\{j_1,\dots,j_r\}$ we also denote

\[\bar{\mathcal{J}}_i^*:=\mathcal{J}_i^*\backslash \{j_1,\dots,j_r\}\]

Given the conclusion of Lemma <ref>, conditional link acceptance probabilities $n^{r/2}\Phi(i,j_1,\dots,j_r|\mathbf{z}_i^*)$ can be approximated by a smooth function of $\hat{I}_{i,j_1,\dots,j_r;n}:=\frac1{J^*}\sum_{j\in\bar{\mathcal{J}}_i^*}\exp\{\tilde{U}_{ij}\}$ at a rate faster than $n^{-1/2}$. Specifically we denote

\[g(\hat{I}_{i,j_1,\dots,j_r;n}):=\frac{r!\prod_{s=1}^{r}\exp\{\tilde{U}_{ij_s}\}}

We also define $\hat{I}_{i;n}:=\frac1{J^*}\sum_{j\in\mathcal{J}_i^*}\exp\{\tilde{U}_{ij}\}$.

First, if $\frac{r}{n^{1/2}}\rightarrow0$, boundedness of the systematic parts from Assumption <ref> implies that
\[n^{1/2}|\hat{I}_{i,j_1,\dots,j_r;n}-\hat{I}_{i,n}|=
\exp\left\{\tilde{U}_{ij}\right\}\right|\]

as asymptotically bounded.

The second approximation error arises from the randomness in $\hat{I}_{i,n}-\hat{\Eta}_n(x_i,s_i)$. To this end we establish that $n^{1/4}(\hat{I}_{i,n}-\hat{\Eta}_n(x_i,s_i))$ is asymptotically normal, so that in particular its asymptotic variance converges to a finite limit. Define $X_{kn}:=\left(\dum\{U_{ki}\geq MC_k\}-\pi_{kn}\right)\exp\{U_{ik}^*(L^*)\}$, where $\pi_{kn}:=\mathbb{P}(U_{ki}\geq MC_k|z_{1i}^*,\dots,z_{ni}^*)$. By Lemma <ref>, $\varepsilon_{ki}$ and $MC_k$ are drawn without replacement from $\varepsilon_{1i},\dots,\varepsilon_{ni}$ and $MC_1,\dots,MC_n$, independent of $z_{1i}^*,\dots,z_{ni}^*$.

Since $\varepsilon_{1i},\dots,\varepsilon_{ni}$ and $MC_1,\dots,MC_n$ are in turn i.i.d. draws from the respective marginal distribution independent of $z_{1i},z_{2i},\dots$, $X_{kn}$ are conditionally independent given $z_{1i}^*,z_{2i}^*,\dots$. We also define $s_n^2:=\sum_{k\neq i}\pi_{kn}(1-\pi_{kn})\exp\{2U_{ik}^*(L^*)\}$. We then consider convergence of $T_n:=\frac1{s_n}\sum_{k\neq i}X_{kn}$ to a standard normal random variable conditional on $z_{1i}^*,\dots,z_{ni}^*$. A sufficient condition for this is the Lyapounov condition
\begin{equation}\label{app_lyapounov_cond}\frac1{s_n^3}\sum_{k\neq i}\mathbb{E}[|X_{kn}|^3]\rightarrow0.\end{equation}

Noting that by Assumption <ref>, $0<\exp\{-\bar{U}\}\leq \exp\{2U_{ik}^*(L^*)\}\leq \exp\{\bar{U}\}<\infty$ for all $k=1,2,\dots$. Furthermore, the (centered) Bernoulli random variable $\mathbb{E}\left|\dum\{U_{ki}\geq MC_k\}-\pi_{kn}\right|^3=\pi_{kn}(1-\pi_{kn})^3 + (1-\pi_{kn})\pi_{kn}^3\leq 2\pi_{kn}$. Hence, we have that

\[\frac1{s_n^3}\sum_{k\neq i}\mathbb{E}[|X_{kn}|^3]\leq 2\exp\{6\bar{U}\}\frac{\sum_{k\neq i}\pi_{kn}}{\left(\sum_{k\neq i}\frac14\pi_{kn}\right)^{3/2}}=\frac14\exp\{6\bar{U}\}\left(\sum_{k\neq i}\pi_{kn}\right)^{-1/2}\]

for $n$ sufficiently large since for each $k$, $\pi_{kn}$ is bounded by a sequence that converges to zero.

Since $\sqrt{n}p_n$ is bounded from above and away from zero, $\sum_{k\neq i}\pi_{kn}$ diverges to infinity so that the Lyapounov condition (<ref>) holds. It then follows from Lyapounov's CLT for triangular arrays that conditional on $z_{1i}^*,z_{2i}^*,\dots$,

\[\frac{\sum_{k\neq i}\left(\dum\{U_{ki}\geq MC_k\}-\pi_{kn}\right)\exp\{U_{ik}^*(L^*)\}}{s_n}\stackrel{d}{\rightarrow}N(0,1)\]

In particular,
\[n^{-1/4}\sum_{k\neq i}\left(\dum\{U_{ki}\geq MC_k\}-\pi_{kn}\right)\exp\{U_{ik}^*(L^*)\}\stackrel{d}{\rightarrow}N(0,V)\]

conditional on $z_{1i}^*,z_{2i}^*,\dots$.

We can now combine these findings with a mean-value expansion to conclude that

\begin{eqnarray}
\nonumber \mathbb{E}[\sqrt{n}g(\hat{I}_{i,j_1,\dots,j_r;n})-g(\hat{\Eta}_n(x_i,s_i))] &=& \sqrt{n}g'(\hat{\Eta}_n(x_i,s_i))\mathbb{E}[\hat{I}_{i,j_1,\dots,j_r;n})-\hat{\Eta}_n(x_i,s_i)]\\
\nonumber&&+\frac12g''(\hat{\Eta}_n(x_i,s_i))\mathbb{E}[(\hat{I}_{i,j_1,\dots,j_r;n}-\hat{\Eta}_n(x_i,s_i))^2]+o(1)\\
\nonumber&=&B_{g,n} + o(1)
\end{eqnarray}

conditional on $\hat{\Eta}_n(x_i,s_i))$, which is asymptotically bounded given the bounds on the bias and variance of $\hat{I}_{i,j_1,\dots,j_r;n}-\hat{\Eta}_n(x_i,s_i)$ established before. Finally, the expectations of $\hat{m}_n(\theta;\eta)$ and $\hat{\Psi}_n(\theta;\eta)$ are Lipschitz in the asymptotic conditional link acceptance probabilities, establishing the conclusion of this Lemma

§.§ Proof of Theorem <ref>
Stacking the moments, we can consider linear functionals of the form

\[Z_n:=\mathbf{a}'\left(n^{1/2}\left[\begin{array}{c}\hat{m}_n(\theta;\eta) - m_0(\theta;\eta)\\\hat{\Psi}_n(\theta;\eta)-\Psi_0(\theta;\eta)  \end{array}\right] - B\right)\]

for an arbitrary conformable column vector $\mathbf{a}$, noting that $B$ is well-defined by Lemma <ref>.

Without loss of generality, we assume that the approximating sequence of arrays $\left(\tilde{Y}_{\mathbf{i}n}\right)_{\mathbf{i}}$ is defined on a common probability space using the construction in Lemma <ref>. Given the convergence rate for $\hat{m}_n$ established in Lemma <ref>, we can apply Lemma <ref> to establish convergence

\[(a'Va)^{-1/2}Z_n\stackrel{d}{\rightarrow}N(0,1)\]

for each $a$. By the Cramér-Wold device we then have that for any $\theta,\eta$,

\[\sqrt{n}V^{-1/2}\left(\begin{array}{c}\hat{m}_n(\theta;\eta) - m_0(\theta;\eta)\\ \hat{\Psi}_n(\theta;\eta)-\Psi_0(\theta;\eta)\end{array}\right) - B\stackrel{d}{\rightarrow} N(0,\mathbf{I})\]

mixing with respect to $\mathcal{F}$.

Since by Assumption <ref>, $\eta_0$ is a regular fixed point of $\Psi_0$ the conclusion follows from a mean-value expansion of $[\hat{m}_n(\theta;\hat{\eta}_n),\hat{\Psi}_n(\theta;\hat{\eta}_n)]'$ about $\eta_0$