Therefore as $N\to\infty$, from (5.1.2) we obtain

$\displaystyle\mathbf{Var}\left(\frac{\widetilde{A}_{2,N}}{N^{R-1/2}}\right)$
$\displaystyle\to\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left(\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\sum_{k=1}^{R}\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\prod_{i=2}^{R}\lambda_{u_{i}}\right)^{2}$
$\displaystyle\equiv\left(\sigma^{[1]}(H,p)\right)^{2}\in(0,\infty),\quad\text{(cf.
\eqref{res-main-1}).}$ (E.22)

The limiting variance $\left(\sigma^{[1]}(H,p)\right)^{2}\in(0,\infty)$, due
to the assumption (A.2) and (A.3).

We will now show that $N^{-R+1/2}\cdot\widetilde{A}_{2,N}$ satisfies the
Lindeberg condition for CLT. We use Lemma 5.5 for this purpose. Note,
$\widetilde{A}_{2,N}$ is a weighted sum of centered independent random
variables $(p^{-1}W_{N,s_{1}}-1)$, with non-random weights,
$a_{s_{1},N}=N^{-R+1/2}\cdot C_{N,s_{1}}$, for $s_{1}=1,\ldots,N$ (see
(5.1.2)). Recall that, $W_{1,N},\ldots,W_{N,N}$, are independent Bernoulli
random variables with $\mathbf{E}(W_{N,i})=p$ for all $i=1,\ldots,N$. From
(5.1.2) we know,
$\mathbf{Var}\left(N^{-(R-1/2)}\cdot\widetilde{A}_{2,N}\right)$ converges to a
positive finite limiting value ${\left(\sigma^{[1]}(H,p)\right)}^{2}$, and the
weights satisfy

$\displaystyle\max_{1\leq s_{1}\leq
N}|a_{s_{1},N}|=\frac{1}{N^{R-1/2}}\max_{1\leq s_{1}\leq
N}\sum_{k=1}^{R}\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\Psi_{H}\left(\bm{\Pi},\left(\bm{\alpha}_{N}\right)_{\mathbf{v}^{[k]}(\mathbf{s})}\right)$
$\displaystyle=\frac{1}{N^{R-1/2}}\max_{1\leq s_{1}\leq
N}\sum_{k=1}^{R}\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\Psi_{H}\left(\bm{\Pi},\left(\bm{\alpha}_{N}\right)_{\mathbf{v}^{[k]}(\mathbf{s})}\right)$
$\displaystyle=\frac{1}{N^{R-1/2}}\max_{1\leq s_{1}\leq
N}\sum_{\mathbf{u}\in[K]^{R}}\sum_{k=1}^{R}\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\mathbf{1}((\bm{\alpha}_{N})_{\mathbf{s}}=\mathbf{u})$
$\displaystyle=\frac{1}{N^{R-1/2}}\max_{1\leq s_{1}\leq
N}\sum_{u_{1}=1}^{K}\mathbf{1}(\alpha_{s_{1}}=u_{1})\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\sum_{k=1}^{R}\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)$
$\displaystyle\qquad\qquad\qquad\times\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\mathbf{1}\left({\left((\bm{\alpha}_{N})_{\mathbf{s}}\right)}^{(1)}={(\mathbf{u})}^{(1)}\right)$
$\displaystyle=\frac{K}{\sqrt{N}}\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\sum_{k=1}^{R}\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\left(\prod_{i=2}^{R}\lambda_{N,u_{i}}+O\left(\frac{1}{N}\right)\right)\rightarrow
0,$ (E.23)

due to assumptions (A.2) and (A.3). Thus, the two sufficient conditions
required for establishing the Lindeberg criteria (see Lemma 5.5) are
satisfied. This implies

$\displaystyle N^{-R+1/2}\cdot\widetilde{A}_{2,N}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left(\sigma^{[1]}(H,p)\right)^{2}\right).$ (E.24)

Now we consider the following expression,

$\displaystyle\mathbf{Var}\left(A_{2,N}-\widetilde{A}_{2,N}\right)=\mathbf{Var}\left[\sum_{\mathbf{s}\in{[N]}_{R}}\left(\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{i}}}{p}-1\right)\right)\cdot\Psi_{H}\left(\bm{\Pi},{(\bm{\alpha}_{N})_{\mathbf{s}}}\right)\right]$
$\displaystyle=\sum_{t=0}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\mathbf{cov}\left(\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{1,i}}}{p}-1\right),\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{2})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{2,i}}}{p}-1\right)\right)$
$\displaystyle\qquad\qquad\qquad\qquad\times\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}})\cdot\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}})$
$\displaystyle=\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\mathbf{cov}\left(\frac{1}{p^{R}}\prod_{i=1}^{R}W_{N,s_{1,i}}-\frac{1}{p}\sum_{i=1}^{R}W_{N,s_{1,i}},\frac{1}{p^{R}}\prod_{i=1}^{R}W_{N,s_{2,i}}-\frac{1}{p}\sum_{i=1}^{R}W_{N,s_{2,i}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\times\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}})\cdot\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}})$
$\displaystyle=\sum_{t=2}^{R}\left[\left(\frac{1}{p^{t}}-1\right)-t\left(\frac{1}{p}-1\right)\right]\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}})\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}})$
$\displaystyle\leq\sum_{t=2}^{R}\left[\left(\frac{1}{p^{t}}-1\right)-t\left(\frac{1}{p}-1\right)\right]\Bigg{(}N^{2R-t}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}\left(u_{j_{r}}=v_{k_{\xi(r)}}\right)$
$\displaystyle\qquad\qquad\times\left\\{\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\cdot\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O\left(\frac{1}{N}\right)\right\\}\Bigg{)}$
$\displaystyle=N^{2R-2}\sum_{t=2}^{R}\left[\left(\frac{1}{p^{t}}-1\right)-t\left(\frac{1}{p}-1\right)\right]\left(\frac{1}{N^{t-2}}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}\left(u_{j_{r}}=v_{k_{\xi(r)}}\right)\right.$
$\displaystyle\qquad\qquad\times\left.\left\\{\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\cdot\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O\left(\frac{1}{N}\right)\right\\}\right),\qquad\text{(similar
to \eqref{cov-val}).}$ (E.25)

In the above expression (5.1.2), when $(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{0}$, then $A(\mathbf{s}_{1})\cap A(\mathbf{s}_{2})=\emptyset$, the
covariance becomes zero due to independence of the sets
$\\{W_{N,s_{1,1}},\ldots,W_{N,s_{1,R}}\\}$ and
$\\{W_{N,s_{2,1}},\ldots,W_{N,s_{2,R}}\\}$. When
$(\mathbf{s}_{1},\mathbf{s}_{2})\in E_{1}$, without loss of generality one can
assume that, $A(\mathbf{s}_{1})\cap A(\mathbf{s}_{2})=\\{s_{1,1}\\}$ that is,
$\mathbf{s}_{1}=(s_{1,1},s_{1,2},\ldots,s_{1,R})^{T}$ and
$\mathbf{s}_{2}=(s_{1,1},s_{2,2},\ldots,s_{2,R})^{T}$. Then, in the expression
(5.1.2), when $t=1$,

$\displaystyle\mathbf{cov}\left(\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{1,i}}}{p}-1\right),\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{2})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{2,i}}}{p}-1\right)\right)$
$\displaystyle=\mathbf{E}\left(\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{1,i}}}{p}-1\right)\right]\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{2})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{2,i}}}{p}-1\right)\right]\right)$
$\displaystyle=\mathbf{E}\left(\mathbf{E}\left(\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{1,i}}}{p}-1\right)\right]\right.\right.$
$\displaystyle\qquad\qquad\left.\left.\times\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{2})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{2,i}}}{p}-1\right)\right]\mid(\mathbf{W}_{N})_{\mathbf{s}_{2}}\right)\right)$
$\displaystyle=\mathbf{E}\left(\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{1,i}}}{p}-1\right)\right]\right.$
$\displaystyle\qquad\qquad\left.\times\mathbf{E}\left(\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{2})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{2,i}}}{p}-1\right)\right]\mid
W_{N,s_{1,1}}\right)\right)$
$\displaystyle=\mathbf{E}\left(\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{1,i}}}{p}-1\right)\right]\right.$
$\displaystyle\qquad\qquad\left.\times\left(\mathbf{E}\left(\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)\mid
W_{N,s_{1,1}}\right)-\left(\frac{W_{N,s_{1,1}}}{p}-1\right)\right)\right)$
$\displaystyle=\mathbf{E}\left(\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{1,i}}}{p}-1\right)\right]\left(\left(\frac{W_{N,s_{1,1}}}{p}-1\right)-\left(\frac{W_{N,s_{1,1}}}{p}-1\right)\right)\right)$
$\displaystyle=\mathbf{E}\left(\left[\left(\frac{\Psi_{H}(\mathbf{U}_{N}^{[1]},\mathbf{s}_{1})}{p^{R}}-1\right)-\sum_{i=1}^{R}\left(\frac{W_{N,s_{1,i}}}{p}-1\right)\right]\cdot
0\right)=0.$

Therefore, from (5.1.2) and (5.1.2) we get,

$\displaystyle\frac{\mathbf{Var}\left(A_{2,N}-\widetilde{A}_{2,N}\right)}{\mathbf{Var}\left(\widetilde{A}_{2,N}\right)}=\frac{O\left(N^{2R-2}\right)}{N^{2R-1}}\to
0,\quad\text{as $N\to\infty$}.$ (E.26)

The above step in (E.26), the asymptotic normality result in (E.24) and Lemma
5.4 can now be combined to claim that,

$\displaystyle\frac{A_{2,N}}{\sqrt{\mathbf{Var}\left(A_{2,N}\right)}}\to
N(0,1).$

Also note that the limiting relation in (E.26) and the arguments used in the
proof of Lemma 5.4 show that,
$\mathbf{Var}\left(A_{2,N}\right)/\mathbf{Var}\left(\widetilde{A}_{2,N}\right)\rightarrow
1$. Finally, using (5.1.2) and Slutsky’s theorem, we obtain

$\displaystyle\frac{A_{2,N}}{N^{R-1/2}}=\frac{A_{2,N}}{\sqrt{\mathbf{Var}(A_{2,N})}}\cdot\sqrt{\frac{\mathbf{Var}(\widetilde{A}_{2,N})}{N^{2R-1}}}\sqrt{\frac{\mathbf{Var}({A}_{2,N})}{\mathbf{Var}(\widetilde{A}_{2,N})}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left(\sigma^{[1]}(H,p)\right)^{2}\right).$ (E.27)

Thus, combining (E.18), (E.27) and using Slutsky’s theorem, we can claim that

$\displaystyle
N^{-R+1/2}\left(\frac{\widehat{S}^{[1]}_{N}(H)}{p^{R}}-S_{N}(H)\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left(\sigma^{[1]}(H,p)\right)^{2}\right),$

where, $\left(\sigma^{[1]}(H,p)\right)^{2}$ is defined in (3.10). ∎

####  5.1.3 Proof of Theorem 3.3: CLT in the ego-centric subgraph case

###### Proof of Theorem 3.3.

Note, $H$ is an undirected, simple and connected graph with $V(H)=[R]$. Recall
that $|E(H)|=T$, and $E(H)=\\{C_{1},\ldots,C_{T}\\}$ with
$C_{t}=\\{i_{t},j_{t}\\}$, for each $t=1,\ldots,T$ (see (3.11)). As $H$ is
assumed to be connected, there does not exist any isolated vertex in $H$.
Recall the definitions of $\mathcal{N}(r)$, $\mathcal{A}(r)$ and
$\mathcal{A}^{c}(r)$ provided in (3.12). Also recall that
$\widetilde{W}_{1},\ldots,\widetilde{W}_{R}$, are i.i.d. Bernoulli $(p)$
random variables (see (3.12)) defined on a common probability space, which are
independent of $\mathbf{Y}_{N}$ and $\mathbf{W}_{N}$. Based on the above, we
define the following random variables,

$\displaystyle\widetilde{Z}_{1,r}$
$\displaystyle\equiv\prod_{v\in\mathcal{N}(r)}\widetilde{W}_{v},\quad\widetilde{V}_{t}\equiv\max\left\\{\widetilde{W}_{i},\widetilde{W}_{j}:i,j\in
C_{t}\right\\},\quad\text{and}\quad\widetilde{Z}_{2,r}=\prod_{t\in\mathcal{A}^{c}(r)}\widetilde{V}_{t},$
(E.28)

for each $r\in[R]$, and $t\in[T]$. Note that, by definition of
$\mathcal{A}^{c}(r)$, if $t\in\mathcal{A}^{c}(r)$, then $r\not\in C_{t}$,
which implies that $\widetilde{W}_{r}$ and $V_{t}$ are independent. This
implies, $\widetilde{W}_{r}$ and $\widetilde{Z}_{2,r}$ will be independent.
Also, as $r\not\in\mathcal{N}(r)$, hence $\widetilde{W}_{r}$ is independent of
$\widetilde{Z}_{1,r}$. Also define the following expected values

$\displaystyle g_{1,r}(p)$
$\displaystyle=\mathbf{E}(\widetilde{Z}_{1,r}\widetilde{Z}_{2,r})\quad\text{and}\quad
g_{2,r}(p)=\mathbf{E}(\widetilde{Z}_{2,r}).$ (E.29)

Note that the expectations defined above always exist and they depend on the
edge structure of the graph $H$, but as $H$ is fixed, we suppress this
dependence. Fix any $r\in[R]$. Using conditional expectations we can write
(see (3.12))

$\displaystyle f(H,p)$ $\displaystyle=\mathbf{E}\left[\prod_{\\{i,j\\}\in
E(H)}\max\left\\{\widetilde{W}_{i},\widetilde{W}_{j}\right\\}\right]=\mathbf{E}\left[\prod_{t=1}^{T}\widetilde{V}_{t}\right]=\mathbf{E}\left[\prod_{t\in\mathcal{A}(r)}\widetilde{V}_{t}\cdot\prod_{t\in\mathcal{A}^{c}(r)}\widetilde{V}_{t}\right]$
$\displaystyle=\mathbf{E}\left[\left\\{\widetilde{W}_{r}+(1-\widetilde{W}_{r})\cdot\prod_{v\in\mathcal{N}(r)}\widetilde{W}_{v}\right\\}\cdot\left\\{\prod_{t\in\mathcal{A}^{c}(r)}\widetilde{V}_{t}\right\\}\right]$
$\displaystyle=\mathbf{E}\left[\widetilde{W}_{r}\cdot\widetilde{Z}_{2,r}+(1-\widetilde{W}_{r})\cdot\widetilde{Z}_{1,r}\cdot\widetilde{Z}_{2,r}\right]$
$\displaystyle=\mathbf{E}\left[\widetilde{W}_{r}\cdot
g_{2,r}(p)+(1-\widetilde{W}_{r})\cdot g_{1,r}(p)\right]=p\cdot
g_{2,r}(p)+(1-p)\cdot g_{1,r}(p),\quad\text{(cf. \eqref{g12-def})}$
$\displaystyle=p\cdot\\{g_{2,r}(p)-g_{1,r}(p)\\}+g_{1,r}(p),\quad\text{for any
$r\in[R]$.}$ (E.30)

Now consider the sequence of i.i.d. Bernoulli $(p)$ node sampling indicator
random variables $W_{N,1},\ldots,W_{N,N}$ (cf. (2.5)), and a vector
$\mathbf{s}=(s_{1},\ldots,s_{R})^{T}\in{[N]}_{R}$. Similar to (E.28), we
define the random variables

$\displaystyle Z_{1,r}(\mathbf{s})=\prod_{v\in\mathcal{N}(r)}W_{N,s_{v}},\quad
V_{t}(\mathbf{s})=\max\left\\{W_{N,s_{i}},W_{N,s_{j}}:i,j\in
C_{t}\right\\}\quad\text{and}\quad
Z_{2,r}(\mathbf{s})=\prod_{t\in\mathcal{A}^{c}(r)}V_{t}(\mathbf{s}),$ (E.31)

for each $r\in[R]$, for each $t\in[T]$, and each $\mathbf{s}\in{[N]}_{R}$.
Notice that as all components $s_{1},\ldots,s_{R}$ are distinct and as
because,

$\displaystyle{\left(W_{N,s_{1}},\ldots,W_{N,s_{R}}\right)}^{T}\stackrel{{\scriptstyle
d}}{{=}}{\left(\widetilde{W}_{1},\ldots,\widetilde{W}_{R}\right)}^{T},\quad\text{for
all $\mathbf{s}\in{[N]}_{R}$,}$ (E.32)

it implies
$\mathbf{E}\left(Z_{1,r}(\mathbf{s})Z_{2,r}(\mathbf{s})\right)=g_{1,r}(p)$ and
$\mathbf{E}\left(Z_{2,r}(\mathbf{s})\right)=g_{2,r}(p)$, for all
$\mathbf{s}\in{[N]}_{R}$. In the ego-centric sampling scheme, the pivotal
quantity written in (3.13), can be rewritten in the following way,

$\displaystyle\widehat{S}^{[2]}_{N}(H)-f(H,p)\cdot
S_{N}(H)=\sum_{\mathbf{s}\in{[N]}_{R}}\left(\Psi_{H}\left(\mathbf{U}^{[2]}_{N},\mathbf{s}\right)-f(H,p)\right)\cdot\left(\Psi\left(\mathbf{Y},\mathbf{s}\right)-\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})\right),$

where, $f(H,p)$ defined in (3.12). Also from (E.12),

$\displaystyle\Psi_{H}\left(\mathbf{U}^{[2]}_{N},\mathbf{s}\right)=\prod_{\\{i,j\\}\in
E(H)}h_{2}(W_{N,s_{i}},W_{N,s_{j}})=\prod_{\\{i,j\\}\in
E(H)}\max\left(W_{N,s_{i}},W_{N,s_{j}}\right).$

We split the pivotal quantity defined in Theorem 3.3 into the two terms
$B_{1,N}$ and $B_{2,N}$ where,

$\displaystyle B_{1,N}$
$\displaystyle=\sum_{\mathbf{s}\in{[N]}_{R}}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}\right)-f(H,p)\right)\cdot\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s})-\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})\right),\quad\text{and}$
(E.33) $\displaystyle B_{2,N}$
$\displaystyle=\sum_{\mathbf{s}\in{[N]}_{R}}\left(\Psi_{H}\left(\mathbf{U}^{[2]}_{N},\mathbf{s}\right)-f(H,p)\right)\cdot\Psi_{H}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}).$

Now, $\mathbf{E}(B_{1,N})=0$ and $\mathbf{Var}\left(B_{1,N}\right)$ can be
upper bounded in the following way,

$\displaystyle\mathbf{Var}(B_{1,N})=\left|\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right)\cdot\mathbf{cov}\left(\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)\right)\right|$
$\displaystyle\leq\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\left|\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right)\right|\left|\mathbf{cov}\left(\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)\right)\right|$
$\displaystyle\leq\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\left[\mathbf{Var}\left(\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right)\right)\mathbf{Var}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right)\right)\right]^{1/2}\left|\mathbf{cov}\left(\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)\right)\right|$
$\displaystyle=\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}f(H,p)(1-f(H,p))\left|\mathbf{cov}\left(\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)\right)\right|$
$\displaystyle=f(H,p)(1-f(H,p))\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\left|\mathbf{cov}\left(\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)\right)\right|$
$\displaystyle=f(H,p)(1-f(H,p))\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\mathbf{cov}\left(\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)\right),$
(E.34)

where, $E_{t}$ defined in (E.60). In the last line of (5.1.3), the absolute
value sign is removed since (E.64) in Lemma 5.1 shows that this covariance is
non-negative. Hence, using Lemma 5.1, we get

$\displaystyle\mathbf{Var}\left(\frac{B_{1,N}}{N^{R-1/2}}\right)$
$\displaystyle\leq
f(H,p)(1-f(H,p))\sum_{t=2}^{R}\left\\{\frac{1}{N^{2R-1}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\mathbf{cov}\left(\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)\right)\right\\}\to
0.$

Therefore,

$\displaystyle B_{1,N}=o_{p}\left(N^{R-1/2}\right)$ (E.35)

For $\mathbf{s}\in{[N]}_{R}$, write,
$\omega_{N}(\mathbf{s})=\Psi_{H}(\bm{\Pi},\mathbf{s})$. Then the Hájek
projection of $B_{2,N}$ will be

$\displaystyle\widetilde{B}_{2,N}=\sum_{l=1}^{N}\mathbf{E}\left(B_{2,N}\mid
W_{N,l}\right)=\sum_{l=1}^{N}\sum_{\mathbf{s}\in{[N]}_{R}}\sum_{r=1}^{R}\omega_{N}(\mathbf{s})\cdot\mathbf{1}(s_{r}=l)\cdot\mathbf{E}\left[\prod_{t=1}^{T}V_{t}(\mathbf{s})-f(H,p)\mid
W_{N,l}\right]$
$\displaystyle=\sum_{l=1}^{N}\sum_{\mathbf{s}\in{[N]}_{R}}\sum_{r=1}^{R}\omega_{N}(\mathbf{s})\cdot\mathbf{1}(s_{r}=l)\cdot\mathbf{E}\left[\prod_{t\in\mathcal{A}(r)}V_{t}(\mathbf{s})\cdot\prod_{t\in\mathcal{A}^{c}(r)}V_{t}(\mathbf{s})-f(H,p)\mid
W_{N,l}\right]$
$\displaystyle=\sum_{l=1}^{N}\sum_{\mathbf{s}\in{[N]}_{R}}\sum_{r=1}^{R}\omega_{N}(\mathbf{s})\mathbf{1}(s_{r}=l)\cdot\left\\{\mathbf{E}\left[\left\\{W_{N,l}+(1-W_{N,l})\cdot
Z_{1,r}(\mathbf{s})\right\\}\cdot Z_{2,r}(\mathbf{s})\mid
W_{N,l}\right]-f(H,p)\right\\}$
$\displaystyle=\sum_{l=1}^{N}\sum_{\mathbf{s}\in{[N]}_{R}}\sum_{r=1}^{R}\omega_{N}(\mathbf{s})\mathbf{1}(s_{r}=l)\cdot\left\\{W_{N,l}\cdot
g_{2,r}(p)+(1-W_{N,l})\cdot g_{1,r}(p)-p\cdot g_{2,r}(p)-(1-p)\cdot
g_{1,r}(p)\right\\}$
$\displaystyle=\sum_{l=1}^{N}(W_{l,N}-p)\cdot\sum_{\mathbf{s}\in{[N]}_{R}}\sum_{r=1}^{R}\omega_{N}(\mathbf{s})\mathbf{1}(s_{r}=l)\cdot\\{g_{2,r}(p)-g_{1,r}(p)\\}$
$\displaystyle=\sum_{l=1}^{N}(W_{N,l}-p)\cdot\sum_{r=1}^{R}\\{g_{2,r}(p)-g_{1,r}(p)\\}\sum_{{(\mathbf{s})}^{(r)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\omega_{N}\left(s_{1},\ldots,s_{r-1},l,s_{r+1},\ldots,s_{R}\right)$
$\displaystyle=\sum_{l=1}^{N}(W_{N,l}-p)\cdot\sum_{r=1}^{R}\delta_{r}(H,p)\sum_{{(\mathbf{s})}^{(r)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\omega_{N}\left(s_{1},\ldots,s_{r-1},l,s_{r+1},\ldots,s_{R}\right)$
$\displaystyle=\sum_{l=1}^{N}(W_{N,l}-p)\sum_{r=1}^{R}\delta_{r}(H,p)\sum_{{(\mathbf{s})}^{(r)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\omega_{N}\left(s_{1},\ldots,s_{r-1},l,s_{r+1},\ldots,s_{R}\right)$
$\displaystyle=\sum_{l=1}^{N}(W_{N,l}-p)\Bigg{[}\delta_{1}(H,p)\sum_{{(\mathbf{s})}^{(1)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\omega_{N}\left(l,s_{2},\ldots,s_{R}\right)+\ldots$
$\displaystyle\qquad\qquad\qquad\ldots+\delta_{R}(H,p)\sum_{{(\mathbf{s})}^{(R)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\omega_{N}\left(s_{1},s_{2},\ldots,s_{R-1},l\right)\Bigg{]}$
$\displaystyle=\sum_{s_{1}=1}^{N}(W_{N,s_{1}}-p)\cdot\delta_{1}(H,p)\sum_{{(\mathbf{s})}^{(1)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\omega_{N}\left(s_{1},s_{2},\ldots,s_{R}\right)+\cdots$
$\displaystyle\qquad\qquad\qquad\cdots+\sum_{s_{R}=1}^{N}(W_{N,s_{R}}-p)\cdot\delta_{R}(H,p)\sum_{{(\mathbf{s})}^{(R)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\omega_{N}\left(s_{1},s_{2},\ldots,s_{R-1},s_{R}\right)$
$\displaystyle=\sum_{s_{1}=1}^{N}(W_{N,s_{1}}-p)\cdot\delta_{1}(H,p)\sum_{{(\mathbf{s})}^{(1)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right)+\cdots$
$\displaystyle\qquad\qquad\qquad\cdots+\sum_{s_{R}=1}^{N}(W_{N,s_{R}}-p)\cdot\delta_{R}(H,p)\sum_{{(\mathbf{s})}^{(R)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right)$
$\displaystyle=\sum_{s_{1}=1}^{N}(W_{N,s_{1}}-p)\cdot\delta_{1}(H,p)\sum_{{(\mathbf{s})}^{(1)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{v}^{[1]}(\mathbf{s})}\right)+\cdots$
$\displaystyle\qquad\qquad\qquad\cdots+\sum_{s_{1}=1}^{N}(W_{N,s_{R}}-p)\cdot\delta_{R}(H,p)\sum_{{(\mathbf{s})}^{(1)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{v}^{[R]}(\mathbf{s})}\right)$
$\displaystyle=\sum_{s_{1}=1}^{N}(W_{N,s_{1}}-p)\sum_{r=1}^{R}\delta_{r}(H,p)\sum_{{(\mathbf{s})}^{(1)}\in{[{[N]}\setminus\\{l\\}]}_{(R-1)}}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{v}^{[r]}(\mathbf{s})}\right)$
$\displaystyle=\sum_{s_{1}=1}^{N}(W_{N,s_{1}}-p)\cdot
C_{N,s_{1}}(p),\quad\text{ where,
$C_{N,s_{1}}(p)=\sum_{k=1}^{R}\delta_{k}(H,p)\sum_{{(\mathbf{s})}^{(1)}\in[[N]\setminus\\{s_{1}\\}]_{R-1}}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{v}^{[k]}(\mathbf{s})}\right)$},$
(E.36)

where, $\delta_{r}(H,p)$ defined in (3.12). Now, as $C_{N,s_{1}}(p)$ is a non
random weights and depending on $p$ and from (5.1.3) we can write,

$\displaystyle\mathbf{Var}(\widetilde{B}_{2,N})=p(1-p)\sum_{s_{1}=1}^{N}\left(C_{N,s_{1}}(p)\right)^{2}$
$\displaystyle=p(1-p)\sum_{s_{1}=1}^{N}\left(\sum_{r=1}^{R}\delta_{r}(H,p)\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{v}^{[r]}(\mathbf{s})}\right)\right)^{2}$
$\displaystyle=p(1-p\sum_{u_{1}=1}^{K}\sum_{s_{1}=1}^{N}\mathbf{1}(\alpha_{s_{1}}=u_{1})\left(\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\sum_{r=1}^{R}\delta_{r}(H,p)\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[r]}(\mathbf{u})\right)\right.$
$\displaystyle\qquad\qquad\left.\times\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\mathbf{1}\left({\left((\bm{\alpha}_{N})_{\mathbf{s}}\right)}^{(1)}={(\mathbf{u})}^{(1)}\right)\right)^{2}$
$\displaystyle=N^{2R-2}\left(p(1-p)\right)\sum_{u_{1}=1}^{K}\sum_{s_{1}=1}^{N}\mathbf{1}(\alpha_{s_{1}}=u_{1})\left(\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\sum_{r=1}^{R}\delta_{r}(H,p)\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[r]}(\mathbf{u})\right)\left(\prod_{i=2}^{R}\lambda_{N,u_{i}}+O\left(\frac{1}{N}\right)\right)\right)^{2}$
$\displaystyle=N^{2R-1}\left(\left(p(1-p)\right)\sum_{u_{1}=1}^{K}\lambda_{N,u_{1}}\left(\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\sum_{r=1}^{R}\delta_{r}(H,p)\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[r]}(\mathbf{u})\right)\prod_{i=2}^{R}\lambda_{N,u_{i}}\right)^{2}+O\left(\frac{1}{N^{2}}\right)\right).$
(E.37)

Therefore, as $N\to\infty$, the expression in (5.1.3) has the following
expression,

$\displaystyle\mathbf{Var}\left(\frac{\widetilde{B}_{2,N}}{N^{R-1/2}}\right)$
$\displaystyle\to
p(1-p)\sum_{u_{1}=1}^{K}\lambda_{N,u_{1}}\left(\sum_{r=1}^{R}\delta_{r}(H,p)\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[r]}(\mathbf{u})\right)\prod_{i=2}^{R}\lambda_{u_{i}}\right)^{2}$
$\displaystyle\equiv(f(H,p))^{2}\left(\sigma^{[2]}(H,p)\right)^{2}.$ (E.38)

Once again, due to the assumption (A.2) and (A.3),
$\sigma^{[2]}(H,p)\in(0,\infty)$, where $\sigma^{[2]}(H,p)$ is defined in
(3.13).

Now we will show that $N^{-R+1/2}\cdot\widetilde{B}_{2,N}$ satisfies Lindeberg
criterion for CLT using the Lemma 5.5 for this purpose. In this case
$\widetilde{B}_{2,N}$ is a weighted sum of centered independent random
variables $(W_{N,s_{1}}-p)$, with non-random weights,
$a_{s_{1},N}=N^{-R+1/2}\cdot C_{N,s_{1}}(p)$, for $s_{1}=1,\ldots,N$ (cf.
(5.1.3)). Now, similar to the proof of Theorem 3.1, we know
$\mathbf{Var}\left(N^{-(R-1/2)}\cdot\widetilde{B}_{2,N}\right)$ converge to a
positive finite limiting value
$\left[f(H,p)\cdot\sigma^{[2]}(H,p)\right]^{2}$, and the weights satisfy

$\displaystyle\max_{1\leq s_{1}\leq
N}|a_{s_{1},N}|=\frac{1}{N^{R-1/2}}\max_{1\leq s_{1}\leq
N}\sum_{r=1}^{R}\delta_{r}(H,p)\cdot\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\Psi_{H}\left(\bm{\Pi},\left(\bm{\alpha}_{N}\right)_{\mathbf{v}^{[r]}(\mathbf{s})}\right)$
$\displaystyle=\frac{K}{\sqrt{N}}\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\sum_{r=1}^{R}\delta_{r}(H,p)\cdot\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[r]}(\mathbf{u})\right)\left(\prod_{i=2}^{R}\lambda_{N,u_{i}}+O\left(\frac{1}{N}\right)\right)\rightarrow
0,$ (E.39)

due to assumption (A.2) and (A.3). Thus two sufficient conditions required for
establishing the Lindeberg criteria (see Lemma 5.5) are satisfied. This
implies

$\displaystyle N^{-R+1/2}\cdot\widetilde{B}_{2,N}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left[f(H,p)\cdot\sigma^{[2]}(H,p)\right]^{2}\right).$
(E.40)

Now we consider the following expression,

$\displaystyle\left|\mathbf{Var}\left(B_{2,N}-\widetilde{B}_{2,N}\right)\right|$
$\displaystyle=\left|\mathbf{Var}\left(\sum_{\mathbf{s}\in{[N]}_{R}}\left[\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}\right)-f(H,p)\right)-\sum_{k=1}^{R}\delta_{k}(H,p)\cdot(W_{N,s_{k}}-p)\right]\cdot\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right)\right)\right|$
$\displaystyle\leq\sum_{t=0}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\Bigg{|}\mathbf{cov}\Bigg{(}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right)-f(H,p)\right)-\sum_{k=1}^{R}\delta_{k}(H,p)\cdot(W_{N,s_{1,k}}-p),$
$\displaystyle\qquad\qquad\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)-f(H,p)\right)-\sum_{k=1}^{R}\delta_{k}(H,p)\cdot(W_{N,s_{2,k}}-p)\Bigg{)}\Bigg{|}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}}\right)\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}}\right)$
$\displaystyle=\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\Bigg{|}\mathbf{cov}\Bigg{(}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right)-f(H,p)\right)-\sum_{k=1}^{R}\delta_{k}(H,p)\cdot(W_{N,s_{1,k}}-p),$
$\displaystyle\qquad\qquad\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)-f(H,p)\right)-\sum_{k=1}^{R}\delta_{k}(H,p)\cdot(W_{N,s_{2,k}}-p)\Bigg{)}\Bigg{|}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}}\right)\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}}\right).$
(E.41)

If $(\mathbf{s}_{1},\mathbf{s}_{2})\in E_{0}$, the covariance term in the
expression (5.1.3) is zero, since the set
$\\{W_{N,s_{1,1}},\ldots,W_{N,s_{1,R}}\\}$ and
$\\{W_{N,s_{2,1}},\ldots,W_{N,s_{2,R}}\\}$ are independent. If
$(\mathbf{s}_{1},\mathbf{s}_{2})\in E_{1}$ then, without loss of generality
assume that the $j_{1}$-th position of $\mathbf{s}_{1}$ and $k_{1}$-th
position of $\mathbf{s}_{2}$ have the same component, i.e.,
$s_{1,j_{1}}=s_{2,k_{1}}$. First note that, using (E.31), for $k\in[R]$,

$\displaystyle\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}\right),W_{N,s_{k}}\right)=\mathbf{E}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}\right)\cdot
W_{N,s_{k}}\right)-p\cdot f(H,p)$
$\displaystyle=p\mathbf{E}\left(Z_{2,k}(\mathbf{s})\right)+p(1-p)\mathbf{E}\left(Z_{2,k}(\mathbf{s})Z_{1,k}(\mathbf{s})\right)-p\left(p\cdot
E(Z_{2,k}(\mathbf{s}))+(1-p)\mathbf{E}(Z_{2,k}(\mathbf{s})Z_{1,k}(\mathbf{s}))\right)$
$\displaystyle=p(1-p)\cdot\mathbf{E}\left(Z_{2,k}(\mathbf{s})\right)=p(1-p)\cdot
g_{2,k}(p),$ (E.42)

and,

$\displaystyle\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right)$
$\displaystyle=\mathbf{E}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right)\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right)-\mathbf{E}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right)\right)\mathbf{E}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right)$
$\displaystyle=\mathbf{E}\left((W_{N,s_{1,j_{1}}}+(1-W_{N,s_{1,j_{1}}})Z_{1,j_{1}}(\mathbf{s}))(W_{N,s_{2,k_{1}}}+(1-W_{N,s_{2,k_{1}}})Z_{1,k_{1}}(\mathbf{s}))Z_{2,j_{1}}(\mathbf{s})Z_{2,k_{1}}(\mathbf{s})\right)$
$\displaystyle\quad{}-\left(p\mathbf{E}(Z_{2,j_{1}}(\mathbf{s}))+(1-p)\mathbf{E}(Z_{2,j_{1}}(\mathbf{s})Z_{1,j_{1}}(\mathbf{s}))\right)\left(p\mathbf{E}(Z_{2,k_{1}}(\mathbf{s}))+(1-p)\mathbf{E}(Z_{2,k_{1}}(\mathbf{s})Z_{1,k_{1}}(\mathbf{s}))\right)$
$\displaystyle=\bigg{[}p\mathbf{E}\left(Z_{2,j_{1}}(\mathbf{s})\right)\mathbf{E}\left(Z_{2,k_{1}}(\mathbf{s})\right)+p(1-p)\mathbf{E}\left(Z_{2,j_{1}}(\mathbf{s})\right)\mathbf{E}\left(Z_{2,k_{1}}(\mathbf{s})Z_{1,k_{1}}(\mathbf{s})\right)$
$\displaystyle\qquad{}+p(1-p)\mathbf{E}\left(Z_{2,k_{1}}(\mathbf{s})\right)\mathbf{E}\left(Z_{2,j_{1}}(\mathbf{s})Z_{1,j_{1}}(\mathbf{s})\right)+(1-p)\mathbf{E}\left(Z_{2,j_{1}}(\mathbf{s})Z_{1,j_{1}}(\mathbf{s})\right)\mathbf{E}\left(Z_{2,k_{1}}(\mathbf{s})Z_{1,k_{1}}(\mathbf{s})\right)\bigg{]}$
$\displaystyle\qquad{}-\left(p\mathbf{E}(Z_{2,j_{1}}(\mathbf{s}))+(1-p)\mathbf{E}(Z_{2,j_{1}}(\mathbf{s})Z_{1,j_{1}}(\mathbf{s}))\right)\left(p\mathbf{E}(Z_{2,k_{1}}(\mathbf{s}))+(1-p)\mathbf{E}(Z_{2,k_{1}}(\mathbf{s})Z_{1,k_{1}}(\mathbf{s}))\right)$
$\displaystyle=p(1-p)\left[\mathbf{E}\left(Z_{2,j_{1}}(\mathbf{s})\right)\mathbf{E}\left(Z_{2,k_{1}}(\mathbf{s})\right)+\mathbf{E}\left(Z_{2,j_{1}}(\mathbf{s})Z_{1,j_{1}}(\mathbf{s})\right)\mathbf{E}\left(Z_{2,k_{1}}(\mathbf{s})Z_{1,k_{1}}(\mathbf{s})\right)\right]$
$\displaystyle=p(1-p)\left[g_{2,j_{1}}(p)\cdot
g_{2,k_{1}}(p)+g_{1,j_{1}}(p)\cdot g_{1,k_{1}}(p)\right].$ (E.43)

Hence, using (5.1.3) and (5.1.3), we can claim that the covariance term
involved in the expression (5.1.3) is equal to zero for
$(\mathbf{s}_{1},\mathbf{s}_{2})\in E_{1}$, in the following way,

$\displaystyle\mathbf{cov}\left(\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right)-f(H,p)\right)-\sum_{i=1}^{R}\delta_{i}(H,p)\left(W_{N,s_{1,i}}-p\right),\right.$
$\displaystyle\qquad\qquad\left.\left(\Psi_{H}(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2})-f(H,p)\right)-\sum_{i=1}^{R}\delta_{i}(H,p)\left(W_{N,s_{2,i}}-p\right)\right)$
$\displaystyle=\mathbf{cov}\left(\Psi_{H}(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1})-\sum_{i=1}^{R}\delta_{i}(H,p)\cdot
W_{N,s_{1,i}},\Psi_{H}(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2})-\sum_{i=1}^{R}\delta_{i}(H,p)\cdot
W_{N,s_{2,i}}\right)$
$\displaystyle=\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right),\Psi_{H}(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2})\right)-\delta_{k_{1}}(H,p)\cdot\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right),W_{N,s_{2,k_{1}}}\right)$
$\displaystyle\qquad\qquad-\delta_{j_{1}}(H,p)\cdot\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right),W_{N,s_{1,j_{1}}}\right)+p(1-p)\cdot\delta_{j_{1}}(H,p)\cdot\delta_{k_{1}}(H,p)$
$\displaystyle=\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right),\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right)-\delta_{k_{1}}(H,p)\cdot\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right),W_{N,s_{2,k_{1}}}\right)$
$\displaystyle\qquad\qquad-\delta_{j_{1}}(H,p)\cdot\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right),W_{N,s_{1,j_{1}}}\right)+p(1-p)\cdot\delta_{j_{1}}(H,p)\cdot\delta_{k_{1}}(H,p)$
$\displaystyle=p(1-p)\left[g_{2,j_{1}}(p)\cdot
g_{2,k_{1}}(p)+g_{1,j_{1}}(p)\cdot
g_{1,k_{1}}(p)\right]-p(1-p)\cdot\delta_{k_{1}}(H,p)\cdot g_{2,j_{1}}(p)$
$\displaystyle\qquad{}-p(1-p)\cdot\delta_{j_{1}}(H,p)\cdot
g_{2,k_{1}}(p)+p(1-p)\cdot\delta_{j_{1}}(H,p)\cdot\delta_{k_{1}}(H,p)$
$\displaystyle=0.$

Also note that, due to i.i.d nature of Bernoulli indicators $\mathbf{W}_{N}$,
for $\mathbf{s}_{1},\mathbf{s}_{2}\in{[N]}_{R}$,

$\mathbf{Var}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right)\right)=\mathbf{Var}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right).$

Now using Cauchy-Schwartz inequality over (5.1.3), and (5.1.3),we get the
following expression,

$\displaystyle\text{l.h.s. of
\eqref{V3-ego}}\leq\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\Bigg{[}\mathbf{Var}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{1}\right)-\sum_{k=1}^{R}\delta_{k}(H,p)\cdot
W_{N,s_{1,k}}\right)$
$\displaystyle\qquad\qquad\times\mathbf{Var}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)-\sum_{k=1}^{R}\delta_{k}(H,p)\cdot
W_{N,s_{2,k}}\right)\Bigg{]}^{1/2}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}}\right)\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}}\right)$
$\displaystyle=\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\left\\{\mathbf{Var}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right)\right)+\sum_{k=1}^{R}\delta_{k}(H,p)^{2}\mathbf{Var}(W_{N,s_{1,k}})\right.$
$\displaystyle\qquad\left.-2\sum_{k=1}^{R}\delta_{k}(H,p)\mathbf{cov}\left(\Psi_{H}\left(\mathbf{U}_{N}^{[2]},\mathbf{s}_{2}\right),W_{N,s_{2,k}}\right)\right\\}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}}\right)\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}}\right)$
$\displaystyle=\sum_{t=2}^{R}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\left\\{f(H,p)(1-f(H,p))+p(1-p)\sum_{k=1}^{R}\delta_{k}(H,p)^{2}-2p(1-p)\sum_{k=1}^{R}\delta_{k}(H,p)\cdot
g_{2,k}(p)\right\\}$
$\displaystyle\qquad\qquad\times\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}}\right)\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}}\right)$
$\displaystyle=\sum_{t=2}^{R}\left\\{f(H,p)(1-f(H,p))+p(1-p)\sum_{k=1}^{R}\delta_{k}(H,p)^{2}-2p(1-p)\sum_{k=1}^{R}\delta_{k}(H,p)\cdot
g_{2,k}(p)\right\\}$
$\displaystyle\qquad\qquad\times\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}}\right)\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}}\right).$
(E.44)

Therefore, in the expression of (5.1.3), the term,
$\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1}}\right)\Psi_{H}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{2}}\right)$
is similar to the term present in (5.1.2), we can say the following,

$\displaystyle\frac{\mathbf{Var}\left(B_{2,N}-\widetilde{B}_{2,N}\right)}{\mathbf{Var}(\widetilde{B}_{2,N})}=\frac{O(N^{2R-2})}{N^{2R-1}}\to
0.$ (E.45)

The above step in (E.45), the asymptotic normality result in (E.40) and Lemma
5.4 can now be combined to claim that,
$B_{2,N}/\sqrt{\mathbf{Var}(B_{2,N})}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N(0,1)$. Also note that the limiting relation in (E.45) and
the argument used in the proof of Lemma 5.4 show that,
$\mathbf{Var}(B_{2,N})/\mathbf{Var}(\widetilde{B}_{2,N})\rightarrow 1$.
Finally, using (5.1.3) and Slutsky’s theorem, we obtain

$\displaystyle\frac{B_{2,N}}{N^{R-1/2}}=\frac{B_{2,N}}{\sqrt{\mathbf{Var}(B_{2,N})}}\cdot\sqrt{\frac{\mathbf{Var}(\widetilde{B}_{2,N})}{N^{2R-1}}}\sqrt{\frac{\mathbf{Var}({B}_{2,N})}{\mathbf{Var}(\widetilde{B}_{2,N})}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left[f(H,p)\cdot\sigma^{[2]}(H,p)\right]^{2}\right).$
(E.46)

Thus, combining (E.35), (E.46) and using Slutsky’s theorem, we can claim that

$\displaystyle
N^{-R+1/2}\left(\frac{\widehat{S}_{N}^{[2]}(H)}{f(H,p)}-S_{N}(H)\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,(\sigma^{[2]}(H,p))^{2}\right),$

where $(\sigma^{[2]}(H,p))^{2}$ defined in (3.13).

∎

####  5.1.4 Proof of Theorem 3.5: CLT for estimated clustering coefficient

###### Proof of the Theorem 3.5.

We use the Delta-method and multivariate CLT provided in Proposition 5.9 to
prove this result. Note that, from the definition provided in (3.14) and
(3.15), we can write (for both $i=1$ and $i=2$),

$\displaystyle\widehat{\Gamma}^{[i]}_{N}-\Gamma_{N}=\begin{cases}\displaystyle\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{3})}{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})}-\frac{S_{N}(\mathbb{K}_{3})}{S_{N}(\mathbb{K}_{1,2})}\displaystyle&\quad\text{if
$S_{N}(\mathbb{K}_{1,2})>0$ and
$\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})>0$},\\\ \qquad{}0&\quad\text{if
$S_{N}(\mathbb{K}_{1,2})=0$},\\\
\displaystyle{}-\frac{S_{N}(\mathbb{K}_{3})}{S_{N}(\mathbb{K}_{1,2})}\displaystyle&\quad\text{if
$S_{N}(\mathbb{K}_{1,2})>0$ and
$\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})=0$.}\end{cases}.$

Define the sets, $D_{1,N}=\left[{S}_{N}(\mathbb{K}_{1,2})=0\right]$ and
$D^{[i]}_{2,N}=\left[\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})=0\right]$, for
each $N\geq 1$, and for $i=1,2$. Then, for any $t\in\mathbb{R}$,

$\displaystyle\mathbf{P}\left(\sqrt{N}\left(\widehat{\Gamma}^{[i]}_{N}-\Gamma_{N}\right)\leq
t\right)$
$\displaystyle=\mathbf{P}\left(\sqrt{N}\left(\widehat{\Gamma}^{[i]}_{N}-\Gamma_{N}\right)\leq
t\mid{\left(D^{[i]}_{2,N}\right)}^{c}\cap
D^{c}_{1,N}\right)\mathbf{P}\left({\left(D^{[i]}_{2,N}\right)}^{c}\cap
D^{c}_{1,N}\right)$
$\displaystyle\quad{}+\mathbf{P}\left(\sqrt{N}\left(\widehat{\Gamma}^{[i]}_{N}-\Gamma_{N}\right)\leq
t\mid D_{1,N}\right)\mathbf{P}\left(D_{1,N}\right)$
$\displaystyle\quad{}+\mathbf{P}\left(\sqrt{N}\left(\widehat{\Gamma}^{[i]}_{N}-\Gamma_{N}\right)\leq
t\mid D_{1,N}^{c}\cap D^{[i]}_{2,N}\right)\mathbf{P}\left(D_{1,N}^{c}\cap
D^{[i]}_{2,N}\right)$ $\displaystyle\equiv
r_{1,N}(t)+r_{2,N}(t)+r_{3,N}(t),\quad\text{(say).}$ (E.47)

Consider the second term in (E.47). Note that, $D_{1,N}\subseteq
D^{[i]}_{2,N}$, for both $i=1,2$, since zero wedges in the population network
$G_{N}$ will imply zero wedges will be observed in the induced (or ego-
centric) subgraph. Now, Proposition 5.9 shows that

$\displaystyle\sqrt{N}\cdot\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})-\mathbf{E}\left[\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right]}{N^{3}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,{\left(\tau^{[i]}_{1,2}\right)}^{2}\right),\quad\text{for
both $i=1,2$,}$ (E.48)

where,

$\displaystyle{\left(\tau^{[i]}_{1,2}\right)}^{2}=\begin{cases}p^{3}(1-p)\sum_{u=1}^{K}\lambda_{u}\left(\sum_{v,w=1}^{K}(\pi_{u,w}+2\pi_{v,w})\pi_{u,v}\cdot\lambda_{v}\lambda_{w}\right)^{2}\quad&\text{if
$i=1$},\\\
p(1-p)^{3}\sum_{u=1}^{K}\lambda_{u}\left(\sum_{v,w=1}^{K}((1+p)\pi_{u,w}+2p\pi_{v,w})\pi_{u,v}\cdot\lambda_{v}\lambda_{w}\right)^{2}\quad&\text{if
$i=2$.}\end{cases}$

Due to assumption (A.2) and (A.3),
${\left(\tau^{[i]}_{1,2}\right)}^{2}\in(0,\infty)$, for each $i=1,2$. Also
note that
$N^{-3}\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)\rightarrow\mu^{[i]}_{1}\in(0,\infty)$
(see (E.52)). Thus,

$\displaystyle\mathbf{P}\left(D^{[i]}_{2,N}\right)=\mathbf{P}\left(\sqrt{N}\cdot\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})-\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)}{N^{3}}={}-\sqrt{N}\cdot\frac{\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)}{N^{3}}\right)$
$\displaystyle=\mathbf{P}\left(\sqrt{N}\cdot\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})-\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)}{N^{3}}={}-\sqrt{N}\cdot\left(\mu^{[i]}_{1,2}+O\left(\frac{1}{N}\right)\right)\right)$
$\displaystyle=\mathbf{P}\left(\sqrt{N}\cdot\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})-\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)}{N^{3}}\leq{}-\sqrt{N}\cdot\left(\mu^{[i]}_{1,2}+O\left(\frac{1}{N}\right)\right)\right)-\Phi\left({}-\frac{\sqrt{N}}{\tau^{[i]}_{1,2}}\cdot\left(\mu^{[i]}_{1,2}+O\left(\frac{1}{N}\right)\right)\right)$
$\displaystyle\quad{}-\left[\mathbf{P}\left(\sqrt{N}\cdot\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})-\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)}{N^{3}}<{}-\sqrt{N}\cdot\left(\mu^{[i]}_{1,2}+O\left(\frac{1}{N}\right)\right)\right)\right.$
$\displaystyle\quad\left.{}-\Phi\left({}-\frac{\sqrt{N}}{\tau^{[i]}_{1,2}}\cdot\left(\mu^{[i]}_{1,2}+O\left(\frac{1}{N}\right)\right)\right)\right]$
$\displaystyle\equiv e_{1,N}-e_{2,N},\quad\text{(say).}$

Since (E.48) holds, it implies using Polya’s theorem (see Lemma 8.2.6 of
Athreya and Lahiri,  (2006))

$\sup_{t\in\mathbb{R}}\left|\mathbf{P}\left(\sqrt{N}\cdot\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})-\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)}{N^{3}}\leq
t\right)-\Phi\left(\frac{t}{\tau^{[i]}_{1,2}}\right)\right|\rightarrow
0,\quad\text{as $N\rightarrow\infty$.}$

This implies, $e_{1,N}\rightarrow 0$, as $N\rightarrow\infty$. In order to
handle $e_{2,N}$, we use Lemma 5.6 (with $G$ being the standard normal cdf).
This implies

$\displaystyle\sup_{t\in\mathbb{R}}\left|\mathbf{P}\left(\sqrt{N}\cdot\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})-\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)}{N^{3}}<t\right)-\Phi\left(\frac{t}{\tau^{[i]}_{1,2}}\right)\right|$
$\displaystyle=\sup_{t\in\mathbb{R}}\left|\mathbf{P}\left(\sqrt{N}\cdot\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})-\mathbf{E}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})\right)}{N^{3}}\leq
t\right)-\Phi\left(\frac{t}{\tau^{[i]}_{1,2}}\right)\right|\rightarrow
0,\quad\text{as $N\rightarrow\infty$,}$

which in turn ensures $e_{2,N}\rightarrow 0$. This shows that
$\mathbf{P}(D_{1,N})\leq\mathbf{P}\left(D^{[i]}_{2,N}\right)\rightarrow 0$, as
$N\rightarrow\infty$, for each $i=1,2$. This implies, $r_{2,N}(t)\rightarrow
0$, as $N\rightarrow\infty$, for each $t\in\mathbb{R}$ (cf. (E.47)). It also
shows that, $\mathbf{P}\left(D^{c}_{1,N}\cap
D^{[i]}_{2,N}\right)\leq\mathbf{P}\left(D^{[i]}_{2,N}\right)\rightarrow 0$, in
the third term of (E.47), thereby ensuring that $r_{3,N}(t)\rightarrow 0$, for
each $t\in\mathbb{R}$. Continuing from (E.47), now we can write

$\displaystyle\mathbf{P}\left(\sqrt{N}\left(\widehat{\Gamma}^{[i]}_{N}-\Gamma_{N}\right)\leq
t\right)=r_{1,N}(t)+o(1)$
$\displaystyle=\mathbf{P}\left(\sqrt{N}\left(\widehat{\Gamma}^{[i]}_{N}-\Gamma_{N}\right)\leq
t,\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})>0,S_{N}(\mathbb{K}_{1,2})>0\right)+o(1)$
$\displaystyle=\mathbf{P}\left(\sqrt{N}\cdot\left[\displaystyle\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{3})}{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})}-\frac{S_{N}(\mathbb{K}_{3})}{S_{N}(\mathbb{K}_{1,2})}\displaystyle\right]\leq
t,~{}\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})>0,S_{N}(\mathbb{K}_{1,2})>0\right)+o(1)$
$\displaystyle=\mathbf{P}\left(\sqrt{N}\cdot\left[\displaystyle\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{3})}{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})}-\frac{S_{N}(\mathbb{K}_{3})}{S_{N}(\mathbb{K}_{1,2})}\displaystyle\right]\leq
t\right)+o(1)\quad\text{(using previous arguments).}$ (E.49)

Now, we use the Delta method to obtain the limiting expression for the first
term on the r.h.s. of the above expression. For each $N\geq 1$, define

$\left.\begin{aligned}
\mathbf{X}^{[i]}_{N}&=\frac{1}{N^{3}}\left(\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2}),\widehat{S}^{[i]}_{N}(\mathbb{K}_{3}),S_{N}(\mathbb{K}_{1,2}),S_{N}(\mathbb{K}_{3})\right)^{{T}},\quad\text{and}\\\
\bm{\mu}^{[i]}_{N}&=\mathbf{E}\left(\mathbf{X}^{[i]}_{N}\right)={\left(f^{[i]}(\mathbb{K}_{1,2},p)\cdot{\theta_{1,N}},~{}f^{[i]}(\mathbb{K}_{3},p)\cdot{\theta_{2,N}},~{}{\theta_{1,N}},~{}{\theta_{2,N}}\right)}^{T},\end{aligned}\right\\},\
\text{for $i=1,2$,}$ (E.50)

where,

$\displaystyle f^{[i]}(\mathbb{K}_{1,2},p)=\begin{cases}p^{3}&\text{if
$i=1$,}\\\ p^{2}(1-p)+p&\text{if $i=2$,}\end{cases}\quad\text{and}\quad\
f^{[i]}(\mathbb{K}_{3},p)=\begin{cases}p^{3}&\text{if $i=1$,}\\\
3p^{2}(1-p)+p^{3}&\text{if $i=2$,}\end{cases}$ (E.51)
$\displaystyle\theta_{1,N}=\frac{\mathbf{E}(S_{N}(\mathbb{K}_{1,2}))}{N^{3}},\quad\text{and}\quad\theta_{2,N}=\frac{\mathbf{E}(S_{N}(\mathbb{K}_{3}))}{N^{3}},\quad\text{for
all $N\geq 1$.}$

Also note that,

$\displaystyle\bm{\mu}^{[i]}_{N}\rightarrow\bm{\mu}^{[i]}={\left(f^{[i]}(\mathbb{K}_{1,2},p)\cdot\theta_{1},~{}f^{[i]}(\mathbb{K}_{3},p)\cdot\theta_{2},~{}\theta_{1},~{}\theta_{2}\right)}^{T},\quad\text{for
$i=1,2$,}$ (E.52)

where $\theta_{i}$, $i=1,2$, are defined in (3.16). Define the sets
$\mathcal{A}=\\{{(x,y,a,b)}^{T}\in\mathbb{R}^{4}:0\leq y\leq x,0\leq b\leq
a,y\leq b,x\leq a\\}$,
$\mathcal{A}^{+}=\\{{(x,y,a,b)}^{T}\in\mathcal{A}:x,y,a,b>0\\}$. We consider a
function $g:\mathcal{A}\to\mathbb{R}$, defined as

$\displaystyle g(x,y,a,b)=\begin{cases}\frac{y}{x}-\frac{b}{a}&\quad\text{if
${(x,y,a,b)}^{T}\in\mathcal{A}$, $x>0$ and $a>0$,}\\\
{}-\frac{b}{a}&\quad\text{if ${(x,y,a,b)}^{T}\in\mathcal{A}$, $x=0$ and
$a>0$,}\\\ 0&\quad\text{if ${(x,y,a,b)}^{T}\in\mathcal{A}$ and
$a=0$.}\end{cases}$ (E.53)

For $(x,y,a,b)^{T}\in\mathcal{A}^{+}$,

$\displaystyle\nabla
g(x,y,a,b)=\left(-\frac{y}{x^{2}},\frac{1}{x},\frac{b}{a^{2}},-\frac{1}{a}\right)^{{T}}.$

Thus $(x,y,a,b)^{T}\mapsto\nabla g(x,y,a,b)$ consists of ratios of polynomials
in each of the components, and the derivative vector will be a continuous map
at $(x,y,a,b)^{T}$ when $(x,y,a,b)^{T}\in\mathcal{A}^{+}$. Note that from the
definition provided in (E.52), using the expression for $f^{[i]}(\cdot,p)$ in
(E.51) (and using Assumption (A.4)) and by noting that
$\theta_{1},\theta_{2}>0$ (see (3.16)) it follows that
$\bm{\mu}^{[i]}\in\mathcal{A}^{+}$. Further, there exists a $\eta>0$, such
that
$B(\bm{\mu}^{[i]},\eta)\equiv\left\\{{(x,y,a,b)}^{T}\in\mathcal{A}:\|{(x,y,a,b)}^{T}-\bm{\mu}^{[i]}\|<\eta\right\\}\subset\text{int}(\mathcal{A}^{+})$,
for both $i=1,2$, where $\text{int}(A)$ denotes the interior of a set $A$.
Hence, all components of the derivative map $\nabla g(x,y,a,b)$ are continuous
at each ${(x,y,a,b)}^{T}\in B(\bm{\mu}^{[i]},\eta)$. This implies, $g$ is
continuously differentiable in a small neighbourhood around $\bm{\mu}^{[i]}$,
for both $i=1,2$. We also know from Proposition 5.9, that
$\sqrt{N}\left(\mathbf{X}^{[i]}_{N}-\bm{\mu}^{[i]}_{N}\right)$ converges to a
Gaussian limiting distribution with mean zero and covariance matrix
$\Delta^{[i]}$ for each $i=1,2$. We can now use Theorem 3.8 of van der Vaart,
(1998) along with the convergence statement in (E.52) and the continuity of
$\nabla g(\cdot)$ in a neighbourhood of $\bm{\mu}^{[i]}$ to claim that

$\sqrt{N}\left(g(\mathbf{X}^{[i]}_{N})-g(\bm{\mu}^{[i]}_{N})\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,{\big{(}\tau^{[i]}(p)\big{)}}^{2}\right),\quad{\left(\tau^{[i]}(p)\right)}^{2}={\left(\nabla
g\left(\bm{\mu}^{[i]}\right)\right)}^{T}\Delta^{[i]}{\left(\nabla
g\left(\bm{\mu}^{[i]}\right)\right)},\ \text{for $i=1,2$.}$

The exact expression for $\tau^{[i]}(p)$, $i=1,2$, are provided in (3.18) and
(3.19) respectively. Now, for large enough $N$,
$\bm{\mu}^{[i]}_{N}\in\mathcal{A}^{+}$, for $i=1,2$. Hence, by substituting
values obtained in (E.50) and (E.51) we get

$g\left(\bm{\mu}^{[1]}_{N}\right)=0.$

This implies,

$\sqrt{N}\left(g(\mathbf{X}^{[1]}_{N})-g(\bm{\mu}^{[1]}_{N})\right)=\sqrt{N}\left(\widehat{\Gamma}^{[1]}_{N}-\Gamma_{N}\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,{\big{(}\tau^{[1]}(p)\big{)}}^{2}\right).$

However, for $i=2$, and large enough $N$, we get (from (E.50), (E.51) and
(E.52)),

$\displaystyle\sqrt{N}\left(g(\bm{\mu}^{[2]}_{N})-g(\bm{\mu}^{[2]})\right)={}-\frac{{(1-p)}^{2}}{p(1-p)+1}\cdot\sqrt{N}\cdot\left[\frac{\theta_{2,N}}{\theta_{1,N}}-\frac{\theta_{2}}{\theta_{1}}\right].$

We know, $N^{-3}\theta_{j,N}\rightarrow\theta_{j}$, where $\theta_{j}$,
$j=1,2$, are defined in (3.16). One can check that,
$\theta_{j,N}=\theta_{j}+O(1/N)$, for $j=1,2$. Thus, by expanding
${\\{\theta_{1}+O(1/N)\\}}^{-1}$, we can write

$\displaystyle\sqrt{N}\cdot\left[\frac{\theta_{2,N}}{\theta_{1,N}}-\frac{\theta_{2}}{\theta_{1}}\right]=O\left(\frac{1}{\sqrt{N}}\right),$

This implies,
$\sqrt{N}\left(g(\bm{\mu}^{[2]}_{N})-g(\bm{\mu}^{[2]})\right)=o(1)$. Hence,

$\displaystyle\sqrt{N}\left(g(\mathbf{X}^{[2]}_{N})-g(\bm{\mu}^{[2]}_{N})\right)$
$\displaystyle=\sqrt{N}\left(\widehat{\Gamma}^{[2]}_{N}-\Gamma_{N}-g(\bm{\mu}^{[2]})\right)+\sqrt{N}\left(g(\bm{\mu}^{[2]})-g(\bm{\mu}^{[2]}_{N})\right)$
$\displaystyle=\sqrt{N}\left(\widehat{\Gamma}^{[2]}_{N}-\Gamma_{N}-\left\\{{}-\frac{{(1-p)}^{2}}{p(1-p)+1}\cdot\frac{\theta_{2}}{\theta_{1}}\right\\}\right)+o(1)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,{\left(\tau^{[2]}(p)\right)}^{2}\right).$

This completes the proof. ∎

####  5.1.5 Proof of Theorem 3.6: CLT in the sparse SBM setup

###### Proof of Theorem 3.6.

Consider the edge probability matrix $\bm{\Pi}_{N}$ defined in (3.22) which
satisfies assumption (A.2†) (cf. (3.23)). As a result,

$\displaystyle\lim_{N\to\infty}\Psi_{H}\left(N^{\beta}\cdot\bm{\Pi}_{N},\mathbf{u}\right)=\Psi_{H}\left(\mathbf{C},\mathbf{u}\right),\
\text{for any $\mathbf{u}\in{[K]}^{R}$.}$ (E.54)

Consider the induced subgraph sampling scheme. Let $H$ be a simple, fixed,
undirected and connected graph, with $|V(H)|=R$ and $|E(H)|=T$. Consider the
pivotal quantity provided in the l.h.s. of (3.27). We can split it into two
parts, similar to the split provided in (E.14) (see proof of Theorem 3.1), and
write these two terms as $L_{1,N}(H)$ and $L_{2,N}(H)$ respectively, where

$\displaystyle
L_{1,N}(H)=\sum_{\mathbf{s}\in{[N]}_{R}}\left(\frac{\Psi_{H}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)}{p^{R}}-1\right)\cdot\left[\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}\right)-\Psi_{H}\left(\bm{\Pi}_{N},(\bm{\alpha}_{N})_{\mathbf{s}}\right)\right],\quad\text{and}$
(E.55) $\displaystyle
L_{2,N}(H)=\sum_{\mathbf{s}\in{[N]}_{R}}\left(\frac{\Psi_{H}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)}{p^{R}}-1\right)\cdot\Psi_{H}\left(\bm{\Pi}_{N},(\bm{\alpha}_{N})_{\mathbf{s}}\right).$

Initially we will attempt to find a sequence $\\{a_{N}:N\geq 1\\}$ such that
$(a_{N})^{-1}\cdot L_{2,N}(H)$ will converge in law to a limiting normal
distribution. We will then try to find conditions under which
$a^{-1}_{N}L_{1,N}(H)$ will converge in probability to zero. Initially we
consider the Hájek projection $\widetilde{L}_{2,N}(H)$ of $L_{2,N}(H)$, which
is same as the expression obtained in (5.1.2). Thus,

$\displaystyle\widetilde{L}_{2,N}(H)=\sum_{s_{1}=1}^{N}\left(\frac{W_{N,s_{1}}}{p}-1\right)\left(\sum_{k=1}^{R}\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\Psi_{H}\left(\bm{\Pi}_{N},\left(\bm{\alpha}_{N}\right)_{\mathbf{v}^{[k]}(\mathbf{s})}\right)\right).$
(E.56)

From (E.56), using $\mathbf{E}(\widetilde{L}_{2,N})=0$, and using expression
(5.1.2), we obtain

$\displaystyle\mathbf{Var}\left({a^{-1}_{N}}\cdot\widetilde{L}_{2,N}(H)\right)$
$\displaystyle=\frac{N^{2R-1}}{a_{N}^{2}}\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{N,u_{1}}\left(\sum_{(\mathbf{u})^{(1)}\in[K]^{R-1}}\sum_{k=1}^{R}\Psi_{H}(\bm{\Pi}_{N},\mathbf{v}^{[k]}(\mathbf{u}))\left(\prod_{i=2}^{R}\lambda_{N,u_{i}}+O\left(\frac{1}{N}\right)\right)\right)^{2}$
$\displaystyle=\frac{N^{2R-1-2T\beta}}{a_{N}^{2}}\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{N,u_{1}}\left(\sum_{(\mathbf{u})^{(1)}\in[K]^{R-1}}\sum_{k=1}^{R}\Psi_{H}(N^{\beta}\cdot\bm{\Pi}_{N},\mathbf{v}^{[k]}(\mathbf{u}))\left(\prod_{i=2}^{R}\lambda_{N,u_{i}}+O\left(\frac{1}{N}\right)\right)\right)^{2}.$

Now, if we consider

$a_{N}=N^{R-1/2-T\beta},\quad\text{for all $N\geq 1$,}$

then the using (E.54), above variance has the finite limit of the form,

$\displaystyle\mathbf{Var}\left(\frac{\widetilde{L}_{2,N}(H)}{N^{R-1/2-T\beta}}\right)\to\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left(\sum_{(\mathbf{u})^{(1)}\in[K]^{R-1}}\sum_{k=1}^{R}\Psi_{H}(\mathbf{C},\mathbf{v}^{[k]}(\mathbf{u}))\prod_{i=2}^{R}\lambda_{u_{i}}\right)^{2}.$
(E.57)

Now, $a^{-1}_{N}\widetilde{L}_{2,N}(H)$ is a weighted sum of centered
independent random variables $(p^{-1}W_{N,s_{1}}-1)$, with non-random weights,

$a_{s_{1},N}=\frac{1}{N^{R-1/2-T\beta}}\left(\sum_{k=1}^{R}\sum_{{(\mathbf{s})}^{(1)}\in{\left[[N]\setminus\\{s_{1}\\}\right]}_{R-1}}\Psi_{H}\left(\bm{\Pi}_{N},\left(\bm{\alpha}_{N}\right)_{\mathbf{v}^{[k]}(\mathbf{s})}\right)\right),\quad\text{for
all $1\leq s_{1}\leq N$ (see \eqref{H-proj-L2}).}$

Therefore, from expression (5.1.2),

$\displaystyle\max_{1\leq s_{1}\leq
N}|a_{s_{1},N}|=\frac{K}{\sqrt{N}}\sum_{{(\mathbf{u})}^{(1)}\in[K]^{R-1}}\sum_{k=1}^{R}\Psi_{H}\left(N^{\beta}\cdot\bm{\Pi}_{N},\mathbf{v}^{[k]}(\mathbf{u})\right)\left(\prod_{i=2}^{R}\lambda_{N,u_{i}}+O\left(\frac{1}{N}\right)\right)\rightarrow
0,$

due to the assumption (A.2†) and (A.3). Thus, the sufficient conditions stated
in Lemma 5.5 hold, and it implies

$\displaystyle\frac{\widetilde{L}_{2,N}(H)}{a_{N}}$
$\displaystyle\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,~{}\left(\tau^{[1]}(H,p)\right)^{2}\right),\quad\text{where,}$
$\displaystyle\left(\tau^{[1]}(H,p)\right)^{2}$
$\displaystyle=\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left(\sum_{(\mathbf{u})^{(1)}\in[K]^{R-1}}\sum_{k=1}^{R}\Psi_{H}(\mathbf{C},\mathbf{v}^{[k]}(\mathbf{u}))\prod_{i=2}^{R}\lambda_{u_{i}}\right)^{2}.$

Also $\tau^{[1]}(H,p)>0$, as per the assumption (A.2†) and by using the
argument provided in Lemma 5.3. Next, we will show that
$\mathbf{Var}\left(\widetilde{L}_{2,N}(H)-L_{2,N}(H)\right)=o\left(\mathbf{Var}\left(\widetilde{L}_{2,N}(H)\right)\right)$.
From the expression (5.1.2), we get the following,

$\displaystyle\mathbf{Var}(\widetilde{L}_{2,N}(H)-L_{2,N}(H))$
$\displaystyle=\sum_{t=2}^{R}\left[\left(\frac{1}{p^{t}}-1\right)-t\left(\frac{1}{p}-1\right)\right]\Bigg{(}N^{2R-t}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\Psi_{H}(\bm{\Pi}_{N},\mathbf{u})\Psi_{H}(\bm{\Pi}_{N},\mathbf{v})\prod_{r=1}^{t}\mathbf{1}\left(u_{j_{r}}=v_{k_{\xi(r)}}\right)$
$\displaystyle\qquad\qquad\times\left\\{\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\cdot\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O\left(\frac{1}{N}\right)\right\\}\Bigg{)}$
$\displaystyle=\sum_{t=2}^{R}\left[\left(\frac{1}{p^{t}}-1\right)-t\left(\frac{1}{p}-1\right)\right]\Bigg{(}N^{2R-t-2T\beta}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\Psi_{H}(N^{\beta}\bm{\Pi}_{N},\mathbf{u})\Psi_{H}(N^{\beta}\bm{\Pi}_{N},\mathbf{v})$
$\displaystyle\qquad\qquad\times\prod_{r=1}^{t}\mathbf{1}\left(u_{j_{r}}=v_{k_{\xi(r)}}\right)\left\\{\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\cdot\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O\left(\frac{1}{N}\right)\right\\}\Bigg{)}$
$\displaystyle\leq
N^{2R-2-2T\beta}\sum_{t=2}^{R}\left[\left(\frac{1}{p^{t}}-1\right)-t\left(\frac{1}{p}-1\right)\right]\Bigg{(}\frac{1}{N^{t-2}}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\left(N^{\beta}\cdot\max_{1\leq
i\leq K}\pi_{i,i}^{N}\right)^{2T}$
$\displaystyle\qquad\qquad\times\prod_{r=1}^{t}\mathbf{1}\left(u_{j_{r}}=v_{k_{\xi(r)}}\right)\left\\{\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\cdot\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O\left(\frac{1}{N}\right)\right\\}\Bigg{)}.$

Using (E.57) and the fact that $\tau^{[1]}(H,p)\in(0,\infty)$, we get
$\mathbf{Var}\left(\widetilde{L}_{2,N}(H)\right)\sim N^{2R-1-2T\beta}$. The
above argument now implies,
$\mathbf{Var}(\widetilde{L}_{2,N}(H)-L_{2,N}(H))=O(N^{2R-2-2T\beta})=o\left(N^{2R-1-2T\beta}\right)$.
Hence, using Lemma 5.4 it follows that,

$\displaystyle N^{-R+1/2+T\beta}L_{2,N}(H)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left(\tau^{[1]}(H,p)\right)^{2}\right).$

Now, we will show that $N^{-R+1/2+T\beta}\cdot L_{1,N}(H)=o_{p}(1)$. Note,
$\mathbf{E}(L_{1,N}(H))=0$ and $\mathbf{Var}(L_{1,N}(H))$ has an expression
which is similar to (5.1.2). Also note that, as per assumption (A.2†), there
exists some $c\in(0,\infty)$, such that, $N^{\beta}\max_{1\leq i\leq
K}\pi^{N}_{i,i}\rightarrow c$, as $N\rightarrow\infty$. Thus,
$N^{\beta}\max_{1\leq i\leq K}\pi^{N}_{i,i}\leq\max\\{1,2c\\}\equiv c_{\star}$
(say), for large enough $N$. Now, using similar arguments as those used in the
proof of Theorem 3.1, we get

$\displaystyle\mathbf{Var}\left(L_{1,N}(H)\right)$
$\displaystyle=\sum_{t=2}^{R}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\bm{\Pi}_{N},\mathbf{u}_{1\cup
2}\right)(1-\widetilde{\Psi}_{H_{1}\cap
H_{2}}\left(\bm{\Pi}_{N},(\mathbf{u})_{\mathbf{j}}\right))\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})$
$\displaystyle\qquad\times\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})\right\\}$
$\displaystyle=\sum_{t=2}^{R}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}N^{-\beta(2T-|E(H_{1})\cap
E(H_{2})|)}\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(N^{\beta}\bm{\Pi}_{N},\mathbf{u}_{1\cup 2}\right)$
$\displaystyle\quad\times\left(1-\widetilde{\Psi}_{H_{1}\cap
H_{2}}\left(\bm{\Pi}_{N},(\mathbf{u})_{\mathbf{j}}\right)\right)\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})$
$\displaystyle\qquad\times\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})\right\\}$
$\displaystyle\leq\sum_{t=2}^{R}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}N^{-\beta(2T-|E(H_{1})\cap
E(H_{2})|)}\left(N^{\beta}\cdot\max_{1\leq i\leq
K}\pi^{N}_{i,i}\right)^{|E(H_{1})\cup E(H_{2})|}$
$\displaystyle\qquad\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})\right\\}$ $\displaystyle\leq
c^{2T}_{\star}\cdot\sum_{t=2}^{R}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}N^{-\beta(2T-|E(H_{1})\cap
E(H_{2})|)}$
$\displaystyle\qquad\times\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})\right\\}.$ (E.58)

Recall the definition of $f_{1}(t,H)$ from (E.87). Then from (5.1.5), we get,

$\displaystyle\mathbf{Var}\left(\frac{L_{1,N}(H)}{N^{R-1/2-T\beta}}\right)\leq
c^{2T}_{\star}\cdot\sum_{t=2}^{R}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\frac{N^{-\beta(2T-f_{1}(t,H))}}{N^{2R-1-2T\beta}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})$
$\displaystyle\qquad\times\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})\right\\}$
$\displaystyle=c^{2T}_{\star}\cdot\sum_{t=2}^{R}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\frac{N^{2R-t-\beta(2T-f_{1}(t,H))}}{N^{2R-1-2T\beta}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})$
$\displaystyle\qquad\times\left\\{\prod_{r=1}^{t}\lambda_{N,a_{r}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O\left(\frac{1}{N}\right)\right\\}.$

The bound on the r.h.s. above converges to zero if

$\displaystyle
0<\beta<c(H)=\min\left\\{\frac{1}{f_{1}(2,H)},\ldots,\frac{R-1}{f_{1}(R,H)}\right\\},\quad\text{(see
\eqref{beta-upp}).}$ (E.59)

This implies $L_{1,N}(H)=o_{p}\left(N^{-R+1/2+T\beta}\right)$. Combining this
with previous steps we complete the proof of the theorem. ∎

####  5.1.6 Statement and proof of Lemma 5.1: a result which is used in
proving Theorem 3.1

######  Lemma 5.1.

Let $H$ be a fixed, undirected, simple and connected graph with
$V(H)=\\{1,\ldots,R\\}$ and edge set $E(H)\subseteq{[V(H)]}^{2}$, where $R\geq
2$, is a fixed positive integer. For each $t\in\\{0,1,\ldots,R\\}$ define the
collection of ordered pairs of vectors

$\displaystyle
E_{t}=\left\\{(\mathbf{s}_{1},\mathbf{s}_{2}):\mathbf{s}_{1},\mathbf{s}_{2}\in{[N]}_{R},~{}R_{1\cap
2}=|A(\mathbf{s}_{1})\cap A(\mathbf{s}_{2})|=t\right\\}.$ (E.60)

Then,

$\displaystyle\frac{1}{N^{2R-1}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\mathbf{cov}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{1}),\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{2})\right)\rightarrow
0,$ (E.61)

where $\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{i})$, $i=1,2$, are defined as per
(E.10).

###### Proof of Lemma 5.1.

We start with the case of $t=0$. As $\mathbf{s}_{1},\mathbf{s}_{2}\in E_{0}$,
it implies, the collections of random variables $\\{Y_{s_{1,i},s_{1,j}}:1\leq
i\neq j\leq R\\}$ and $\\{Y_{s_{2,i},s_{2,j}}:1\leq i\neq j\leq R\\}$ do not
have any overlap and these two collections are independent, as per the
assumptions of the SBM (see (2.3)). This implies, the sum (in the l.h.s. of
(E.61)) over $E_{0}$ is zero.

If $\mathbf{s}_{1},\mathbf{s}_{2}\in E_{1}$, there exists one common component
between $\mathbf{s}_{1}$ and $\mathbf{s}_{2}$. Suppose, $A(\mathbf{s}_{1})\cap
A(\mathbf{s}_{2})=\\{a\\}$, for some $a\in[N]$. Then, edge independence in SBM
implies $Y_{a,s_{1,j}}$ and $Y_{a,s_{2,j}}$ are independent for
$j=2,\ldots,R$, which leads to conclusion in (E.61) for $t=1$.

Now we consider the case where $R_{1\cap 2}=t\in\\{2,\ldots,R\\}$. Thus, there
are $t$ common elements between $\mathbf{s}_{1}$ and $\mathbf{s}_{2}$. Fix any
$\mathbf{j}=(j_{1},\ldots,j_{t})^{T}\in[R]_{t}$ and
$\mathbf{k}=(k_{1},\ldots,k_{t})^{T}\in[R]_{t}$, where $\mathbf{j}$ and
$\mathbf{k}$ denote the positions at which the common elements appear within
$\mathbf{s}_{1}$ and $\mathbf{s}_{2}$ respectively. As per earlier convention
(see (E.7)), we assume, $1\leq j_{1}<\cdots<j_{t}\leq R$ and $1\leq
k_{1}<\cdots<k_{t}\leq R$. Consider the set $[t]=\\{1,\ldots,t\\}$, and define

$\displaystyle\mathcal{S}_{t}=\left\\{\xi:[t]\mapsto(\xi(1),\ldots,\xi(t))\in{[t]}_{t}~{}\big{|}~{}\text{$\xi$
is a permutation of $\\{1,\ldots,t\\}$}\right\\},$ (E.62)

as the collection of all possible permutations of $\\{1,\ldots,t\\}$. Note
$|\mathcal{S}_{t}|=t!$. For each fixed $\mathbf{j},\mathbf{k}\in{[R]}_{t}$ and
any permutation $\xi\in\mathcal{S}_{t}$, define the following subsets of
$E_{t}$ (see (E.60)),

$\displaystyle E_{t}(\mathbf{j},\mathbf{k})$
$\displaystyle=\left\\{(\mathbf{s}_{1},\mathbf{s}_{2}):\mathbf{s}_{1},\mathbf{s}_{2}\in
E_{t},\
A\left({(\mathbf{s}_{1})}_{\mathbf{j}}\right)=A\left({(\mathbf{s}_{2})}_{\mathbf{k}}\right)\right\\},\quad\text{and}$
(E.63) $\displaystyle E_{t}(\mathbf{j},\mathbf{k},\xi)$
$\displaystyle=\left\\{(\mathbf{s}_{1},\mathbf{s}_{2}):\mathbf{s}_{1},\mathbf{s}_{2}\in
E_{t},\
s_{1,j_{1}}=s_{2,k_{\xi(1)}},\ldots,s_{1,j_{t}}=s_{2,k_{\xi(t)}}\right\\}.$

For a fixed $t\in\\{2,\ldots,R\\}$, the sets $E_{t}(\mathbf{j},\mathbf{k})$
and $E_{t}(\mathbf{j}^{\prime},\mathbf{k}^{\prime})$ are disjoint if,
$\mathbf{j}\neq\mathbf{j}^{\prime}$ or $\mathbf{k}\neq\mathbf{k}^{\prime}$.
Also note that,

$\displaystyle
E_{t}=\bigcup_{\mathbf{j},\mathbf{k}\in[R]_{t}}E_{t}(\mathbf{j},\mathbf{k})\quad\text{and}\quad
E_{t}(\mathbf{j},\mathbf{k})=\bigcup_{\xi\in\mathcal{S}_{t}}E_{t}(\mathbf{j},\mathbf{k},\xi).$

Fix any $(\mathbf{s}_{1},\mathbf{s}_{2})\in E_{t}(\mathbf{j},\mathbf{k},\xi)$
and write, $\mathbf{s}_{i}=\left(s_{i,1},\ldots,s_{i,R}\right)^{T}\in[N]_{R}$,
for $i=1,2$. Let $A(\mathbf{s}_{1})\cap
A(\mathbf{s}_{2})=\\{a_{1},\ldots,a_{t}\\}$, where $a_{i}$’s are distinct and
$a_{i}\in[N]$ for each $i=1,\ldots,t$. For any
$\mathbf{j}=(j_{1},\ldots,j_{t})^{T}\in{[R]}_{t}$ and
$\mathbf{k}=(k_{1},\ldots,k_{t})^{T}\in{[K]}_{t}$, we write
$A(\mathbf{j})=\\{j_{1},\ldots,j_{t}\\}$, and
$A(\mathbf{k})=\\{k_{1},\ldots,k_{t}\\}$. For simplicity of notation, we will
write the product,

$\prod_{j\in{[R]}\setminus\\{j_{1},\ldots,j_{t}\\}}\equiv\prod_{j\in{[R]}\setminus
A(\mathbf{j})},$

and similarly for the case of the product over
$k\in{[R]}\setminus\\{k_{1},\ldots,k_{t}\\}$. Consider two simple, connected
graphs $H_{1}$ and $H_{2}$ with $V(H_{i})=A(\mathbf{s}_{i})$, for $i=1,2$.
Also assume, $H_{1}$ and $H_{2}$ are both isomorphic to $H$, i.e.,
$H_{i}\simeq H$, for $i=1,2$. Then, the covariance term in the l.h.s of (E.61)
can be written as,

$\displaystyle\mathbf{cov}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{1}),\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{2})\right)$
$\displaystyle=\mathbf{E}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{1})\cdot\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{2})\right)-\mathbf{E}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{1})\right)\mathbf{E}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{2})\right)$
$\displaystyle=\mathbf{E}\left(\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\mathbf{Y}_{N},\mathbf{s}_{1\cup
2}\right)\right)-\mathbf{E}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{1})\right)\mathbf{E}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{2})\right)\quad\text{(cf.
\eqref{s1cup2}, \eqref{s1cap2} and \eqref{psi-G-def-1}),}$
$\displaystyle=\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1\cup
2}}\right)-\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1\cup
2}}\right)\cdot\widetilde{\Psi}_{H_{1}\cap
H_{2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1\cap 2}}\right)$
$\displaystyle=\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1\cup
2}}\right)(1-\widetilde{\Psi}_{H_{1}\cap
H_{2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}_{1\cap 2}}\right))$
$\displaystyle=\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\bm{\Pi},\mathbf{u}_{1\cup 2}\right)(1-\widetilde{\Psi}_{H_{1}\cap
H_{2}}\left(\bm{\Pi},(\mathbf{u})_{\mathbf{j}}\right))$
$\displaystyle\qquad\times\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{r=1}^{t}\mathbf{1}\left(\alpha_{a_{r}}=v_{k_{\xi(r)}}\right)\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})\right\\},$ (E.64)

where, $\mathbf{u}_{1\cup 2}=\left(u_{1\cup 2,1},\ldots,u_{1\cup
2,2R-t}\right)^{T}=(\mathbf{u}^{T},((\mathbf{v})_{\mathbf{m}})^{T})^{T}\in[K]^{2R-t}$,
where $\mathbf{m}=(m_{1},\ldots,m_{R-t})^{T}$ with $m_{i}\in[R]_{t}\setminus
A(\mathbf{k})$. The term

$\widetilde{\Psi}_{H_{1}\cup H_{2}}\left(\bm{\Pi},\mathbf{u}_{1\cup
2}\right)(1-\widetilde{\Psi}_{H_{1}\cap
H_{2}}\left(\bm{\Pi},(\mathbf{u})_{\mathbf{j}}\right))\in[0,1],$

due to assumption (A.2). Then, using (E.64), we get

$\displaystyle\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\mathbf{cov}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{1}),\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{2})\right)$
$\displaystyle=\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\bm{\Pi},\mathbf{u}_{1\cup 2}\right)(1-\widetilde{\Psi}_{H_{1}\cap
H_{2}}\left(\bm{\Pi},(\mathbf{u})_{\mathbf{j}}\right))$
$\displaystyle\qquad\times\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=v_{k_{\xi(r)}})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})\right\\}$
$\displaystyle\leq\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})$
$\displaystyle\qquad\qquad\times\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=v_{k_{\xi(r)}})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})$
$\displaystyle=\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}(\mathbf{j},\mathbf{k},\xi)}\Bigg{\\{}\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})$
$\displaystyle\qquad\qquad\times\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})\Bigg{\\}}$
$\displaystyle=\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\sum_{\begin{subarray}{c}1\leq
a_{1}\neq\ldots\neq a_{t}\neq s_{1,j}\leq N\\\ 1\leq a_{1}\neq\ldots\neq
a_{t}\neq s_{2,k}\leq N\\\ j\in[R]\setminus A(\mathbf{j})\\\ k\in[R]\setminus
A(\mathbf{k})\end{subarray}}\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})$
$\displaystyle\qquad\times\prod_{j\in[R]\setminus
A(\mathbf{j})}\mathbf{1}(\alpha_{s_{1,j}}=u_{j})\prod_{k\in[R]\setminus
A(\mathbf{k})}\mathbf{1}(\alpha_{s_{2,k}}=v_{k})$
$\displaystyle=\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\mathbf{1}(m_{1}\neq\cdots\neq
m_{R-t}\in[R]\setminus A(\mathbf{j}),\ l_{1}\neq\cdots\neq
l_{R-t}\in[R]\setminus A(\mathbf{k}))$
$\displaystyle\qquad\times\sum_{\begin{subarray}{c}1\leq a_{1}\neq\ldots\neq
a_{t}\neq s_{1,m_{1}}\neq\ldots\neq s_{1,m_{R-t}}\leq N\\\ 1\leq
a_{1}\neq\ldots\neq a_{t}\neq s_{2,l_{1}}\neq\ldots\neq s_{2,l_{R-t}}\leq
N\end{subarray}}\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{x=1}^{R-t}\mathbf{1}(\alpha_{s_{1,m_{x}}}=u_{m_{x}})\mathbf{1}(\alpha_{s_{2,l_{x}}}=v_{l_{x}})\right\\}$
$\displaystyle=\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\mathbf{1}(m_{1}\neq\cdots\neq
m_{R-t}\in[R]\setminus A(\mathbf{j}),\ l_{1}\neq\cdots\neq
l_{R-t}\in[R]\setminus A(\mathbf{k}))$
$\displaystyle\qquad\times\sum_{\begin{subarray}{c}1\leq a_{1}\neq\ldots\neq
a_{t}\neq s_{1,m_{1}}\neq\ldots\neq s_{1,m_{R-t}}\leq N\\\ 1\leq
a_{1}\neq\ldots\neq a_{t}\neq s_{2,l_{1}}\neq\ldots\neq s_{2,l_{R-t}}\leq
N\end{subarray}}\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{x=1}^{R-t}\mathbf{1}(\alpha_{s_{1,m_{x}}}=u_{m_{x}})\mathbf{1}(\alpha_{s_{2,l_{x}}}=v_{l_{x}})\right\\}$
$\displaystyle=\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\mathbf{1}(m_{1}\neq\cdots\neq
m_{R-t}\in[R]\setminus A(\mathbf{j}),\ l_{1}\neq\cdots\neq
l_{R-t}\in[R]\setminus A(\mathbf{k}))$
$\displaystyle\qquad\times\left[\sum_{\begin{subarray}{c}1\leq
a_{1},\ldots,a_{t},s_{1,m_{1}},\ldots,s_{1,m_{R-t}}\leq N\\\ 1\leq
a_{1},\ldots,a_{t},s_{2,l_{1}},\ldots,s_{2,l_{R-t}}\leq
N\end{subarray}}\left\\{\prod_{r=1}^{t}\mathbf{1}(\alpha_{a_{r}}=u_{j_{r}})\prod_{x=1}^{R-t}\mathbf{1}(\alpha_{s_{1,m_{x}}}=u_{m_{x}})\mathbf{1}(\alpha_{s_{2,l_{x}}}=v_{l_{x}})\right\\}+O\left(N^{2R-t-1}\right)\right]$
$\displaystyle=\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\mathbf{1}(m_{1}\neq\cdots\neq
m_{R-t}\in[R]\setminus A(\mathbf{j}),\ l_{1}\neq\cdots\neq
l_{R-t}\in[R]\setminus A(\mathbf{k}))$
$\displaystyle\qquad\qquad\times\left\\{N^{2R-t}\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{x=1}^{R-t}\lambda_{N,u_{m_{x}}}\cdot\lambda_{N,v_{l_{x}}}+O(N^{2R-t-1})\right\\}$
$\displaystyle=\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\left\\{N^{2R-t}\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\cdot\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O(N^{2R-t-1})\right\\}$
$\displaystyle=N^{2R-t}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\left\\{\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\cdot\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O\left(\frac{1}{N}\right)\right\\}.$ (E.65)

Therefore, for $t\in\\{2,\ldots,R\\}$ and from (5.1.6) we say that,

$\displaystyle
0\leq\quad\frac{1}{N^{2R-1}}\sum_{(\mathbf{s}_{1},\mathbf{s}_{2})\in
E_{t}}\mathbf{cov}\left(\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{1}),\Psi_{H}(\mathbf{Y}_{N},\mathbf{s}_{2})\right)$
$\displaystyle\leq\frac{1}{N^{t-1}}\sum_{\mathbf{j},\mathbf{k}\in[R]_{t}}\sum_{\xi\in\mathcal{S}_{t}}\sum_{\mathbf{u},\mathbf{v}\in[K]^{R}}\prod_{r=1}^{t}\mathbf{1}(u_{j_{r}}=v_{k_{\xi(r)}})\left\\{\prod_{r=1}^{t}\lambda_{N,u_{j_{r}}}\prod_{j\in[R]\setminus
A(\mathbf{j})}\lambda_{N,u_{j}}\cdot\prod_{k\in[R]\setminus
A(\mathbf{k})}\lambda_{N,v_{k}}+O\left(\frac{1}{N}\right)\right\\}$
$\displaystyle\to 0,$

due to assumptions (A.4) and (A.2). Hence the result follows. ∎

####  5.1.7 Statement and proof of Lemma 5.2: expressing
$\widetilde{\Psi}_{H_{1}\cup H_{2}}(\cdot,\mathbf{s}_{1\cup 2})$ in terms of
$\Psi_{H}(\cdot,\mathbf{s}_{1})$ and $\Psi_{H}(\cdot,\mathbf{s}_{2})$

######  Lemma 5.2.

Let $H$ be a fixed, undirected, simple and connected graph with vertex set
$V(H)=[R]$. Consider a graph $H_{i}$ with vertex set
$V(H_{i})=A(\mathbf{s}_{i})$ and $H_{i}\simeq H$ for $i=1,2$. Then

$\displaystyle\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right)\cdot\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)=\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\mathbf{Y}_{N},\mathbf{s}_{1\cup 2}\right),$ (E.66)

where, $\mathbf{s}_{1\cup 2}$ defined in (E.8) and
$\Psi_{\cdot}(\cdot,\cdot),\widetilde{\Psi}_{\cdot}(\cdot,\cdot)$ defined as
per (E.10) and (E.11) respectively.

###### Proof of Lemma 5.2.

If $H_{1}$ and $H_{2}$ are disjoint, then $\mathbf{s}_{1\cup
2}=(\mathbf{s}_{1},\mathbf{s}_{2})$ from the definition (E.8), Then,

$\displaystyle\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\mathbf{Y}_{N},\mathbf{s}_{1\cup
2}\right)=\prod_{\begin{subarray}{c}\\{i,j\\}\in E(G)\\\ G\simeq H_{1}\cup
H_{2}\end{subarray}}Y_{s_{1\cup 2,i},s_{1\cup 2,j}}=\prod_{\\{i,j\\}\in
E(H_{1}\cup H_{2})}Y_{i,j}$ $\displaystyle=\prod_{\\{i,j\\}\in
E(H_{1})}Y_{i,j}\prod_{\\{i,j\\}\in E(H_{2})}Y_{i,j}=\prod_{\\{i,j\\}\in
H}Y_{s_{1,i},s_{1,j}}\prod_{\\{i,j\\}\in
H}Y_{s_{2,i},s_{2,j}}=\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right)\cdot\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right)$

If $H_{1}$ and $H_{2}$ are not disjoint, that is $|A(\mathbf{s}_{1})\cap
A(\mathbf{s}_{2})|=R_{1\cap 2}>0$ and note that, $R_{1\cup 2}=2R-R_{1\cap 2}$.
Then, $\mathbf{s}_{1\cup 2}=(s_{1\cup 2,1},\ldots,s_{1\cup 2,R_{1\cup
2}})^{T}={\left(\mathbf{s}^{T}_{1},{\left((\mathbf{s}_{2})_{\mathbf{m}}\right)}^{T}\right)}^{T}$
where, $\mathbf{m}=\left(m_{1},\ldots,m_{R-R_{1\cap 2}}\right)^{T}$, with
components arranged in increasing order, such that,
$A(\mathbf{m})=\\{k\in[R]:s_{2,k}\not\in A(\mathbf{s}_{1})\\}$. Therefore,

$\displaystyle\widetilde{\Psi}_{H_{1}\cup
H_{2}}\left(\mathbf{Y}_{N},\mathbf{s}_{1\cup
2}\right)=\prod_{\begin{subarray}{c}\\{i,j\\}\in E(G)\\\ G\simeq H_{1}\cup
H_{2}\end{subarray}}Y_{s_{1\cup 2,i},s_{1\cup 2,j}}$

We define the graph $G$ with vertex set $V(G)=[R_{1\cup 2}]$ and the
isomorphism $\phi\colon V(G)\to V(H_{1}\cup H_{2})$. From Remark 7, we can
choose any isomorphic map $\phi^{-1}$, which can generate different $G$.
Therefore, without loss of generality, we define,

$\displaystyle\phi(i)=\begin{cases}s_{1\cup 2,i}=s_{1,i},&\quad\text{if $1\leq
i\leq R$,}\\\ s_{1\cup 2,i}=s_{2,i-R},&\quad\text{if $R+1\leq i\leq R_{1\cup
2}$}\end{cases}$

Based on this one can obtain,

$\displaystyle\prod_{\begin{subarray}{c}\\{i,j\\}\in E(G)\\\ G\simeq H_{1}\cup
H_{2}\end{subarray}}Y_{s_{1\cup 2,i},s_{1\cup
2,j}}=\prod_{\begin{subarray}{c}\\{i,j\\}\in E(G)\\\ 1\leq i,j\leq
R\end{subarray}}Y_{s_{1\cup 2,i},s_{1\cup
2,j}}\prod_{\begin{subarray}{c}\\{i,j\\}\in E(G)\\\ 1\leq i\leq R,\ R<j\leq
R_{1\cup 2}\end{subarray}}Y_{s_{1\cup 2,i},s_{1\cup
2,j}}\prod_{\begin{subarray}{c}\\{i,j\\}\in E(G)\\\ R<i,j\leq R_{1\cup
2}\end{subarray}}Y_{s_{1\cup 2,i},s_{1\cup 2,j}}$

Now, if $\\{s_{1\cup 2,i},s_{1\cup 2,j}\\}\in E(H_{1}\cup H_{2})$ with $1\leq
i,j\leq R$, then $\\{s_{1,i},s_{1,j}\\}\in E(H_{1})$, if $R<i,j\leq R_{1\cup
2}$, then $\\{s_{2,i},s_{2,j}\\}\in E(H_{2})$. if $\\{s_{1\cup 2,i},s_{1\cup
2,j}\\}\in E(H_{1}\cup H_{2})$ with $1\leq i\leq R,\ R<j\leq R_{1\cup 2}$,
then there exist some $s_{2,i}\in A(\mathbf{s}_{1\cup 2})\setminus
A((\mathbf{s}_{2})_{\mathbf{m}})$ and $s_{2,l}\in
A((\mathbf{s}_{2})_{\mathbf{m}})$ where, $l=j-R$, such that
$\\{s_{2,i},s_{2,l}\\}\in E(H_{2})$. Therefore,

$\displaystyle\prod_{\begin{subarray}{c}\\{i,j\\}\in E(G)\\\ G\simeq H_{1}\cup
H_{2}\end{subarray}}Y_{s_{1\cup 2,i},s_{1\cup
2,j}}=\left\\{\prod_{\begin{subarray}{c}\\{i,j\\}\in E(G)\\\ 1\leq i,j\leq
R\end{subarray}}Y_{s_{1,i},s_{1,j}}\right\\}\left\\{\prod_{\begin{subarray}{c}\\{i,j\\}\in
E(G)\\\ 1\leq i\leq R,\ R<j\leq R_{1\cup
2}\end{subarray}}Y_{s_{2,i},s_{2,j-R}}\prod_{\begin{subarray}{c}\\{i,j\\}\in
E(G)\\\ R<i,j\leq R_{1\cup 2}\end{subarray}}Y_{s_{2,i},s_{2,j}}\right\\}$
$\displaystyle=\left\\{\prod_{\\{i,j\\}\in
E(H_{1})}Y_{i,j}\right\\}\left\\{\prod_{\\{i,j\\}\in
E(H_{2})}Y_{i,j}\right\\}=\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{1}\right)\cdot\Psi_{H}\left(\mathbf{Y}_{N},\mathbf{s}_{2}\right).$

Hence the proof. ∎

####  5.1.8 Statement and proof of Lemma 5.3: showing that asymptotic
variances are strictly positive

######  Lemma 5.3.

Consider the asymptotic variance expressions
${\left(\sigma^{[1]}(H,p)\right)}^{2}$ and
${\left(\sigma^{[2]}(H,p)\right)}^{2}$ obtained in Theorems 3.1 and 3.3
respectively for the induced and ego-centric sampling cases. Suppose,
assumptions (A.1)-(A.4) hold. Then, $\sigma^{[i]}(H,p)>0$, for $i=1,2$, for
any fixed, connected, simple, undirected graph $H$ and any $p\in(0,1)$.

###### Proof of Lemma 5.3.

Consider $\sigma^{[1]}(H,p)$ (see (3.10)). As per assumption (A.2), there
exists some $j_{0}\in\\{1,\ldots,K\\}$, such that $\pi_{j_{0},j_{0}}>0$. Also
note assumption (A.3) implies $\lambda_{j}\in(0,1)$, for each
$j\in\\{1,\ldots,K\\}$ and (3.7) implies $\Psi_{H}(\bm{\Pi},\mathbf{u})\geq 0$
for any $\mathbf{u}\in{[K]}^{R}$ and any graph $H$. If we ignore the terms
involving $p$ in the expression for ${\left(\sigma^{[1]}(H,p)\right)}^{2}$ in
(3.10), then

$\displaystyle\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{k=1}^{R}\left\\{\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}{\Psi}_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right\\}\right]^{2}$
$\displaystyle\geq\lambda_{j_{0}}{\left[\sum_{k=1}^{R}\left\\{{\Psi}_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}((j_{0},j_{0},\cdots,j_{0})^{T})\right)\cdot\lambda^{R-1}_{j_{0}}\right\\}\right]}^{2},\quad\text{(with
$u_{2}=\cdots=u_{R}=j_{0}=u_{1}$),}$
$\displaystyle\geq\lambda_{j_{0}}{\left[\left\\{{\Psi}_{H}\left(\bm{\Pi},\mathbf{v}^{[1]}((j_{0},j_{0},\cdots,j_{0})^{T})\right)\cdot\lambda^{R-1}_{j_{0}}\right\\}\right]}^{2}=\lambda_{j_{0}}{\left[{\Psi}_{H}\left(\bm{\Pi},(j_{0},j_{0},\cdots,j_{0})^{T}\right)\cdot\lambda^{R-1}_{j_{0}}\right]}^{2}$
$\displaystyle=\lambda^{2R-1}_{j_{0}}\cdot\prod_{\\{i,j\\}\in
E(H)}\pi^{2}_{u_{i},u_{j}},\quad\text{with
$\mathbf{u}={(u_{1},\ldots,u_{R})}^{T}={(j_{0},\cdots,j_{0})}^{T}$,}$
$\displaystyle=\lambda^{2R-1}_{j_{0}}\cdot\pi^{2|E(H)|}_{j_{0},j_{0}}>0,$
(E.67)

as per the given assumptions, implying that the $\sigma^{[1]}(H,p)>0$.

Now consider the variance expression ${\left(\sigma^{[2]}(H,p)\right)}^{2}$ in
(3.13). If we ignore the $p(1-p)/{(f(H,p))}^{2}$ factor (which is always
positive), then as shown above we can write

$\displaystyle\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{k=1}^{R}\delta_{k}(H,p)\cdot\left\\{\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}{\Psi}_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right\\}\right]^{2}$
$\displaystyle\geq\lambda_{j_{0}}{\left[\sum_{k=1}^{R}\delta_{k}(H,p)\cdot\sum_{{(u_{2},\ldots,u_{R})^{T}\in{[K]}^{R-1}}}\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right]}^{2}$
$\displaystyle\geq\lambda^{2R-1}_{j_{0}}{\left[\sum_{k=1}^{R}\delta_{k}(H,p)\cdot\Psi_{H}\left(\bm{\Pi},\mathbf{v}^{[k]}\left((j_{0},\ldots,j_{0})^{T}\right)\right)\right]}^{2},\quad\text{(using
$u_{2}=\cdots=u_{R}=j_{0}=u_{1}$),}$
$\displaystyle=\lambda^{2R-1}_{j_{0}}\cdot\pi^{2|E(H)|}_{j_{0},j_{0}}\cdot{\left[\sum_{k=1}^{R}\delta_{k}(H,p)\right]}^{2}.$
(E.68)

In case there exists a $r\in V(H)$ such that $\mathcal{A}^{c}(r)=\emptyset$
(cf. (3.12)), then $\delta_{r}(H,p)=\left[1-\mathbf{E}\left(\prod_{v(\neq
r)=1}^{R}\widetilde{W}_{v}\right)\right]=1-\prod_{v\neq
r}\mathbf{P}(\widetilde{W}_{v}=1)=1-p^{R-1}>0$, whenever $p\in(0,1)$. Using
this lower bound in (E.68) we would obtain,
${\left[\sum_{k=1}^{R}\delta_{k}(H,p)\right]}^{2}\geq{\left[\delta_{r}(H,p)\right]}^{2}\geq
1-p^{R-1}>0$, ensuring that $\sigma^{[2]}(H,p)>0$.

Now consider the case where $\mathcal{A}^{c}(r)\neq\emptyset$, for all $r\in
V(H)$. As $H$ is connected this implies $T=|E(H)|\geq 2$. Without loss of
generality, fix $r=1$. Now, there must exist some $v_{0}\in\mathcal{N}(1)$,
such that $v_{0}\in\cup_{t\in\mathcal{A}^{c}(1)}C_{t}$. If not, then we can
partition the vertex set $V(H)$ into two disjoint (non-empty) parts,
$\left(\\{1\\}\cup\mathcal{N}(1)\right)$ and
$\cup_{t\in\mathcal{A}^{c}(1)}C_{t}$, with vertices in the first partition
connected only with vertex $1$, with no edges among themselves, thus forming a
star graph centered at vertex $1$. The vertices in the second partition have
edges among themselves, but have no edges with vertex $1$, and there are no
edges with any other vertex in $\mathcal{N}(1)$. This implies that $H$ is a
disconnected graph, which is a contradiction to the original assumption about
$H$. Then, with $C_{t}=\\{i_{t},j_{t}\\}$, for each $t\in[T]$, we can write

$\displaystyle\delta_{1}(H,p)=\mathbf{P}\left(\prod_{t\in\mathcal{A}^{c}(1)}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}=1,~{}\prod_{v\in\mathcal{N}(1)}\widetilde{W}_{v}=0\right)$
$\displaystyle=\mathbf{P}\left(\prod_{t\in\mathcal{A}^{c}(1)}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}=1,~{}\bigcup_{v\in\mathcal{N}(1)}\left[\widetilde{W}_{v}=0\right]\right)$
$\displaystyle\geq\mathbf{P}\left(\prod_{t\in\mathcal{A}^{c}(1)}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}=1,\widetilde{W}_{v_{0}}=0\right),\quad\text{where
$v_{0}\in\mathcal{N}(1)\cap\cup_{t\in\mathcal{A}^{c}(1)}C_{t}$,}$
$\displaystyle=\mathbf{P}\left(\prod_{t\in\mathcal{A}^{c}(1):v_{0}\in
C_{t}}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}=1,~{}\prod_{t\in\mathcal{A}^{c}(1):v_{0}\not\in
C_{t}}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}=1,\widetilde{W}_{v_{0}}=0\right)$
$\displaystyle=\mathbf{P}\left(\widetilde{W}_{v_{0}}+\left(1-\widetilde{W}_{v_{0}}\right)\cdot\prod_{r\in\mathcal{N}(v_{0})\setminus\\{1\\}}\widetilde{W}_{r}=1,\prod_{t\in\mathcal{A}^{c}(1):v_{0}\not\in
C_{t}}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}=1,\widetilde{W}_{v_{0}}=0\right)$
$\displaystyle=\mathbf{P}\left(\prod_{r\in\mathcal{N}(v_{0})\setminus\\{1\\}}\widetilde{W}_{r}=1,\prod_{t\in\mathcal{A}^{c}(1):v_{0}\not\in
C_{t}}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}=1,\widetilde{W}_{v_{0}}=0\right).$

Note that we can partition

$V(H)\setminus\\{v_{0}\\}=\\{1\\}\bigcup\left(\mathcal{N}(v_{0})\setminus\\{1\\}\right)\bigcup\left\\{v\in
V(H):\not\exists\ \text{$t\in[T]$, such that $C_{t}=\\{v_{0},v\\}$}\right\\}.$

Also note that,
$\max\left\\{\widetilde{W}_{i},\widetilde{W}_{j}\right\\}\geq\widetilde{W}_{i}\widetilde{W}_{j}$,
for any $i\neq j$. Thus, with $C_{t}=\\{i_{t},j_{t}\\}$,

$\displaystyle\prod_{t\in\mathcal{A}^{c}(1):v_{0}\not\in
C_{t}}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}\geq\prod_{t\in\mathcal{A}^{c}(1):v_{0}\not\in
C_{t}}\widetilde{W}_{i_{t}}\widetilde{W}_{j_{t}}=\prod_{u\not\in\mathcal{N}(v_{0})}\widetilde{W}_{u}$
$\displaystyle\Rightarrow\ $
$\displaystyle\left[\prod_{t\in\mathcal{A}^{c}(1):v_{0}\not\in
C_{t}}\max\left\\{\widetilde{W}_{i_{t}},\widetilde{W}_{j_{t}}\right\\}=1\right]\supseteq\left[\prod_{u\not\in\mathcal{N}(v_{0})}\widetilde{W}_{u}=1\right].$

This implies,

$\displaystyle\delta_{1}(H,p)$
$\displaystyle\geq\mathbf{P}\left(\prod_{r\in\mathcal{N}(v_{0})\setminus\\{1\\}}\widetilde{W}_{r}=1,\prod_{t\in\mathcal{A}^{c}(1):v_{0}\not\in
C_{t}}\max\left\\{\widetilde{W}_{i},\widetilde{W}_{j}:i,j\in
C_{t}\right\\}=1,\widetilde{W}_{v_{0}}=0\right)$
$\displaystyle\geq\mathbf{P}\left(\prod_{r\in\mathcal{N}(v_{0})\setminus\\{1\\}}\widetilde{W}_{r}=1,\prod_{u\not\in\mathcal{N}(v_{0})}\widetilde{W}_{u}=1,\widetilde{W}_{v_{0}}=0\right)$
$\displaystyle=p^{|\mathcal{N}(v_{0})|-1}\cdot
p^{R-|\mathcal{N}(v_{0})|}\cdot(1-p)=p^{R-1}(1-p_{0})>0,$

which in turn ensures that the lower bound in (E.68) is strictly positive. ∎

####  5.1.9 Some miscellaneous results: Lemma 5.4, Lemma 5.5 and Lemma 5.6

######  Lemma 5.4 (Lemma 11.3.1 of Lehmann and Romano,  (2022)).

Consider two sequences of random variables $\\{T_{N}:N\geq 1\\}$ and
$\\{\widetilde{T}_{N}:N\geq 1\\}$, which satisfy,
$\left(\widetilde{T}_{N}-\mathbf{E}(\widetilde{T}_{N})\right)/\sqrt{\mathbf{Var}(\widetilde{T}_{N})}\stackrel{{\scriptstyle
d}}{{\rightarrow}}T$, and
$\mathbf{E}{(T_{N}-\widetilde{T}_{N})}^{2}=o\left(\mathbf{Var}(\widetilde{T}_{N})\right)$,
as $N\rightarrow\infty$. Then,
$(T_{N}-\mathbf{E}(T_{N}))/{\sqrt{\mathbf{Var}(T_{N})}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}T$.

###### Proof of Lemma 5.4.

We can write,

$\displaystyle\frac{T_{N}-\mathbf{E}(T_{N})}{\sqrt{\mathbf{Var}(T_{N})}}$
$\displaystyle=\frac{\widetilde{T}_{N}-\mathbf{E}(\widetilde{T}_{N})}{\sqrt{\mathbf{Var}(\widetilde{T}_{N})}}\cdot\sqrt{\frac{\mathbf{Var}(\widetilde{T}_{N})}{\mathbf{Var}(T_{N})}}+\frac{T_{N}-\widetilde{T}_{N}}{\sqrt{\mathbf{Var}(\widetilde{T}_{N})}}\sqrt{\frac{\mathbf{Var}(\widetilde{T}_{N})}{\mathbf{Var}(T_{N})}}+\frac{\mathbf{E}(\widetilde{T}_{N}-T_{N})}{\sqrt{\mathbf{Var}(\widetilde{T}_{N})}}\sqrt{\frac{\mathbf{Var}(\widetilde{T}_{N})}{\mathbf{Var}(T_{N})}}$
$\displaystyle=H_{N,1}+H_{N,2}+H_{N,3},\quad\text{(say).}$ (E.69)

Now, from the given condition
$\mathbf{E}(T_{N}-\widetilde{T}_{N})^{2}=o\left(\mathbf{Var}\left(\widetilde{T}_{N}\right)\right)$
as $N\to\infty$, we can say that,

$\displaystyle\mathbf{Var}\left(T_{N}-\widetilde{T}_{N}\right)\leq\mathbf{E}(T_{N}-\widetilde{T}_{N})^{2}=o\left(\mathbf{Var}\left(\widetilde{T}_{N}\right)\right).$
(E.70)

Using (E.70) and Cauchy-Schwartz inequality,

$\displaystyle\left(\sqrt{\frac{\mathbf{Var}(\widetilde{T}_{N})}{\mathbf{Var}(T_{N})}}-1\right)^{2}$
$\displaystyle=1+\frac{\mathbf{Var}(T_{N})}{\mathbf{Var}(\widetilde{T}_{N})}-\frac{[2\mathbf{Var}(T_{N})\mathbf{Var}(\widetilde{T}_{N})]^{1/2}}{\mathbf{Var}(\widetilde{T}_{N})}$
$\displaystyle\leq
1+\frac{\mathbf{Var}(T_{N})}{\mathbf{Var}(\widetilde{T}_{N})}-\frac{2\mathbf{cov}(T_{N},\widetilde{T}_{N})}{\mathbf{Var}(\widetilde{T}_{N})}=\frac{\mathbf{Var}(\widetilde{T}_{N}-T_{N})}{\mathbf{Var}(\widetilde{T}_{N})}=o(1).$
(E.71)

Moreover, using Jensen’s inequality, we can say that,

$\displaystyle{[\mathbf{E}(T_{N}-\widetilde{T}_{N})]}^{2}\leq\mathbf{E}(T_{N}-\widetilde{T}_{N})^{2}=o\left(\mathbf{Var}\left(\widetilde{T}_{N}\right)\right)\quad\Rightarrow\quad\mathbf{E}(T_{N}-\widetilde{T}_{N})=o\left(\sqrt{\mathbf{Var}\left(\widetilde{T}_{N}\right)}\right).$
(E.72)

Now, using (5.1.9) and Slutsky’s theorem,

$\displaystyle
H_{N,1}=\frac{\widetilde{T}_{N}-\mathbf{E}(\widetilde{T}_{N})}{\sqrt{\mathbf{Var}(\widetilde{T}_{N})}}\cdot\sqrt{\frac{\mathbf{Var}(\widetilde{T}_{N})}{\mathbf{Var}(T_{N})}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}T.$

Also, using (E.72) and (5.1.9), as $N\to\infty$, $H_{N,3}=o(1)$. Now, we show
that
$(T_{N}-\widetilde{T}_{N})/\sqrt{\mathbf{Var}\left(\widetilde{T}_{N}\right)}=o_{p}(1)$
using (E.70) and Chebyshev’s inequality. For all $\varepsilon>0$,

$\displaystyle\mathbf{P}\left(\frac{\left|T_{N}-\widetilde{T}_{N}\right|}{\sqrt{\mathbf{Var}\left(\widetilde{T}_{N}\right)}}>\varepsilon\right)\leq\frac{1}{\varepsilon^{2}}\frac{\mathbf{Var}\left(T_{N}-\widetilde{T}_{N}\right)}{\mathbf{Var}\left(\widetilde{T}_{N}\right)}=o(1),\quad\text{as
$N\to\infty$.}$

Hence, using (5.1.9), we can say that, $H_{N,2}=o_{p}(1)$. Therefore using the
above steps and Slutsky’s theorem, we can say that, as $N\to\infty$,

$\displaystyle\frac{T_{N}-\mathbf{E}(T_{N})}{\sqrt{\mathbf{Var}(T_{N})}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}T$

∎

######  Lemma 5.5 (A CLT for weighted sums of independent Bernoulli random
variables).

Consider a triangular array of non-random weights,
$\\{a_{i,N}:i=1,\ldots,N:~{}N\geq 1\\}$. Let $\\{X_{1,N},\ldots,X_{N,N}:N\geq
1\\}$ be a triangular array of independent Bernoulli random variables, with
$\mathbf{E}(X_{i,N})=p_{i,N}$, for each $i=1,\ldots,N$, and all $N\geq 1$.
Define,
$\sigma^{2}_{N}=\mathbf{Var}\left(\sum_{i=1}^{N}a_{i,N}X_{i,N}\right)=\sum_{i=1}^{N}a^{2}_{i,N}\cdot
p_{i,N}\\{1-p_{i,N}\\}$, for each $N\geq 1$. Assume,

$\displaystyle\sigma_{N}^{2}\to\sigma^{2}\in(0,\infty)\quad\text{and}\quad\max_{1\leq
i\leq N}|a_{i,N}|\to 0.$ (E.73)

Then,

$\displaystyle\frac{\sum_{i=1}^{N}a_{i,N}(X_{i,N}-p_{i,N})}{\sigma_{N}}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N(0,1).$

###### Proof of Lemma 5.5.

It is enough to verify the Lindeberg-Feller criterion for CLT. Fix any
$\epsilon>0$. Then, on the basis of the conditions provided in (E.73), we can
claim that there exists a $N_{0}(\epsilon)\in\mathbb{N}$, such that

$\sigma_{N}>\frac{\sigma}{2}\quad\text{and}\quad\frac{\epsilon\sigma}{2\max_{1\leq
j\leq N}|a_{j,N}|}>2,\quad\text{for all $N\geq N_{0}$.}$

Thus, for $N\geq N_{0}$ and for each $i=1,\ldots,N$, we obtain

$\mathbf{1}\left(|a_{i,N}|\cdot|X_{i,N}-p_{i,N}|>\epsilon\cdot\sigma_{N}\right)\leq\mathbf{1}\left(|X_{i,N}-p_{i,N}|>\frac{\epsilon\sigma}{2\cdot\max_{1\leq
j\leq N}|a_{j,N}|}\right)=0,$

since $|X_{i,N}-p_{i,N}|\leq 2$, w.p. $1$, for each $i=1,\ldots,N$, and $N\geq
1$. This implies,

$\displaystyle\frac{1}{\sigma_{N}^{2}}\sum_{i=1}^{N}\mathbf{E}\left[a^{2}_{i,N}\cdot{\left(X_{i,N}-p_{i,N}\right)}^{2}\cdot\mathbf{1}\left(|a_{i,N}|\cdot|X_{i,N}-p_{i,N}|>\epsilon\sigma_{N}\right)\right]\rightarrow
0,\quad\text{as $N\rightarrow\infty$.}$

This verifies the Lindeberg-Feller criterion for CLT for sums of independent
random variables and completes the proof. ∎

######  Lemma 5.6.

Assume $F$ and $G$ are both cumulative distribution functions (cdf’s) on
$\mathbb{R}$, with, $G(x)=\int_{-\infty}^{x}g(u)~{}du$, for each
$x\in\mathbb{R}$, where $g(\cdot)$ is a probability density function (pdf).
Then,

$\displaystyle\sup_{x\in\mathbb{R}}|F(x)-G(x)|=\sup_{x\in\mathbb{R}}|F(x^{-})-G(x^{-})|,$

where, $F(x^{-})=\lim_{h\rightarrow 0}F(x-h)$ and $G(x^{-})$ is also defined
similarly.

###### Proof of Lemma 5.6.

Firstly we establish uniform continuity of $G(\cdot)$. Fix $\epsilon>0$, we
need to show that there exists a $\delta>0$, such that $|x-y|<\delta$, will
imply $|G(x)-G(y)|<\epsilon$. Let $x<y$, with $(y-x)<\delta$. Note that,

$\displaystyle|G(y)-G(x)|$
$\displaystyle=\int_{x}^{y}g(u)~{}du=\int_{x}^{y}g(u)\cdot\mathbf{1}\left(g(u)>M\right)~{}du+\int_{x}^{y}g(u)\cdot\mathbf{1}\left(g(u)\leq
M\right)~{}du,$

for any $M>0$, which is to be chosen later. As $g$ is integrable, hence by
DCT, there exists a $M_{\epsilon}>0$, such that

$\int_{\mathbb{R}}g(u)\cdot\mathbf{1}(g(u)>M_{\epsilon})~{}du<\frac{\epsilon}{2},$

where the choice of $M_{\epsilon}$ does not depend on $x$ and $y$, and only
depends on $\epsilon$. Thus, with this choice of $M=M_{\epsilon}$, we obtain

$|G(y)-G(x)|<\frac{\epsilon}{2}+M_{\epsilon}\cdot|y-x|\leq\frac{\epsilon}{2}+M_{\epsilon}\cdot\delta.$

Choose, $\delta=\epsilon/(2M_{\epsilon})$. This leads to

$G(y)-G(x)<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.$

As $x$ and $y$ are arbitrary and $\delta$ depends only on $\epsilon$, we have
established uniform continuity of $G$.

Let $X$ and $Y$ denote the underlying random variables corresponding to the
cdf’s $F$ and $G$ respectively. Then for each $\epsilon>0$,

$F(x^{-})=\mathbf{P}(X<x)\leq
F(x)\leq\mathbf{P}(X<x+\epsilon)=F({(x+\epsilon)}^{-}).$

Uniform continuity implies continuity, i.e., $G(x^{-})=\lim_{h\rightarrow
0}G(x-h)=G(x)$. Write

$\Delta=\sup_{x}|F(x^{-})-G(x^{-})|.$

Then,

$\displaystyle F(x^{-})-G(x)\leq F(x)-G(x)\leq F({(x+\epsilon)}^{-})-G(x)$
$\displaystyle\Rightarrow$ $\displaystyle\ -|F(x^{-})-G(x)|\leq
F(x^{-})-G(x)\leq F(x)-G(x)\leq
F({(x+\epsilon)}^{-})-G(x+\epsilon)+G(x+\epsilon)-G(x)$
$\displaystyle\Rightarrow$ $\displaystyle\ -\Delta\leq
F(x)-G(x)\leq\sup_{t}|F(t^{-})-G(t^{-})|+\sup_{t}|G(t+\epsilon)-G(t)|=\Delta+\Delta_{1,\epsilon}\quad\text{(say),}$
$\displaystyle\Rightarrow$ $\displaystyle\
-\Delta-\Delta_{1,\epsilon}\leq-\Delta\leq
F(x)-G(x)\leq\Delta+\Delta_{1,\epsilon},\quad\text{for all $x$ and all
$\epsilon>0$,}$ $\displaystyle\Rightarrow$ $\displaystyle\
\sup_{x}|F(x)-G(x)|\leq\Delta+\Delta_{1,\epsilon}.$

As $G$ is uniformly continuous, letting $\epsilon\downarrow 0$,
$\Delta_{1,\epsilon}\rightarrow 0$. This implies,

$\sup_{x}|F(x)-G(x)|\leq\Delta.$

To show the reverse inequality, we write for each $x$ and $\epsilon>0$,

$\displaystyle\mathbf{P}(X\leq
x-\epsilon)\leq\mathbf{P}(X<x)\leq\mathbf{P}(X\leq x)\leq\mathbf{P}(X\leq
x+\epsilon)$ $\displaystyle\Rightarrow$ $\displaystyle\ F(x-\epsilon)\leq
F(x^{-})\leq F(x+\epsilon)$ $\displaystyle\Rightarrow$ $\displaystyle\
F(x-\epsilon)-G(x)\leq F(x^{-})-G(x)\leq F(x+\epsilon)-G(x)$
$\displaystyle\Rightarrow$ $\displaystyle\
F(x-\epsilon)-G(x-\epsilon)+G(x-\epsilon)-G(x)\leq F(x^{-})-G(x)\leq
F(x+\epsilon)-G(x+\epsilon)+G(x+\epsilon)-G(x)$ $\displaystyle\Rightarrow$
$\displaystyle\ -\sup_{t}|F(t)-G(t)|-\sup_{t}|G(t-\epsilon)-G(t)|\leq
F(x^{-})-G(x)\leq\sup_{t}|F(t)-G(t)|+\sup_{t}|G(t+\epsilon)-G(t)|$
$\displaystyle\Rightarrow$ $\displaystyle\
-\sup_{t}|F(t)-G(t)|-\Delta_{2,\epsilon}\leq
F(x^{-})-G(x^{-})\leq\sup_{t}|F(t)-G(t)|+\Delta_{1,\epsilon}\quad\text{(say),}$
$\displaystyle\Rightarrow$ $\displaystyle\
-\sup_{t}|F(t)-G(t)|-\Delta_{\epsilon}\leq
F(x^{-})-G(x^{-})\leq\sup_{t}|F(t)-G(t)|+\Delta_{\epsilon},\quad\text{where
$\Delta_{\epsilon}=\max\\{\Delta_{1,\epsilon},\Delta_{2,\epsilon}\\}$}$
$\displaystyle\Rightarrow$ $\displaystyle\
\Delta=\sup_{x}|F(x^{-})-G(x^{-})|\leq\sup_{t}|F(t)-G(t)|+\Delta_{\epsilon},$
$\displaystyle\Rightarrow$ $\displaystyle\
\Delta\leq\sup_{x}|F(x)-G(x)|,\quad\text{as $\Delta_{\epsilon}\rightarrow 0$
when $\epsilon\downarrow 0$.}$

This establishes the reverse inequality, indicating the equality of both
suprema. ∎

###  5.2 Appendix B: Some additional theoretical results

####  5.2.1 Corollary 5.7: limit law for complete and star graphs in the
induced subgraph case

######  Corollary 5.7 (Limit law for $\mathbb{K}_{R}$ and $\mathbb{K}_{1,R-1}$
under induced sampling).

Suppose, the conditions of Theorem 3.1 holds.

  1. (a)

When $H=\mathbb{K}_{R}$ (complete graph on $R$ vertices), then

$\displaystyle
N^{-R+1/2}\left(\frac{\widehat{S}^{[1]}_{N}(\mathbb{K}_{R})}{p^{R}}-S_{N}(\mathbb{K}_{R})\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left(\sigma^{[1]}(\mathbb{K}_{R},p)\right)^{2}\right),\quad\text{where,}$
$\displaystyle\left(\sigma^{[1]}(\mathbb{K}_{R},p)\right)^{2}=R^{2}\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left(\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\Psi_{\mathbb{K}_{R}}(\bm{\Pi},\mathbf{u})\cdot\prod_{i=2}^{R}\lambda_{u_{i}}\right)^{2}.$

  2. (b)

When $H=\mathbb{K}_{1,R-1}$ ($(R-1)$-star graph on $R$ vertices), then

$\displaystyle
N^{-R+1/2}\left(\frac{\widehat{S}^{[1]}_{N}(\mathbb{K}_{1,R-1})}{p^{R}}-S_{N}(\mathbb{K}_{1,R-1})\right)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left(\sigma^{[1]}(\mathbb{K}_{1,R-1},p)\right)^{2}\right),\quad\text{where,}$
$\displaystyle{\left(\sigma^{[1]}(\mathbb{K}_{1,R-1},p)\right)}^{2}$
$\displaystyle=\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\left(\prod_{l(\neq
1)=1}^{R}\pi_{u_{1},u_{l}}+(R-1)\prod_{l(\neq
2)=1}^{R}\pi_{u_{2},u_{l}}\right)\cdot\prod_{i=2}^{R}\lambda_{u_{i}}\right]^{2}.$

###### Proof of Corollary 5.7.

This corollary is the direct application of the Theorem 3.1. Here, we need to
find the explicit expression of variance for particular choices of $H$, that
is complete graph or $\mathbb{K}_{R}$ or $R$-star graph or $\mathbb{K}_{1,R}$.

  1. (a)

When $H=\mathbb{K}_{R}$ with $|V(\mathbb{K}_{R})|=R$, then, from (3.13)

$\displaystyle\left(\sigma^{[1]}(\mathbb{K}_{R},p)\right)^{2}=\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\left(\sum_{k=1}^{R}\Psi_{\mathbb{K}_{R}}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right]^{2}.$

Now, in the above expression, for $k\in[R]$

$\displaystyle\Psi_{\mathbb{K}_{R}}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)=\prod_{\\{i,j\\}\in
E(\mathbb{K}_{R})}\pi_{u_{\rho_{k}(i)},u_{\rho_{k}(j)}}=\prod_{i=1}^{R}\prod_{j=i}^{R}\pi_{u_{\rho_{k}(i)},u_{\rho_{k}(j)}}$
$\displaystyle=\left\\{\prod_{j=1}^{R}\pi_{u_{\rho_{k}(1)},u_{\rho_{k}(j)}}\right\\}\left\\{\prod_{i=2}^{k-1}\prod_{j=i}^{R}\pi_{u_{\rho_{k}(i)},u_{\rho_{k}(j)}}\right\\}\left\\{\prod_{j=k}^{R}\pi_{u_{\rho_{k}(k)},u_{\rho_{k}(j)}}\right\\}\left\\{\prod_{i=k+1}^{R}\prod_{j=i}^{R}\pi_{u_{\rho_{k}(i)},u_{\rho_{k}(j)}}\right\\}$
$\displaystyle=\left\\{\pi_{u_{k},u_{1}}\prod_{j=2}^{R}\pi_{u_{k},u_{j}}\right\\}\left\\{\prod_{i=2}^{k-1}\pi_{u_{i},u_{1}}\prod_{j(\neq
k)=i}^{R}\pi_{u_{i},u_{j}}\right\\}\left\\{\pi_{u_{1},u_{1}}\prod_{j=k+1}^{R}\pi_{u_{1},u_{j}}\right\\}\left\\{\prod_{i=k+1}^{R}\prod_{j=i}^{R}\pi_{u_{i},u_{j}}\right\\}$
$\displaystyle=\left\\{\prod_{j=1}^{R}\pi_{u_{1},u_{j}}\right\\}\left\\{\prod_{j=2}^{R}\pi_{u_{k},u_{j}}\right\\}\left\\{\prod_{i=2}^{k-1}\prod_{j(\neq
k)=i}^{R}\pi_{u_{i},u_{j}}\right\\}\left\\{\prod_{i=k+1}^{R}\prod_{j=i}^{R}\pi_{u_{i},u_{j}}\right\\}$
$\displaystyle=\left\\{\prod_{j=1}^{R}\pi_{u_{1},u_{j}}\right\\}\left\\{\prod_{j=2}^{k-1}\pi_{u_{k},u_{j}}\prod_{j=k}^{R}\pi_{u_{k},u_{j}}\right\\}\left\\{\prod_{i=2}^{k-1}\prod_{j(\neq
k)=i}^{R}\pi_{u_{i},u_{j}}\right\\}\left\\{\prod_{i=k+1}^{R}\prod_{j=i}^{R}\pi_{u_{i},u_{j}}\right\\}$
$\displaystyle=\left\\{\prod_{j=1}^{R}\pi_{u_{1},u_{j}}\right\\}\left\\{\prod_{j=k}^{R}\pi_{u_{k},u_{j}}\right\\}\left\\{\prod_{i=2}^{k-1}\prod_{j=i}^{R}\pi_{u_{i},u_{j}}\right\\}\left\\{\prod_{i=k+1}^{R}\prod_{j=i}^{R}\pi_{u_{i},u_{j}}\right\\}$
$\displaystyle=\left\\{\prod_{j=1}^{R}\pi_{u_{1},u_{j}}\right\\}\left\\{\prod_{i=2}^{k}\prod_{j=i}^{R}\pi_{u_{i},u_{j}}\right\\}\left\\{\prod_{i=k+1}^{R}\prod_{j=i}^{R}\pi_{u_{i},u_{j}}\right\\}$
$\displaystyle=\prod_{i=1}^{R}\prod_{j=i}^{R}\pi_{u_{i},u_{j}}=\Psi_{H}\left(\bm{\Pi},\mathbf{u}\right).$
(E.74)

Therefore, from ((a)),

$\displaystyle\left(\sigma^{[1]}(\mathbb{K}_{R},p)\right)^{2}$
$\displaystyle=\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\left(\sum_{k=1}^{R}\Psi_{\mathbb{K}_{R}}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right]^{2}$
$\displaystyle=\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\left(\sum_{k=1}^{R}\Psi_{\mathbb{K}_{R}}\left(\bm{\Pi},\mathbf{u}\right)\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right]^{2}$
$\displaystyle=R^{2}\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\left(\Psi_{\mathbb{K}_{R}}\left(\bm{\Pi},\mathbf{u}\right)\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right]^{2}.$

  2. (b)

In the case of $(R-1)$-star graph or $H=\mathbb{K}_{1,R-1}$ with
$|V(\mathbb{K}_{1,R-1})|=R$, we get the following,

$\displaystyle\sum_{k=1}^{R}\Psi_{\mathbb{K}_{1,R-1}}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)=\Psi_{\mathbb{K}_{1,R-1}}\left(\bm{\Pi},\mathbf{v}^{[1]}(\mathbf{u})\right)+\sum_{k=2}^{R}\Psi_{\mathbb{K}_{1,R-1}}\left(\Pi,\mathbf{v}^{[k]}(\mathbf{u})\right)$
(E.75) $\displaystyle=\prod_{\\{i,j\\}\in
E(\mathbb{K}_{1,R-1})}\pi_{u_{\rho_{1}(i)},u_{\rho_{1}(j)}}+\sum_{k=2}^{R}\prod_{\\{i,j\\}\in
E(\mathbb{K}_{1,R-1})}\pi_{u_{\rho_{k}(i)},u_{\rho_{k}(j)}}$
$\displaystyle=\prod_{j=2}^{R}\pi_{u_{\rho_{1}(1)},u_{\rho_{1}(j)}}+\sum_{k=2}^{R}\prod_{j=2}^{R}\pi_{u_{\rho_{k}(1)},u_{\rho_{k}(j)}}=\prod_{j=2}^{R}\pi_{u_{1},u_{j}}+\sum_{k=2}^{R}\prod_{j=2}^{R}\pi_{u_{\rho_{k}(1)},u_{\rho_{k}(j)}}$
$\displaystyle=\prod_{j=2}^{R}\pi_{u_{\rho_{1}(1)},u_{\rho_{1}(j)}}+\sum_{k=2}^{R}\prod_{j=2}^{R}\pi_{u_{\rho_{k}(1)},u_{\rho_{k}(j)}}$
$\displaystyle=\prod_{j=2}^{R}\pi_{u_{1},u_{j}}+\sum_{k=2}^{R}\pi_{u_{\rho_{k}(1)},u_{\rho_{k}(2)}}\cdots\pi_{u_{\rho_{k}(1)},u_{\rho_{k}(k)}}\cdots\pi_{u_{\rho_{k}(1)},u_{\rho_{k}(R)}}$
$\displaystyle=\prod_{j=2}^{R}\pi_{u_{1},u_{j}}+\sum_{k=2}^{R}\pi_{u_{k},u_{2}}\cdots\pi_{u_{k},u_{1}}\cdots\pi_{u_{k},u_{R}}=\prod_{j=2}^{R}\pi_{u_{1},u_{j}}+\sum_{k=2}^{R}\prod_{j(\neq
k)=1}^{R}\pi_{u_{k},u_{j}}.$ (E.76)

Therefore, from the expression of variance defined in (3.13), we get

$\displaystyle\left(\sigma^{[1]}(\mathbb{K}_{1,R-1},p)\right)^{2}=\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\left(\sum_{k=1}^{R}\Psi_{\mathbb{K}_{1,R-1}}\left(\bm{\Pi},\mathbf{v}^{[k]}(\mathbf{u})\right)\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right]^{2}$
$\displaystyle=\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\left(\prod_{j=2}^{R}\pi_{u_{1},u_{j}}+\sum_{k=2}^{R}\prod_{j(\neq
k)=1}^{R}\pi_{u_{k},u_{j}}\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right]^{2}$
$\displaystyle=\left(\frac{1}{p}-1\right)\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}\left(\prod_{j=2}^{R}\pi_{u_{1},u_{j}}+(R-1)\prod_{j(\neq
2)=1}^{R}\pi_{u_{2},u_{j}}\right)\cdot\prod_{j=2}^{R}\lambda_{u_{j}}\right]^{2}.$

∎

###### Proof of Corollary 3.2.

Proof of this Corollary directly follows from the Corollary 5.7. ∎

####  5.2.2 Corollary 5.8: limit law for complete and star graphs in the
egocentric subgraph case

######  Corollary 5.8 (Limit laws for $\mathbb{K}_{R}$ and
$\mathbb{K}_{1,R-1}$ under ego-centric sampling).

Suppose, the conditions of Theorem 3.3 hold.

  * (a)

When $H=\mathbb{K}_{R}$ (complete graph on $R$ vertices), then

$\displaystyle
N^{-R+1/2}\left(\frac{\widehat{S}^{[2]}_{N}(\mathbb{K}_{R})}{Rp^{R-1}(1-p)+p^{R}}-S_{N}(\mathbb{K}_{R})\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left(\sigma^{[2]}(\mathbb{K}_{R},p)\right)^{2}\right),\quad\text{where,}$
$\displaystyle\left(\sigma^{[2]}(\mathbb{K}_{R},p)\right)^{2}=\frac{R^{2}(R-1)^{2}p^{(2R-3)}(1-p)^{3}}{{\left(Rp^{R-1}(1-p)+p^{R}\right)}^{2}}\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left(\sum_{(u_{2},\ldots,u_{R})^{T}\in[K]^{R-1}}{\Psi}_{\mathbb{K}_{R}}\left(\bm{\Pi},\mathbf{u}\right)\cdot\prod_{i=2}^{R}\lambda_{u_{i}}\right)^{2}.$

  * (b)

When $H=\mathbb{K}_{1,R-1}$ ($(R-1)$-star graph on $R$ vertices), then

$\displaystyle
N^{-R+1/2}\left(\frac{\widehat{S}^{[2]}_{N}(\mathbb{K}_{1,R-1})}{p^{(R-1)}(1-p)+p}-S_{N}(\mathbb{K}_{1,R-1})\right)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{\rightarrow}}N\left(0,\left(\sigma^{[2]}(\mathbb{K}_{1,R-1},p)\right)^{2}\right),\quad\text{where,}$
$\left(\sigma^{[2]}(\mathbb{K}_{1,R-1},p)\right)^{2}=\frac{p(1-p)}{(p^{(R-1)}(1-p)+p)^{2}}\sum_{u_{1}=1}^{K}\lambda_{u_{1}}\left[\sum_{{(u_{2},\ldots,u_{R})}^{T}\in[K]^{R-1}}\left((1-p^{R-1})\prod_{l(\neq
1)=1}^{R}\pi_{u_{1},u_{l}}\right.\right.\\\
\left.\left.\quad{}+(R-1)p^{R-2}(1-p)\prod_{l(\neq
2)=1}^{R}\pi_{u_{2},u_{l}}\right)\cdot\prod_{i=2}^{R}\lambda_{u_{i}}\right]^{2}.$

###### Proof of Corollary 5.8.

This corollary is a direct application of the Theorem 3.3. For the variance
expressions, we have to write the explicit expression for $f(H,p)$ and
$\delta_{k}(H,p)$ defined in (3.12) for both the cases. In case of
$H=\mathbb{K}_{R}$ or for complete graph with $R$-vertices, using (E.12),
(E.10) and (E.32), we can write the following, number of edges of
$\mathbb{K}_{R}$ is $T=R(R-1)/2$ and edge set is

$\displaystyle E(\mathbb{K}_{R})$
$\displaystyle=\left\\{\\{1,2\\},\ldots,\\{1,R\\},\\{2,3\\},\ldots,\\{2,R\\},\ldots,\\{R-1,R\\}\right\\}$
$\displaystyle=\\{C_{1},\ldots,C_{R-1},C_{R},\ldots,C_{2R-3},\ldots,C_{R(R-1)/2}\\}$

In general, edges connected to $r$-th vertex are
$\\{\\{1,r\\},\ldots,\\{r-1,r\\},\\{r,r+1\\},\ldots,\\{r,R\\}\\}$. Now, from
the definition (3.12), and for $r=1,\ldots,R$

$\displaystyle\mathcal{N}(r)=\\{1,\ldots,r-1,r+1,\ldots,R\\}\quad\text{and}\quad\mathcal{A}^{c}(r)=\left\\{\text{set
of $t\in[R(R-1)/2]$ such that $r\notin C_{t}$}\right\\}.$

Note that, if $r$-th vertex and edge related to that vertex is removed from
the $\mathbb{K}_{R}$, then also the graph is complete on $R-1$ vertices. Then,
from (3.12), and for $r=2,\ldots,R$,

$\displaystyle\delta_{r}(\mathbb{K}_{R},p)=\mathbf{E}\left[\left\\{\prod_{t\in\mathcal{A}^{c}(r)}\max\\{\widetilde{W}_{i},\widetilde{W}_{j}:i,j\in
C_{t}\\}\right\\}\left\\{1-\prod_{v\in\mathcal{N}(r)}\widetilde{W}_{v}\right\\}\right]$
$\displaystyle=\mathbf{E}\left[\left\\{\sum_{i(\neq
r)=1}^{R}\widetilde{W}_{1}\ldots\widetilde{W}_{i-1}\widetilde{W}_{i+1}\ldots,\widetilde{W}_{R}(1-\widetilde{W}_{i})+\prod_{i(\neq
r)}^{R}\widetilde{W}_{i}\right\\}\left\\{1-\prod_{v(\neq
r)=1}^{R}\widetilde{W}_{v}\right\\}\right]$
$\displaystyle=\mathbf{E}\left[\left\\{\sum_{i(\neq
r)=1}^{R}\widetilde{W}_{1}\ldots\widetilde{W}_{i-1}\widetilde{W}_{i+1}\ldots,\widetilde{W}_{R}(1-\widetilde{W}_{i})+\prod_{i(\neq
r)}^{R}\widetilde{W}_{i}\right\\}\right]$
$\displaystyle-\mathbf{E}\left[\left\\{\sum_{i(\neq
r)=1}^{R}\widetilde{W}_{1}\ldots\widetilde{W}_{i-1}\widetilde{W}_{i+1}\ldots,\widetilde{W}_{R}(1-\widetilde{W}_{i})+\prod_{i(\neq
r)=1}^{R}\widetilde{W}_{i}\right\\}\left\\{\prod_{v(\neq
r)=1}^{R}\widetilde{W}_{v}\right\\}\right]$ (E.77)
$\displaystyle=\\{(R-1)p^{R-2}(1-p)+p^{R-1}\\}-p^{R-1}$
$\displaystyle=(R-1)p^{R-2}(1-p)$

Note that, if we denote the random variable under expectation in the second
term of (5.2.2) by $V_{r}$ and $V_{r}$ is a Bernoulli random variable, then

$\displaystyle\mathbf{P}(V_{r}=1)$
$\displaystyle=\mathbf{P}\left(\left\\{\sum_{i(\neq
r)=1}^{R}\widetilde{W}_{1}\ldots\widetilde{W}_{i-1}\widetilde{W}_{i+1}\ldots,\widetilde{W}_{R}(1-\widetilde{W}_{i})+\prod_{i(\neq
r)=1}^{R}\widetilde{W}_{i}\right\\}\left\\{\prod_{v(\neq
r)=1}^{R}\widetilde{W}_{v}\right\\}=1\right)$
$\displaystyle=\mathbf{P}\left(\widetilde{W}_{1}=1,\ldots,\widetilde{W}_{r-1}=1,\widetilde{W}_{r+1}=1,\ldots,\widetilde{W}_{R}=1\right)=p^{R-1}=1-\mathbf{P}(V_{r}=0)$

Now, from the definition (3.12),

$\displaystyle f(\mathbb{K}_{R},p)$
$\displaystyle=\mathbf{E}\left(\prod_{\\{i,j\\}\in
E(H)}\max\\{\widetilde{W}_{i},\widetilde{W}_{j}\\}\right)=\mathbf{E}\left(\sum_{i=1}^{R}\left(1-\widetilde{W}_{i}\right)\prod_{j(\neq
i)=1}^{R}\widetilde{W}_{j}+\prod_{j=1}^{R}\widetilde{W}_{j}\right)=R\cdot
p^{R-1}(1-p)+p^{R}.$

In case of $H=\mathbb{K}_{1,R-1}$ or for star graph with $R$-vertices, using
(E.12), (E.10) and (E.32), we can write the following, number of edges of
$\mathbb{K}_{1,R-1}$ is $T=R-1$ and edge set is

$\displaystyle E(\mathbb{K}_{1,R-1})$
$\displaystyle=\left\\{\\{1,2\\},\ldots,\\{1,R\\}\right\\}=\\{C_{1},\ldots,C_{R-1}\\}$

Now, from the definition (3.12), and for $r=1,\ldots,R$,

$\displaystyle\mathcal{N}(r)=\begin{cases}\\{2,\ldots,R\\},&\quad\text{if
$r=1$}\\\ \\{1\\},&\quad\text{if $r\neq
1$}.\end{cases}\quad\text{and}\quad\mathcal{A}^{c}(r)=\begin{cases}\emptyset,&\quad\text{if
$r=1$}\\\ [R-1]\setminus\\{r-1\\},&\quad\text{if $r\neq 1$}.\end{cases}$

Then, from (3.12), when $r=1$,

$\displaystyle\delta_{1}(\mathbb{K}_{1,R-1},p)$
$\displaystyle=\mathbf{E}\left[\left\\{\prod_{t\in\mathcal{A}^{c}(1)}\max\\{\widetilde{W}_{i},\widetilde{W}_{j}:i,j\in
C_{t}\\}\right\\}\left\\{1-\prod_{v\in\mathcal{N}(1)}\widetilde{W}_{v}\right\\}\right]$
$\displaystyle=\mathbf{E}\left[\left\\{1-\prod_{v=2}^{R}\widetilde{W}_{v}\right\\}\right]=\left(1-p^{R-1}\right),$

and when $r\neq 1$,

$\displaystyle\delta_{r}(\mathbb{K}_{1,R-1},p)=\mathbf{E}\left[\left\\{\prod_{t\in\mathcal{A}^{c}(r)}\max\\{\widetilde{W}_{i},\widetilde{W}_{j}:i,j\in
C_{t}\\}\right\\}\left\\{1-\prod_{v\in\mathcal{N}(r)}\widetilde{W}_{v}\right\\}\right]$
$\displaystyle=\mathbf{E}\left[\left\\{\prod_{t\in[R-1]\setminus\\{r-1\\}}\max\\{\widetilde{W}_{i},\widetilde{W}_{j}:i,j\in
C_{t}\\}\right\\}\left\\{1-\widetilde{W}_{1}\right\\}\right]=\mathbf{E}\left[\left\\{\prod_{t(\neq
r)=2}^{R}\max\\{\widetilde{W}_{1},\widetilde{W}_{t}\\}\right\\}\left\\{1-\widetilde{W}_{1}\right\\}\right]$
$\displaystyle=\mathbf{E}\left[\prod_{t(\neq
r)=2}^{R}\widetilde{W}_{t}\mid\widetilde{W}_{1}=0\right]\mathbf{P}\left(\widetilde{W}_{1}=0\right)=p^{R-2}(1-p).$

Now, from the definition (3.12),

$\displaystyle f(\mathbb{K}_{1,R-1},p)$
$\displaystyle=\mathbf{E}\left(\prod_{\\{i,j\\}\in
E(\mathbb{K}_{1,R-1})}\max\\{\widetilde{W}_{i},\widetilde{W}_{j}\\}\right)=\mathbf{E}\left(\prod_{i=2}^{R}\max\\{\widetilde{W}_{1},\widetilde{W}_{j}\\}\right)$
$\displaystyle=\mathbf{E}\left(\widetilde{W}_{1}+(1-\widetilde{W}_{1})\prod_{i=2}^{R}\widetilde{W}_{i}\right)=p+p^{R-1}(1-p).$

Hence, $f(\mathbb{K}_{1,R-1},p)=p+p^{R-1}(1-p)$. ∎

###### Proof of the Corollary 3.4.

Proof of this corollary directly follows from the Corollary 5.8. ∎

####  5.2.3 A multivariate CLT for estimated and population based edge, wedge
and triangle counts

######  Proposition 5.9 (Asymptotic joint distribution of edge, wedge and
triangle).

Suppose, assumptions (A1)-(A5) holds. Consider the estimated edge, wedge and
triangle count $\widehat{S}^{[i]}_{N}(\mathbb{K}_{2})$,
$\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})$ and
$\widehat{S}^{[i]}_{N}(\mathbb{K}_{3})$ respectively (cf. (3.4)) and the
population based count of same quantities, $S_{N}(\mathbb{K}_{2})$,
$S_{N}(\mathbb{K}_{1,2})$ and $S_{N}(\mathbb{K}_{3})$, respectively (cf.
(3.2)). Consider a vector

$\mathbf{Z}_{N}^{[i]}=\left(\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{2})}{N^{2}},\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{1,2})}{N^{3}},\frac{\widehat{S}^{[i]}_{N}(\mathbb{K}_{3})}{N^{3}},\frac{S_{N}(\mathbb{K}_{2})}{N^{2}},\frac{S_{N}(\mathbb{K}_{1,2})}{N^{3}},\frac{S_{N}(\mathbb{K}_{3})}{N^{3}}\right)^{{T}},\quad\text{for
$i=1,2$}.$

Then

  1. (a)

Under induced sampling scheme or $i=1$,

$\displaystyle\sqrt{N}\left(\mathbf{Z}_{N}^{[1]}-\mathbf{E}\left(\mathbf{Z}_{N}^{[1]}\right)\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N_{6}\left(\mathbf{0},\bm{\Sigma}^{[1]}\right),\quad\text{where,}\
\bm{\Sigma}^{[1]}=p^{3}(1-p)\sum_{u=1}^{K}\lambda_{u}\cdot\mathbf{a}_{u}(p)\mathbf{a}_{u}(p)^{{T}},\quad\text{and,}$
(E.78)
$\displaystyle\mathbf{a}_{u}(p)=\left(2\sum_{v=1}^{K}\pi_{u,v}\lambda_{v},p\sum_{v,w=1}^{K}\left(\pi_{u,w}+2\pi_{v,w}\right)\pi_{u,v}\lambda_{v}\lambda_{w},3p\sum_{v,w=1}^{K}\pi_{u,v}\pi_{v,w}\pi_{w,u}\lambda_{v}\lambda_{w},0,0,0\right)^{{T}}.$

  2. (b)

Under ego-centric sampling scheme or $i=2$,

$\displaystyle\sqrt{N}\left(\mathbf{Z}_{N}^{[2]}-\mathbf{E}\left(\mathbf{Z}_{N}^{[2]}\right)\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}N_{6}\left(\mathbf{0},\bm{\Sigma}^{[2]}\right),\quad,\text{where,}\
\bm{\Sigma}^{[2]}=p(1-p)^{3}\sum_{u=1}^{K}\lambda_{u}\cdot\mathbf{b}_{u}(p)\mathbf{b}_{u}(p)^{{T}},\quad\text{and,}$
(E.79)
$\displaystyle\mathbf{b}_{u}(p)=\left(2\sum_{v=1}^{K}\pi_{u,v}\lambda_{v},\sum_{v,w=1}^{K}((1+p)\cdot\pi_{u,w}+2p\cdot\pi_{v,w})\pi_{u,v}\lambda_{v}\lambda_{w},6p\sum_{v,w=1}^{K}\pi_{u,v}\pi_{v,w}\pi_{w,u}\lambda_{v}\lambda_{w},0,0,0\right)^{{T}}$

###### Proof of the Proposition 5.9.

First we proof for the induced subgraph sampling. To prove the joint
asymptotic normality we use Cramér-Wold device (see. van der Vaart,  (1998)
p.16). Let $\mathbf{t}=(t_{1},\ldots,t_{6})^{T}\in\mathbb{R}^{6}$ be a six
dimensional real vector, then,

$\displaystyle\mathbf{t}^{{T}}\left(\mathbf{Z}_{N}^{[1]}-\mathbf{E}\left(\mathbf{Z}_{N}^{[1]}\right)\right)=\frac{t_{1}}{N^{2}}\left(\sum_{s_{1}\neq
s_{2}}\Psi_{\mathbb{K}_{2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)\Psi_{\mathbb{K}_{2}}(\mathbf{Y}_{N},\mathbf{s})-p^{2}\sum_{s_{1}\neq
s_{2}}\Psi_{\mathbb{K}_{2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})\right)$
$\displaystyle\qquad{}+\frac{t_{2}}{N^{3}}\left(\sum_{s_{1}\neq s_{2}\neq
s_{3}}\Psi_{\mathbb{K}_{1,2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)\Psi_{\mathbb{K}_{1,2}}(\mathbf{Y}_{N},\mathbf{s})-p^{3}\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\Psi_{\mathbb{K}_{1,2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})\right)$
$\displaystyle\qquad{}+\frac{t_{3}}{N^{3}}\left(\sum_{s_{1}\neq s_{2}\neq
s_{3}}\Psi_{\mathbb{K}_{3}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)\Psi_{\mathbb{K}_{3}}(\mathbf{Y}_{N},\mathbf{s})-p^{3}\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\Psi_{\mathbb{K}_{3}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})\right)$
$\displaystyle+\frac{t_{4}}{N^{2}}\left(\sum_{s_{1}\neq
s_{2}}\Psi_{\mathbb{K}_{2}}(\mathbf{Y}_{N},\mathbf{s})-\sum_{s_{1}\neq
s_{2}}\Psi_{\mathbb{K}_{2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})\right)+\frac{t_{5}}{N^{3}}\left(\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\Psi_{\mathbb{K}_{1,2}}(\mathbf{Y}_{N},\mathbf{s})-\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\Psi_{\mathbb{K}_{1,2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})\right)$
$\displaystyle\qquad{}+\frac{t_{6}}{N^{3}}\left(\sum_{s_{1}\neq s_{2}\neq
s_{3}}\Psi_{\mathbb{K}_{3}}(\mathbf{Y}_{N},\mathbf{s})-p^{3}\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\Psi_{\mathbb{K}_{3}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})\right)$
$\displaystyle=\Bigg{\\{}\frac{1}{N^{2}}\sum_{s_{1}\neq
s_{2}}\left(t_{1}\Psi_{\mathbb{K}_{2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)+t_{4}\right)(\Psi_{\mathbb{K}_{2}}(\mathbf{Y}_{N},\mathbf{s})-\Psi_{\mathbb{K}_{2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}))$
$\displaystyle\qquad+\frac{1}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}\left(t_{2}\Psi_{\mathbb{K}_{1,2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)+t_{5}\right)(\Psi_{\mathbb{K}_{1,2}}(\mathbf{Y}_{N},\mathbf{s})-\Psi_{\mathbb{K}_{1,2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}))$
$\displaystyle\qquad+\frac{1}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}\left(t_{3}\Psi_{\mathbb{K}_{3}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)+t_{6}\right)(\Psi_{\mathbb{K}_{3}}(\mathbf{Y}_{N},\mathbf{s})-\Psi_{\mathbb{K}_{3}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}))\Bigg{\\}}+$
$\displaystyle\qquad{}+\Bigg{\\{}\frac{t_{1}}{N^{2}}\sum_{s_{1}\neq
s_{2}}\left(\Psi_{\mathbb{K}_{2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{2}\right)\Psi_{\mathbb{K}_{2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right)+\frac{t_{2}}{N^{2}}\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\left(\Psi_{\mathbb{K}_{1,2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{3}\right)\Psi_{\mathbb{K}_{1,2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right)$
$\displaystyle\qquad{}+\frac{t_{3}}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}\left(\Psi_{\mathbb{K}_{3}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{3}\right)\Psi_{\mathbb{K}_{3}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right)\Bigg{\\}}$
$\displaystyle=M^{[1]}_{1,N}+M^{[1]}_{2,N},\quad\text{(say)},$

where,

$\displaystyle M^{[1]}_{1,N}$ $\displaystyle=\frac{1}{N^{2}}\sum_{s_{1}\neq
s_{2}}\left(t_{1}\Psi_{\mathbb{K}_{2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)+t_{4}\right)(\Psi_{\mathbb{K}_{2}}(\mathbf{Y}_{N},\mathbf{s})-\Psi_{\mathbb{K}_{2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}))$
$\displaystyle\qquad+\frac{1}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}\left(t_{2}\Psi_{\mathbb{K}_{1,2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)+t_{5}\right)(\Psi_{\mathbb{K}_{1,2}}(\mathbf{Y}_{N},\mathbf{s})-\Psi_{\mathbb{K}_{1,2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}))$
$\displaystyle\qquad+\frac{1}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}\left(t_{3}\Psi_{\mathbb{K}_{3}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)+t_{6}\right)(\Psi_{\mathbb{K}_{3}}(\mathbf{Y}_{N},\mathbf{s})-\Psi_{\mathbb{K}_{3}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}))$
$\displaystyle=N^{-2}M^{[1]}_{1,1,N}+N^{-3}M^{[1]}_{1,2,N}+N^{-3}M^{[1]}_{1,3,N},\quad\text{(say)}$
$\displaystyle M^{[1]}_{2,N}$
$\displaystyle=\frac{t_{1}}{N^{2}}\sum_{s_{1}\neq
s_{2}}\left(\Psi_{\mathbb{K}_{2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{2}\right)\Psi_{\mathbb{K}_{2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right)+\frac{t_{2}}{N^{2}}\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\left(\Psi_{\mathbb{K}_{1,2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{3}\right)\Psi_{\mathbb{K}_{1,2}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right)$
$\displaystyle\qquad{}+\frac{t_{3}}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}\left(\Psi_{\mathbb{K}_{3}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{3}\right)\Psi_{\mathbb{K}_{3}}\left(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}}\right).$
(E.80)

Here, we will show that $M^{[1]}_{1,N}=o_{p}(\sqrt{N})$. Here,
$\mathbf{E}\left(M^{[1]}_{1,N}\right)=0$, immediately from the expression, and

$\displaystyle\mathbf{Var}\left(\sqrt{N}\cdot
M^{[1]}_{1,N}\right)=\mathbf{E}\left(\left[\sqrt{N}\cdot
M^{[1]}_{1,N}\right]^{2}\right)=\mathbf{E}\left(\left[N^{-3/2}\cdot
M^{[1]}_{1,1,N}+N^{-5/2}\cdot M^{[1]}_{1,2,N}+N^{-5/2}\cdot
M^{[1]}_{1,3,N}\right]^{2}\right)$
$\displaystyle=N^{-3}\mathbf{E}\left(\left(M^{[1]}_{1,1,N}\right)^{2}\right)+N^{-5}\mathbf{E}\left(\left(M^{[1]}_{1,2,N}\right)^{2}\right)+N^{-5}\mathbf{E}\left(\left(M^{[1]}_{1,3,N}\right)^{2}\right)$
$\displaystyle+2N^{-4}\mathbf{E}\left(M^{[1]}_{1,1,N}\cdot
M^{[1]}_{1,2,N}\right)+2N^{-5}\mathbf{E}\left(M^{[1]}_{1,2,N}\cdot
M^{[1]}_{1,3,N}\right)+2N^{-5}\mathbf{E}\left(M^{[1]}_{1,3,N}\cdot
M^{[1]}_{1,1,N}\right)$ $\displaystyle\leq
N^{-3}\mathbf{E}\left(\left(M^{[1]}_{1,1,N}\right)^{2}\right)+N^{-5}\mathbf{E}\left(\left(M^{[1]}_{1,2,N}\right)^{2}\right)+N^{-5}\mathbf{E}\left(\left(M^{[1]}_{1,3,N}\right)^{2}\right)$
$\displaystyle+2N^{-4}\left[\mathbf{E}\left(\left(M^{[1]}_{1,1,N}\right)^{2}\right)\cdot\mathbf{E}\left(\left(M^{[1]}_{1,2,N}\right)^{2}\right)\right]^{1/2}+2N^{-5}\left[\mathbf{E}\left(\left(M^{[1]}_{1,2,N}\right)^{2}\right)\cdot\mathbf{E}\left(\left(M^{[1]}_{1,3,N}\right)^{2}\right)\right]^{1/2}$
$\displaystyle\quad{}+2N^{-4}\left[\mathbf{E}\left(\left(M^{[1]}_{1,3,N}\right)^{2}\right)\cdot\mathbf{E}\left(\left(M^{[1]}_{1,1,N}\right)^{2}\right)\right]^{1/2}$
$\displaystyle=N^{-3}\mathbf{Var}\left(M^{[1]}_{1,1,N}\right)+N^{-5}\mathbf{Var}\left(M^{[1]}_{1,2,N}\right)+N^{-5}\mathbf{Var}\left(M^{[1]}_{1,3,N}\right)+2N^{-4}\left[\mathbf{Var}\left(M^{[1]}_{1,1,N}\right)\cdot\mathbf{Var}\left(M^{[1]}_{1,2,N}\right)\right]^{1/2}$
$\displaystyle+2N^{-5}\left[\mathbf{Var}\left(M^{[1]}_{1,2,N}\right)\cdot\mathbf{Var}\left(M^{[1]}_{1,3,N}\right)\right]^{1/2}+2N^{-4}\left[\mathbf{Var}\left(M^{[1]}_{1,3,N}\right)\cdot\mathbf{Var}\left(M^{[1]}_{1,1,N}\right)\right]^{1/2}$
$\displaystyle\leq
N^{-3}O(N^{2})+N^{-5}O(N^{4})+N^{-5}O(N^{4})+N^{-4}\left[O(N^{2})\cdot
O(N^{4})\right]^{1/2},\quad\text{(using Lemma \ref{lem-3} and \eqref{V3})}$
$\displaystyle+N^{-5}\left[O(N^{4})\cdot
O(N^{4})\right]^{1/2}+N^{-4}\left[O(N^{4})\cdot O(N^{2})\right]^{1/2}$

Therefore, $M^{[1]}_{1,N}=o_{p}\left(\sqrt{N}\right)$. Now, we consider the
Hájek projection of $M^{[1]}_{2,N}$ defined in (5.2.3) and we get the
following expression

$\displaystyle\widetilde{M}^{[1]}_{2,N}=\sum_{l=1}^{N}\mathbf{E}\left(J^{[1]}_{2,N}\mid
W_{N,l}\right)$
$\displaystyle=\frac{t_{1}}{N^{2}}\sum_{l=1}^{N}\mathbf{E}\left(\sum_{s_{1}\neq
s_{2}}\left(\Psi_{\mathbb{K}_{2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{2}\right)\mid
W_{N,l}\right)\cdot\Psi_{\mathbb{K}_{2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})$
$\displaystyle+\frac{t_{2}}{N^{3}}\sum_{l=1}^{N}\mathbf{E}\left(\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\left(\Psi_{\mathbb{K}_{1,2}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{3}\right)\mid
W_{N,l}\right)\cdot\Psi_{\mathbb{K}_{1,2}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})$
$\displaystyle+\frac{t_{3}}{N^{3}}\sum_{l=1}^{N}\mathbf{E}\left(\sum_{s_{1}\neq
s_{2}\neq
s_{3}}\left(\Psi_{\mathbb{K}_{3}}\left(\mathbf{U}_{N}^{[1]},\mathbf{s}\right)-p^{3}\right)\mid
W_{N,l}\right)\cdot\Psi_{\mathbb{K}_{3}}(\bm{\Pi},(\bm{\alpha}_{N})_{\mathbf{s}})$
$\displaystyle=\frac{t_{1}}{N^{2}}\sum_{s_{1}\neq
s_{2}}\left((pW_{N,s_{1}}-p^{2})+(pW_{N,s_{2}}-p^{2})\right)\cdot\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}$
$\displaystyle+\frac{t_{2}}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}\left((p^{2}W_{N,s_{1}}-p^{3})+(p^{2}W_{N,s_{2}}-p^{3})+(p^{2}W_{N,s_{3}}-p^{3})\right)\cdot\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}$
$\displaystyle+\frac{t_{3}}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}\left((p^{2}W_{N,s_{1}}-p^{3})+(p^{2}W_{N,s_{2}}-p^{3})+(p^{2}W_{N,s_{3}}-p^{3})\right)\cdot\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\pi_{\alpha_{s_{3}},\alpha_{s_{1}}}$
(E.81) $\displaystyle=2\frac{t_{1}}{N^{2}}\sum_{s_{1}\neq
s_{2}}(pW_{N,s_{1}}-p^{2})\cdot\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}$
$\displaystyle+\frac{t_{2}}{N^{3}}\left(\sum_{s_{1}\neq s_{2}\neq
s_{3}}(p^{2}W_{N,s_{1}}-p^{3})\cdot\pi_{\alpha_{s_{2}},\alpha_{s_{1}}}\pi_{\alpha_{s_{1}},\alpha_{s_{3}}}+2\sum_{s_{1}\neq
s_{2}\neq
s_{3}}(p^{2}W_{N,s_{2}}-p^{3})\cdot\pi_{\alpha_{s_{2}},\alpha_{s_{1}}}\pi_{\alpha_{s_{1}},\alpha_{s_{3}}}\right)$
$\displaystyle+3\frac{t_{3}}{N^{3}}\sum_{s_{1}\neq s_{2}\neq
s_{3}}(p^{2}W_{N,s_{1}}-p^{3})\cdot\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\pi_{\alpha_{s_{3}},\alpha_{s_{1}}}$
$\displaystyle=2p\frac{t_{1}}{N^{2}}\sum_{s_{1}=1}^{N}(W_{N,s_{1}}-p)\sum_{\begin{subarray}{c}s_{2}=1\\\
s_{2}\neq
s_{1}\end{subarray}}^{N}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}+p^{2}\frac{t_{2}}{N^{3}}\sum_{s_{1}=1}^{N}(W_{N,s_{1}}-p)\left(\sum_{\begin{subarray}{c}s_{2}\neq
s_{3}\\\ s_{2},s_{3}\neq
s_{1}\end{subarray}}\\{\pi_{\alpha_{s_{1}},\alpha_{s_{3}}}+2\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\\}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\right)$
$\displaystyle+3p^{2}\frac{t_{3}}{N^{3}}\sum_{s_{1}=1}^{N}(W_{N,s_{1}}-p)\sum_{\begin{subarray}{c}s_{2}\neq
s_{3}\\\ s_{2},s_{3}\neq
s_{1}\end{subarray}}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\pi_{\alpha_{s_{3}},\alpha_{s_{1}}}.$
(E.82)

Now the variance of $\sqrt{N}\cdot\widetilde{M}^{[1]}_{2,N}$ has the following
expression,

$\displaystyle\mathbf{Var}\left(\sqrt{N}\cdot\widetilde{M}^{[1]}_{2,N}\right)$
$\displaystyle=4t_{1}^{2}N^{-3}p^{3}(1-p)\sum_{s_{1}=1}^{N}\left(\sum_{\begin{subarray}{c}s_{2}=1\\\
s_{2}\neq
s_{1}\end{subarray}}^{N}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\right)^{2}+t_{2}^{2}N^{-5}p^{5}(1-p)\sum_{s_{1}=1}^{N}\left(\sum_{\begin{subarray}{c}s_{2}\neq
s_{3}\\\ s_{2},s_{3}\neq
s_{1}\end{subarray}}\\{\pi_{\alpha_{s_{1}},\alpha_{s_{3}}}+2\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\\}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\right)^{2}$
$\displaystyle+9t_{3}^{2}N^{-5}p^{5}(1-p)\sum_{s_{1}=1}^{N}\left(\sum_{\begin{subarray}{c}s_{2}\neq
s_{3}\\\ s_{2},s_{3}\neq
s_{1}\end{subarray}}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\pi_{\alpha_{s_{3}},\alpha_{s_{1}}}\right)^{2}$
$\displaystyle+4t_{1}t_{2}N^{-4}p^{4}(1-p)\sum_{s_{1}=1}^{N}\left(\sum_{\begin{subarray}{c}s_{2}=1\\\
s_{2}\neq
s_{1}\end{subarray}}^{N}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\right)\left(\sum_{\begin{subarray}{c}s_{2}\neq
s_{3}\\\ s_{2},s_{3}\neq
s_{1}\end{subarray}}\\{\pi_{\alpha_{s_{1}},\alpha_{s_{3}}}+2\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\\}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\right)$
$\displaystyle+6t_{2}t_{3}N^{-5}p^{5}(1-p)\sum_{s_{1}=1}^{N}\left(\sum_{\begin{subarray}{c}s_{2}\neq
s_{3}\\\ s_{2},s_{3}\neq
s_{1}\end{subarray}}\\{\pi_{\alpha_{s_{1}},\alpha_{s_{3}}}+2\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\\}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\right)\left(\sum_{\begin{subarray}{c}s_{2}\neq
s_{3}\\\ s_{2},s_{3}\neq
s_{1}\end{subarray}}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\pi_{\alpha_{s_{1}},\alpha_{s_{3}}}\right)$
$\displaystyle+12t_{3}t_{1}N^{-4}p^{4}(1-p)\sum_{s_{1}=1}^{N}\left(\sum_{\begin{subarray}{c}s_{2}\neq
s_{3}\\\ s_{2},s_{3}\neq
s_{1}\end{subarray}}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\pi_{\alpha_{s_{3}},\alpha_{s_{1}}}\right)\left(\sum_{\begin{subarray}{c}s_{2}=1\\\
s_{2}\neq s_{1}\end{subarray}}^{N}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\right)$
$\displaystyle\to
4t_{1}^{2}p^{3}(1-p)\sum_{u=1}^{K}\lambda_{u}\left(\sum_{v=1}^{K}\pi_{u,v}\lambda_{v}\right)^{2}+t_{2}^{2}p^{5}(1-p)\sum_{u=1}^{K}\lambda_{u}\left(\sum_{v,w=1}^{K}\\{\pi_{u,w}+2\pi_{v,w}\\}\pi_{u,v}\lambda_{v}\lambda_{w}\right)^{2}$
$\displaystyle+9t_{3}^{2}p^{5}(1-p)\sum_{u=1}^{K}\lambda_{u}\left(\sum_{v,w=1}^{K}\pi_{u,v}\pi_{v,w}\pi_{w,u}\lambda_{v}\lambda_{w}\right)^{2}$
$\displaystyle+4t_{1}t_{2}p^{4}(1-p)\sum_{u=1}^{K}\lambda_{u}\left(\sum_{v=1}^{K}\pi_{u,v}\lambda_{v}\right)\left(\sum_{v,w=1}^{K}\\{\pi_{u,w}+2\pi_{v,w}\\}\pi_{u,v}\lambda_{v}\lambda_{w}\right)$
$\displaystyle+6t_{2}t_{3}p^{5}(1-p)\sum_{u=1}^{K}\lambda_{u}\left(\sum_{v,w=1}^{K}\\{\pi_{u,w}+2\pi_{v,w}\\}\pi_{u,v}\lambda_{v}\lambda_{w}\right)\left(\sum_{v,w=1}^{K}\pi_{u,v}\pi_{v,w}\pi_{w,u}\lambda_{v}\lambda_{w}\right)$
$\displaystyle+12t_{3}t_{1}p^{4}(1-p)\sum_{u=1}^{K}\lambda_{u}\left(\sum_{v,w=1}^{K}\pi_{u,v}\pi_{v,w}\pi_{w,u}\lambda_{v}\lambda_{w}\right)\left(\sum_{v=1}^{K}\pi_{u,v}\lambda_{v}\right)=\mathbf{t}^{{T}}\bm{\Sigma}^{[1]}\mathbf{t}$

where, $\bm{\Sigma}^{[1]}$ and $\mathbf{a}_{u}(p)$ defined in (LABEL:joint-
ind) and the variance is finite due to assumption (A.2) and (A.3). Now, from
the expression (E.82),

$\displaystyle\sqrt{N}\cdot\widetilde{M}^{[1]}_{2,N}=\sum_{s_{1}=1}^{N}\left(W_{N,s_{1}}-p\right)\frac{1}{N^{3/2}}\Bigg{\\{}2t_{1}p\sum_{s_{2}=1}^{N}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}$
$\displaystyle+\frac{p^{2}t_{2}}{N}\sum_{s_{2}\neq
s_{3}}\left(\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}+2\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\right)+\frac{3t_{3}}{N}\sum_{s_{2}\neq
s_{3}}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\pi_{\alpha_{s_{3}},\alpha_{s_{1}}}\Bigg{\\}}$

Note that, $\sqrt{N}\cdot\widetilde{M}^{[1]}_{2,N}$ is a weighted sum of
centered, independent random variables with non-nonrandom weights,

$\displaystyle a_{s_{1},N}$
$\displaystyle=\frac{1}{N^{3/2}}\Bigg{\\{}2t_{1}p\sum_{s_{2}=1}^{N}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}+\frac{p^{2}t_{2}}{N}\sum_{s_{2}\neq
s_{3}}\left(\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{1}},\alpha_{s_{3}}}+2\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\right)$
$\displaystyle+\frac{3t_{3}}{N}\sum_{s_{2}\neq
s_{3}}\pi_{\alpha_{s_{1}},\alpha_{s_{2}}}\pi_{\alpha_{s_{2}},\alpha_{s_{3}}}\pi_{\alpha_{s_{3}},\alpha_{s_{1}}}\Bigg{\\}}$

for $s_{1}=1,\ldots,N$. Recall that, $W_{N,1},\ldots,W_{N,N}$ are independent
Bernoulli random variables with $\mathbf{E}(W_{N,i})=p$ for all
$i=1,\ldots,N$. Then,