with marginals ${\rm P}^{\mathbf{Z}}$ and ${\rm P}^{\mathbf{U}}$ and support
the graph
$\\{({\mathbf{z}},\mathbf{F}(\mathbf{z}))|\,\mathbf{z}\in{\mathcal{S}}^{d-1}\\}$
of $\mathbf{F}$. Its sample version $\mathbf{F}^{(n)}$ similarly induces the
transport plan

$\gamma^{(n)}\coloneqq({\bf I}_{\rm d}\times{\bf F}^{(n)})\\#{\rm
P}^{(n)}\in\Gamma({\rm P}^{(n)},{\rm P}_{\mathbf{U}}^{(n)}),$ (B.4)

with marginals the empirical distribution ${\rm P}^{(n)}$ of the sample
$\mathbf{Z}_{1},\ldots,\mathbf{Z}_{n}$ and the uniform distribution ${\rm
P}_{\mathbf{U}}^{(n)}$ over the grid ${\mathfrak{G}}^{(n)}$; $\gamma^{(n)}$ is
supported on the (discrete) graph
$\\{(\mathbf{Z}^{(n)}_{i},\mathbf{F}^{(n)}(\mathbf{Z}^{(n)}_{i}))|\,i=1,\ldots,n\\}$
of $\mathbf{F}^{(n)}$. The proof is based on two lemmas.

######  Lemma B.1.

Let $\gamma^{(n)}$ and $\gamma$ as defined in (B.3) and (B.4). We have that
$\gamma^{(n)}$ converges weakly to $\gamma$ as $n\rightarrow\infty$.

Proof of Lemma B.1. By construction, $\gamma^{(n)}$, with marginals ${\rm
P}^{(n)}$ and ${\rm P}_{\mathbf{U}}^{(n)}$, has $c$-cyclically monotone
support. Since ${\rm P}^{(n)}$ converges weakly to ${\rm P}^{\mathbf{Z}}$ and
${\rm P}_{\mathbf{U}}^{(n)}$ has uniformly bounded support, $\gamma^{(n)}$ is
tight. Hence, by the Prohorov Theorem, there exist subsequences
$\gamma^{(n_{k})}$ converging weakly to some probability measure
$\gamma^{\infty}$, say, where $\gamma^{\infty}\in\Gamma({\rm
P}^{\mathbf{Z}},{\rm P}^{\mathbf{U}})$. For all $n_{k}$, $\gamma^{(n_{k})}$
has $c$-cyclically monotone support. It thus follows from Villani [88, pages
64-65] that the support of $\gamma^{\infty}$ also is $c$-cyclically monotone
and therefore coincides with the unique optimal transport plan of Proposition
2.3, which is $\gamma$. Assume that $\gamma^{(n)}$ does not converge weakly to
$\gamma$. Then, either (a) there exists no weakly convergent subsequence, or
(b) there exists a subsequence $\gamma^{(n_{k})}$ converging to
$\gamma^{\infty}\neq\gamma$. Because of tightness, (a) is impossible; (b) is
impossible as well in view of the $c$-cyclical monotonicity argument above;
the claim follows. $\square$

Now, Proposition 2.3 entails that

${\bf F}(\mathbf{x})=\exp_{\mathbf{x}}(-\nabla\psi(\mathbf{x})),$

where (see (2) in [62]) the $c$-concave optimal potential $\psi$ and its
$c$-transform $\phi$ are such that

$\psi(\mathbf{x})+\phi({\bf F}(\mathbf{x}))=c(\mathbf{x},{\bf
F}(\mathbf{x})).$

For a $c$-concave function $g$, define the $c$-superdifferential of $g$ as

$\partial
g\coloneqq\left\\{(\mathbf{x},\mathbf{y})\in{\mathcal{S}}^{d-1}\times{\mathcal{S}}^{d-1}{\Big{|}}g(\mathbf{z})\leq
g(\mathbf{x})+c(\mathbf{z},\mathbf{y})-c(\mathbf{x},\mathbf{y})\text{ for all
}\mathbf{z}\in{\mathcal{S}}^{d-1}\right\\}.$

Write $\partial g(\mathbf{x})$ for the set of points $\mathbf{y}\in{\cal
S}^{d-1}$ such that $(\mathbf{x},\mathbf{y})\in\partial g$. Since ${\bf F}$ is
a.s. unique, we have that

${\bf
F}(\mathbf{x})=\exp_{\mathbf{x}}(-\nabla\psi(\mathbf{x}))=\partial\psi(\mathbf{x})$

is a singleton. Writing ${\rm Supp}({\rm P})$ for the support of a probability
measure ${\rm P}$, it follows from Section 2 in [31] that the
$c$-superdifferential of the optimal potential $\psi$ is such that ${\rm
Supp}(\gamma)\subset\partial\psi$.

A potential $\psi^{(n)}$ such that $\mathbf{F}^{(n)}=\exp(-\nabla\psi^{(n)})$
similarly is associated with $\mathbf{F}^{(n)}$ (see, e.g., Theorem 5.10 (ii)
in [88]). The next lemma establishes the convergence of $\partial\psi^{(n)}$
to $\partial\psi$ in the Painlevé-Kuratowski sense. Recall that the sequence
$\partial g^{(n)}$ of $c$-superdifferentials converges to $\partial g$ in the
Painlevé-Kuratowski sense (notation: $\partial g^{(n)}\to_{\rm PK}\partial g$)
if (i) the outer limit set $\lim\sup_{n\to\infty}\partial g^{(n)}$—namely, the
set of points
$(\mathbf{x},\mathbf{y})\in{\mathcal{S}}^{d-1}\times{\mathcal{S}}^{d-1}$ for
which there exists a sequence $(\mathbf{x}_{n},\mathbf{y}_{n})\in\partial
g^{(n)}$ admitting a subsequence converging to $(\mathbf{x},\mathbf{y})$—and
(ii) the inner limit set $\lim\inf_{n\to\infty}\partial g^{(n)}$—the set of
points
$(\mathbf{x},\mathbf{y})\in{\mathcal{S}}^{d-1}\times{\mathcal{S}}^{d-1}$ which
are limits of sequences $(\mathbf{x}_{n},\mathbf{y}_{n})\in\partial
g^{(n)}$—coincide with $\partial g$.

######  Lemma B.2.

Denoting by $\partial\psi^{(n)}$ and $\partial\psi$ the $c$-subdifferentials
of the empirical and population potential $\psi^{(n)}$ and $\psi$,
respectively, we have that $\partial\psi^{(n)}\to_{\rm PK}\partial\psi$ as
$n\rightarrow\infty$.

Proof of Lemma B.2. Fix $\mathbf{z}_{0}\in{\mathcal{S}}^{d-1}$. In view of
Proposition 3.2, $\mathbf{z}\mapsto\mathbf{F}(\mathbf{z})$ is continuous at
$\mathbf{z}_{0}$. Therefore, denoting by $C_{\delta}(\mathbf{z})$ the
hyperspherical cap obtained as the intersection between ${\cal S}^{d-1}$ and
the ball $B_{\delta}(\mathbf{z})$ with radius $\delta$ centered at
$\mathbf{z}$, for each $\epsilon>0$, there exists a
$\delta=\delta(\epsilon)>0$ such that for all $\mathbf{z}$ with
$c(\mathbf{z},\mathbf{z}_{0})<\delta$,
$\mathbf{F}(\mathbf{z})=\partial\psi(\mathbf{z})\subset C_{\epsilon}({\bf
F}(\mathbf{z}_{0}))$. Since ${\rm Supp}(\gamma)\subset\partial\psi$, we have
that

$\gamma\left(B_{\delta}(\mathbf{z}_{0})\times C_{\epsilon}({\bf
F}(\mathbf{z}_{0}))\right)=\gamma\left(B_{\delta}(\mathbf{z}_{0})\times{\cal
S}^{d-1}\right)={\rm
P}^{\mathbf{Z}}\left(B_{\delta}(\mathbf{z}_{0})\right)\eqqcolon\eta>0.$

Taking a decreasing sequence $\epsilon_{k}\downarrow 0$ and the corresponding
sequence $\delta_{k}=\delta(\epsilon_{k})\downarrow 0$, we therefore have that

$\gamma\left(B_{\delta_{k}}(\mathbf{z}_{0})\times C_{\epsilon_{k}}({\bf
F}(\mathbf{z}_{0}))\right)={\rm
P}^{\mathbf{Z}}\left(B_{\delta_{k}}(\mathbf{z}_{0})\right)=:\eta_{k}>0.$

Now, Lemma B.1 entails that there exists $N_{k}$ such that for $n>N_{k}$,

$\gamma^{(n)}\left(B_{\delta_{k}}(\mathbf{z}_{0})\times C_{\epsilon_{k}}({\bf
F}(\mathbf{z}_{0}))\right)\geq\eta_{k}/2.$

Therefore, since ${\rm Supp}(\gamma^{(n)})\subset\partial\psi^{(n)}$, there
exists a sequence

$(\mathbf{x}_{n_{k}},\mathbf{y}_{n_{k}})\in\partial\psi^{(n)}\cap(B_{\delta_{k}}(\mathbf{z}_{0})\times
C_{\epsilon_{k}}({\bf F}(\mathbf{z}_{0})))$

converging as $k\to\infty$ to $(\mathbf{z}_{0},{\bf F}(\mathbf{z}_{0}))$. This
entails that (i) $\partial\psi^{(n)}$ does not escape to the horizon in the
sense of [9] and (ii) we can extract a subsequence of $\partial\psi^{(n)}$
(which are $c$-cyclically monotone sets) converging to $(\mathbf{z}_{0},{\bf
F}(\mathbf{z}_{0}))$ for any arbitrarily fixed $\mathbf{z}_{0}\in{\cal
S}^{d-1}$. The Painlevé-Kuratowski convergence then follows from Lemma 2.9 and
Theorem 2.8 in [9]. $\square$

To conclude with the proof of Proposition 4.1, assume that there exists a
sequence $(\mathbf{x}_{n},\mathbf{y}_{n})\in{\cal
S}^{d-1}\times\partial\psi^{(n)}(\mathbf{x}_{n})$ such that, for all $n$,

$c(\mathbf{y}_{n},\partial\psi(\mathbf{x}_{n}))>0.$ (B.5)

Since ${\cal S}^{d-1}$ is compact, there exists a subsequence of
$\mathbf{x}_{n}$ that converges to some $\mathbf{x}\in{\cal S}^{d-1}$. The
continuity of $\mathbf{z}\to{\bf F}(\mathbf{z})=\partial\psi(\mathbf{z})$
together with Lemma B.2 (which entails that $\mathbf{y}_{n}$ converges to
$\partial\psi(\mathbf{x})={\bf F}(\mathbf{x})$) contradict (B.5). The result
follows. $\square$

Proof of Proposition A.1. Assume that $n_{0}=0$ or $1$. First note that an
element
${\scriptstyle{\mathfrak{G}}}^{(n)}_{ij}({{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}})$
of
${\mathfrak{G}}^{(n)}_{ij}({{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}})$
decomposes into

$\displaystyle{\scriptstyle{\mathfrak{G}}}^{(n)}_{ij}({{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}})$
$\displaystyle=$
$\displaystyle\left(({\scriptstyle{\mathfrak{G}}}^{(n)}_{ij}({{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}}))^{\top}{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}\right){\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}+\left({\bf
I}_{p}-{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}({\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}})^{\top}\right){\scriptstyle{\mathfrak{G}}}^{(n)}_{ij}({{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}})$
(B.6) $\displaystyle=$ $\displaystyle
Q_{*}\left(1-\frac{i}{n_{R}+1}\right){\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}+\left(1-Q^{2}_{*}\left(1-\frac{i}{n_{R}+1}\right)\right)^{1/2}{\boldsymbol{\Gamma}}_{{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}}{{\bf
s}}_{j}$

Therefore, since
${{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}}$ is
$\mathbf{Z}_{(\cdot)}^{(n)}$-measurable, we have that, conditionally on
$\mathbf{Z}_{(\cdot)}^{(n)}$, the $n$-tuple of vectors

$\mathbf{F}^{(n)}(\mathbf{Z}^{(n)})=(\mathbf{F}^{(n)}(\mathbf{Z}_{1}^{(n)}),\ldots,\mathbf{F}^{(n)}(\mathbf{Z}_{n}^{(n)}))$

is uniform over the $n!$ permutations of the $n$ gridpoints—irrespective of
the value of $\mathbf{Z}_{(\cdot)}^{(n)}$; therefore,
$\mathbf{F}^{(n)}(\mathbf{Z}^{(n)})$ is also unconditionally uniform over
these permutations. If $n_{0}>1$, the situation is exactly the same, except
that the $n_{0}!$ permutations involving the $n_{0}$ copies of
${\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}$ are
undistinguishable so that the $n!$ permutations reduce to $n!/n_{0}!$
permutations with repetitions. This, however, can be avoided by breaking the
$n_{0}$ ties at the origin as in [40]. Point (ii) follows. Points (iii), (iv),
and (v) are direct consequences of (B.6) and the fact that the uniform over
the grid ${\mathfrak{G}}^{(n)}$ factorizes into the product of uniform
measures over the two grids involved in its construction (step (2a)).
$\square$

Proof of Proposition A.2. Points (i) and (ii) follow exactly along the same
lines as in the proof of Proposition A.1. We therefore focus on point (iii)
assuming that the grid does not exhibit any multiplicity at
${\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}$
(either $n_{0}=0$ or 1, or the tie-breaking device has been performed). In
view of [10] (see Proposition E.4 in Appendix E of [40]), since
$\mathbf{Z}_{(\cdot)}^{(n)}$ is sufficient and complete, we only have to show
that the sigma-field ${\sigma}(\mathbf{Z}_{(\cdot)}^{(n)},{\bf
R}^{(n)},\tilde{\mathbf{S}}^{(n)})$ is essentially equivalent to the Borel
$\sigma$-field ${\cal B}^{d-1}$. Since
${{\widehat{\boldsymbol{\theta}}}^{\phantom{n}}{\phantom{\theta}}^{{(n)}}}$ is
$\mathbf{Z}_{(\cdot)}^{(n)}$-measurable and in view of (B.6), the map-ping
$\mathbf{z}\in{\cal S}^{d-1}\mapsto\left(\mathbf{z}_{(}\cdot),{\bf
R}(\mathbf{z}),\mathbf{S}(\mathbf{z})\right)$ clearly is injective. The claim
follows. $\square$

Proof of Proposition A.3. Since the Riemannian distance is invariant under
orthogonal transformations, it holdsthat
$c\left(\mathbf{z}_{i}^{(n)},\scriptstyle\mathfrak{G}^{(n)}_{j}\right)=c\left({\bf
O}\mathbf{z}_{i}^{(n)},{\bf O}\scriptstyle\mathfrak{G}^{(n)}_{j}\right)$ for
all $i,j\in...$ and all orthogonal matrix $\bf O$. It follows that the
empirical optimal transport pushing
$n^{-1}\sum_{i=1}^{n}\delta(\mathbf{z}_{i}^{(n)})$ forward to
$n^{-1}\sum_{i=1}^{n}\delta(\scriptstyle\mathfrak{G}^{(n)}_{i})$ (where
$\delta({\bf z})$ is the Dirac distribution at $\bf z$) and the empirical
optimal transport pushing $n^{-1}\sum_{i=1}^{n}\delta({\bf
O}\mathbf{Z}_{i}^{(n)})$ forward to $n^{-1}\sum_{i=1}^{n}\delta({\bf
O}{\scriptstyle\mathfrak{G}^{(n)}_{i}})$ coincide up to the orthogonal
transformation ${\bf O}$, that is, if ${\bf F}^{(n)}_{\bf z}({\bf
z}_{i})=\scriptstyle\mathfrak{G}^{(n)}_{j}$, then ${\bf F}^{(n)}_{{\bf
Oz}}({\bf Oz}_{i})={\bf O}\scriptstyle\mathfrak{G}^{(n)}_{j}$. Point (i) of
the proposition follows. Point (ii) is a straightforward corollary to Point
(i). $\square$

Proof of Proposition 5.1. Let
$\mathbf{Z}_{1}^{(n)},\ldots,\mathbf{Z}_{n}^{(n)}$ be i.i.d. with directional
distribution function ${\bf F}$. Since ${\bf F}^{(n)}(\mathbf{Z}_{i}^{(n)})$
and ${\bf F}_{0}(\mathbf{Z}_{i}^{(n)})$ are unit vectors,

$T_{n}=n^{-1}\sum_{i=1}^{n}\|{\bf F}^{(n)}(\mathbf{Z}_{i}^{(n)})-{\bf
F}_{0}(\mathbf{Z}_{i}^{(n)})\|^{2}=2\left(1-n^{-1}\sum_{i=1}^{n}({\bf
F}^{(n)}(\mathbf{Z}_{i}^{(n)}))^{\top}{\bf
F}_{0}(\mathbf{Z}_{i}^{(n)})\right).$

Therefore, we just need to study the asymptotic behavior of

$W_{n}\coloneqq n^{-1}\sum_{i=1}^{n}({\bf
F}^{(n)}(\mathbf{Z}_{i}^{(n)}))^{\top}{\bf F}_{0}(\mathbf{Z}_{i}^{(n)}).$

We readily have that

$\displaystyle W_{n}$ $\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}({\bf F}^{(n)}(\mathbf{Z}_{i}^{(n)}))^{\top}{\bf
F}_{0}(\mathbf{Z}_{i}^{(n)})$ $\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}({\bf F}^{(n)}(\mathbf{Z}_{i}^{(n)})-{\bf
F}(\mathbf{Z}_{i}^{(n)}))^{\top}{\bf
F}_{0}(\mathbf{Z}_{i}^{(n)})+n^{-1}\sum_{i=1}^{n}({\bf
F}(\mathbf{Z}_{i}^{(n)}))^{\top}{\bf F}_{0}(\mathbf{Z}_{i}^{(n)})$
$\displaystyle\eqqcolon$ $\displaystyle W_{1n}+W_{2n},\text{ say.}$

The Cauchy-Schwartz inequality and Proposition 4.1 jointly entail that

$\displaystyle|W_{1n}|$ $\displaystyle\leq$ $\displaystyle
n^{-1}\sum_{i=1}^{n}|({\bf F}^{(n)}(\mathbf{Z}_{i}^{(n)})-{\bf
F}(\mathbf{Z}_{i}^{(n)}))^{\top}{\bf F}_{0}(\mathbf{Z}_{i}^{(n)})|$
$\displaystyle\leq$ $\displaystyle n^{-1}\sum_{i=1}^{n}\|{\bf
F}_{0}(\mathbf{Z}_{i}^{(n)})\|\|({\bf F}^{(n)}(\mathbf{Z}_{i}^{(n)})-{\bf
F}(\mathbf{Z}_{i}^{(n)})\|$ $\displaystyle=$ $\displaystyle
n^{-1}\sum_{i=1}^{n}\|({\bf F}^{(n)}(\mathbf{Z}_{i}^{(n)})-{\bf
F}(\mathbf{Z}_{i}^{(n)})\|$ $\displaystyle\leq$ $\displaystyle{\rm max}_{1\leq
i\leq n}\|({\bf F}^{(n)}(\mathbf{Z}_{i}^{(n)})-{\bf
F}(\mathbf{Z}_{i}^{(n)})\|,$

which is $o_{\rm P}(1)$ as $n\rightarrow\infty$. Now, if ${\bf F}={\bf
F}_{0}$, we have that $W_{2n}=1$ almost surely so that $W_{n}$ converges to
one in probability. As as result, $T_{n}$ converges to zero in probability
when ${\bf F}={\bf F}_{0}$. When ${\bf F}\neq{\bf F}_{0}$, the law of large
numbers entails that $W_{2n}$ converges in probability to ${\rm E}[({\bf
F}(\mathbf{Z}_{1}^{(n)}))^{\top}{\bf F}_{0}(\mathbf{Z}_{1}^{(n)})]<1$. The
result follows. $\square$

Proof of Proposition 6.1. In this proof, we write $\mathbf{F}_{\ell}$ for
$\mathbf{F}(\mathbf{Y}^{(n)}_{\ell})$ and $\mathbf{F}^{(n)}_{\ell}$ for
$\mathbf{F}^{(n)}(\mathbf{Y}^{(n)}_{\ell})$. Since
$\sum_{{\ell}=1}^{n}(a^{(n)}_{i\ell}-\bar{a}_{i}^{(n)})=0$,

$\sum_{{\ell}=1}^{n}(a^{(n)}_{i\ell}-\bar{a}_{i}^{(n)})^{2}=-\sum_{j\neq{\ell}}(a^{(n)}_{ij}-\bar{a}_{i}^{(n)})(a^{(n)}_{i\ell}-\bar{a}_{i}^{(n)}).$

Letting ${\mbox{\boldmath$\Gamma$}}_{i;\mathbf{J}}^{(n_{i})}\coloneqq
n^{-1/2}\sum_{{\ell}=1}^{n}(a^{(n)}_{i\ell}-\bar{a}_{i}^{(n)})\mathbf{J}(\mathbf{F}_{\ell})$
and
$\mbox{\boldmath$\gamma$}_{\ell}^{(n)}\coloneqq\mathbf{J}(\mathbf{F}^{(n)}_{\ell})-\mathbf{J}(\mathbf{F}_{\ell})$,
we have that

$\displaystyle{\rm
E}[\|{\mbox{\boldmath$\Delta$}}_{i;\mathbf{J}}^{(n_{i})}-{\mbox{\boldmath$\Gamma$}}_{i;\mathbf{J}}^{(n_{i})}\|^{2}]$
$\displaystyle=$ $\displaystyle
n^{-1}\sum_{j,{\ell}=1}^{n}(a^{(n)}_{ij}-\bar{a}_{i}^{(n)})(a^{(n)}_{i\ell}-\bar{a}_{i}^{(n)}){\rm
E}[(\mbox{\boldmath$\gamma$}_{j}^{(n)})^{\top}\mbox{\boldmath$\gamma$}_{\ell}^{(n)}]$
$\displaystyle=$ $\displaystyle
n^{-1}\sum_{{\ell}=1}^{n}(a^{(n)}_{i\ell}-\bar{a}_{i}^{(n)})^{2}({\rm
E}[\|\mbox{\boldmath$\gamma$}_{1}^{(n)}]\|^{2}-{\rm
E}[(\mbox{\boldmath$\gamma$}_{1}^{(n)})^{\top}\mbox{\boldmath$\gamma$}_{2}^{(n)}])$
$\displaystyle\leq$ $\displaystyle 2{\rm
E}[\|\mbox{\boldmath$\gamma$}_{1}^{(n)}]\|^{2}.$

It follows from the continuity of $\mathbf{J}$ and Proposition 4.1 that
$\mbox{\boldmath$\gamma$}_{1}^{(n)}$ converges to $0$ a.s. Since ${\rm
E}\|\mathbf{J}(\mathbf{F}^{(n)}_{1})\|^{2}\rightarrow{\rm
E}\|\mathbf{J}(\mathbf{F}_{1})\|^{2}<\infty$ as $n\rightarrow\infty$, ${\rm
E}\|\mbox{\boldmath$\gamma$}_{1}^{(n)}\|^{2}=o(1)$ as $n\rightarrow\infty$
(see, e.g., Theorem 5.5 in Chapter 3 of [84]). As a result,

${Q}_{\mathbf{J}}^{(n)}=({\mbox{\boldmath$\Gamma$}}_{\mathbf{J}}^{(n)})^{\top}({\bf
I}_{m}\otimes{\bf
D}_{\mathbf{J}}^{-1}){\mbox{\boldmath$\Gamma$}}_{\mathbf{J}}^{(n)}+o_{\rm
P}(1)$

as $n\rightarrow\infty$, where
${\mbox{\boldmath$\Gamma$}}_{\mathbf{J}}^{(n)}\coloneqq(({\mbox{\boldmath$\Gamma$}}_{1;\mathbf{J}}^{(n_{1})})^{\top},\ldots,({\mbox{\boldmath$\Gamma$}}_{m;\mathbf{J}}^{(n_{m})})^{\top})^{\top}$.
It follows immediately from the Central Limit Theorem that
${\mbox{\boldmath$\Gamma$}}_{\mathbf{J}}^{(n)}$ is asymptotically normal with
mean zero and covariance $({\bf I}_{m}-{\bf C}_{m})\otimes{\bf
D}_{\mathbf{J}}$, where ${\bf C}_{m}=(r_{i}^{1/2}r_{j}^{1/2})_{1\leq i,j\leq
m}=\mathbf{v}_{m}\mathbf{v}_{m}^{\top}$ with
$\mathbf{v}_{m}\coloneqq(r_{1}^{1/2},\ldots,r_{m}^{1/2})^{\top}$. Note that
${\bf C}_{m}$ is idempotent, since

${\bf
C}_{m}^{2}=\mathbf{v}_{m}\mathbf{v}_{m}^{\top}\mathbf{v}_{m}\mathbf{v}_{m}^{\top}=\mathbf{v}_{m}(\sum_{i=1}^{m}r_{i})\mathbf{v}_{m}^{\top}=\mathbf{v}_{m}\mathbf{v}_{m}^{\top}={\bf
C}_{m}.$

The result then follows from noticing that $({\bf I}_{m}\otimes{\bf
D}_{\mathbf{J}}^{-})[({\bf I}_{m}-{\bf C}_{m})\otimes{\bf D}_{\mathbf{J}}]$ is
idempotent with trace $(m-1)d^{*}$. $\square$

Proof of Proposition 6.2. Using the proof of Proposition 6.1, we easily obtain
that under ${\rm P}^{(n)}_{{\underline{{\scriptstyle{\bm{\omega}}}}}_{0}}$,
$(({\mbox{\boldmath$\Delta$}}_{i;{\bf
J}}^{(n_{i})})^{\top},\Lambda^{(n)})^{\top}$ is asymptotically normal with
mean $({\bf
0}^{\top},-\frac{1}{2}\mbox{\boldmath$\tau$}^{\top}{\bm{\Gamma}}_{{\scriptstyle{\bm{\omega}}}}\mbox{\boldmath$\tau$})^{\top}$
and covariance matrix

$\begin{pmatrix}(1-r_{i}){\bf D}_{{\bf J}}&{\bf K}_{{\bf
J},{\scriptstyle{\bm{\omega}}}_{0}}(\sum_{j=1}^{m}(1-r_{j})\mbox{\boldmath$\tau$}_{j})\\\
(\sum_{j=1}^{m}(1-r_{j})\mbox{\boldmath$\tau$}_{j}^{\top}){\bf K}_{{\bf
J},{\scriptstyle{\bm{\omega}}}_{0}}^{\top}&\mbox{\boldmath$\tau$}^{\top}{\bm{\Gamma}}_{{\scriptstyle{\bm{\omega}}}}\mbox{\boldmath$\tau$}\end{pmatrix}.$

The desired result then follows from applying the Le Cam’s third Lemma.

$\square$