vec}\left(P_{k}V_{k}V_{k}^{\prime}P_{k}^{\prime}\right){\rm
vec}\left(P_{k}V_{k}V_{k}^{\prime}P_{k}^{\prime}\right)^{\prime}\right]$
$\displaystyle=$ $\displaystyle\left(P_{k}\otimes
P_{k}\right)\mathbb{E}\left[W_{k}^{2}{\rm
vec}\left(V_{k}V_{k}^{\prime}\right){\rm
vec}\left(V_{k}V_{k}^{\prime}\right)^{\prime}\right]\left(P_{k}^{\prime}\otimes
P_{k}^{\prime}\right)$ $\displaystyle=$
$\displaystyle\phi_{k}\left(P_{k}\otimes P_{k}\right)\left[H_{n_{k}}+{\rm
vec}(I_{n_{k}}){\rm vec}(I_{n_{k}})^{\prime}\right]\left(P_{k}^{\prime}\otimes
P_{k}^{\prime}\right)$ $\displaystyle=$
$\displaystyle\phi_{k}\left[J_{k\otimes}H_{n}+{\rm vec}(J_{k}){\rm
vec}(J_{k})^{\prime}\right].$

Finally we arrive at the expression,

$\displaystyle\mathbb{E}\left[\varphi\left(k,k\right)\right]$
$\displaystyle=\psi_{k}\left(I_{n}-J_{k}\right)\otimes
J_{k}+\phi_{k}\left[J_{k\otimes}H_{n}+{\rm vec}(J_{k}){\rm
vec}(J_{k})^{\prime}\right].$

#### Expectation of $\varphi\left(k,l\right)$ when $k\neq l$

When $k\neq l$, we have

$\displaystyle\mathbb{E}\left[\varphi\left(k,l\right)\right]=$
$\displaystyle\mathbb{E}\left[W_{k}W_{l}\sum_{p=1}^{G}{\rm
vec}\left(P_{k}V_{k}V_{p}^{\prime}P_{p}^{\prime}\right)\sum_{q=1}^{G}{\rm
vec}\left(P_{l}V_{l}V_{q}^{\prime}P_{q}^{\prime}\right)^{\prime}\right]$
$\displaystyle=$ $\displaystyle\mathbb{E}\left[W_{k}W_{l}{\rm
vec}\left(P_{k}V_{k}V_{k}^{\prime}P_{k}^{\prime}\right){\rm
vec}\left(P_{l}V_{l}V_{l}^{\prime}P_{l}^{\prime}\right)^{\prime}+W_{k}W_{l}{\rm
vec}\left(P_{k}V_{k}V_{l}^{\prime}P_{l}^{\prime}\right){\rm
vec}\left(P_{l}V_{l}V_{k}^{\prime}P_{k}^{\prime}\right)^{\prime}\right].$

From Lemma A.1 with $p=2$ and $q=2$, we have

$\displaystyle\mathbb{E}\left[W_{k}W_{l}{\rm
vec}\left(P_{k}V_{k}V_{k}^{\prime}P_{k}^{\prime}\right){\rm
vec}\left(P_{l}V_{l}V_{q}^{\prime}P_{q}^{\prime}\right)^{\prime}\right]$
$\displaystyle=$ $\displaystyle{\rm
vec}\left(P_{k}\mathbb{E}\left[W_{k}V_{k}V_{k}^{\prime}\right]P_{k}^{\prime}\right){\rm
vec}\left(P_{l}\mathbb{E}\left[W_{l}V_{l}V_{l}^{\prime}\right]P_{l}^{\prime}\right)^{\prime}$
$\displaystyle=$ $\displaystyle{\rm vec}\left(J_{k}\right){\rm
vec}\left(J_{l}\right)^{\prime},$

and we also have

$\displaystyle\mathbb{E}\left[W_{k}W_{l}{\rm
vec}\left(P_{k}V_{k}V_{l}^{\prime}P_{l}^{\prime}\right){\rm
vec}\left(P_{l}V_{l}V_{k}^{\prime}P_{k}^{\prime}\right)^{\prime}\right]$
$\displaystyle=$ $\displaystyle\mathbb{E}\left[W_{k}W_{l}{\rm
vec}\left(P_{k}V_{k}V_{l}^{\prime}P_{l}^{\prime}\right){\rm
vec}\left(P_{k}V_{k}V_{l}^{\prime}P_{l}^{\prime}\right)^{\prime}K_{n}\right]$
$\displaystyle=$ $\displaystyle\left(P_{l}\otimes
P_{k}\right)\mathbb{E}\left[{\rm
vec}\left(\tilde{V}_{k}\tilde{V}_{l}^{\prime}\right){\rm
vec}\left(\tilde{V}_{k}\tilde{V}_{l}^{\prime}\right)^{\prime}\right]\left(P_{l}^{\prime}\otimes
P_{k}^{\prime}\right)K_{n}$ $\displaystyle=$ $\displaystyle\left(P_{l}\otimes
P_{k}\right)\left(P_{l}^{\prime}\otimes P_{k}^{\prime}\right)K_{n}$
$\displaystyle=$ $\displaystyle\left(J_{l}\otimes J_{k}\right)K_{n},$

where $\tilde{V}_{k}\sim N(0,I_{n_{k}})$. Finally we arrive at

$\mathbb{E}\left[\varphi\left(k,l\right)\right]={\rm
vec}\left(J_{k}\right){\rm vec}\left(J_{l}\right)^{\prime}+\left(J_{l}\otimes
J_{k}\right)K_{n}.$

#### The Expression for $\mathbb{E}\left(\nabla_{s}\nabla_{s}^{\prime}\right)$

We have the following expression,

$\displaystyle\mathbb{E}\left(\nabla_{s}\nabla_{s}^{\prime}\right)$
$\displaystyle=\mathbb{E}\left[\sum_{k=1}^{G}\sum_{l=1}^{G}W_{k}W_{l}{\rm
vec}\left(P_{k}V_{k}U^{\prime}\right){\rm
vec}\left(P_{l}V_{l}U^{\prime}\right)^{\prime}\right]-{\rm
vec}\left(I_{n}\right){\rm vec}\left(I_{n}\right)^{\prime}$
$\displaystyle=\sum_{k=1}^{G}\sum_{l=1}^{G}\left[{\rm
vec}\left(J_{k}\right){\rm vec}\left(J_{l}\right)^{\prime}+\left(J_{l}\otimes
J_{k}\right)K_{n}\right]-{\rm vec}\left(I_{n}\right){\rm
vec}\left(I_{n}\right)^{\prime}$
$\displaystyle\quad+\sum_{k=1}^{G}\left[\mathbb{E}\left[\varphi\left(k,k\right)\right]-{\rm
vec}\left(J_{k}\right){\rm
vec}\left(J_{k}\right)^{\prime}+J_{k\otimes}K_{n}\right]$
$\displaystyle=K_{n}+\Upsilon_{G}.$

where $\Upsilon_{G}=\sum_{k=1}^{G}\Psi_{k}$ with $\Psi_{k}$ given by

$\displaystyle\Psi_{k}$ $\displaystyle=\psi_{k}\left(I_{n}-J_{k}\right)\otimes
J_{k}+\phi_{k}\left[J_{k\otimes}H_{n}+{\rm vec}\left(J_{k}\right){\rm
vec}\left(J_{k}\right)^{\prime}\right]-{\rm vec}\left(J_{k}\right){\rm
vec}\left(J_{k}\right)^{\prime}-J_{k\otimes}K_{n}$
$\displaystyle=\psi_{k}\left(I_{n}-J_{k}\right)\otimes
J_{k}+\phi_{k}J_{k\otimes}+\left(\phi_{k}-1\right)\left[J_{k\otimes}K_{n}+{\rm
vec}\left(J_{k}\right){\rm vec}\left(J_{k}\right)^{\prime}\right]$
$\displaystyle=\psi_{k}\left(I_{n}\otimes
J_{k}\right)+\left(\phi_{k}-\psi_{k}\right)J_{k\otimes}+\left(\phi_{k}-1\right)\left[J_{k\otimes}K_{n}+{\rm
vec}\left(J_{k}\right){\rm vec}\left(J_{k}\right)^{\prime}\right].$

So, the final formula for information matrix $\mathcal{I}$ is given by

$\displaystyle\mathcal{I}$
$\displaystyle=M^{\prime}\Omega\mathbb{E}\left(\nabla_{s}\nabla_{s}^{\prime}\right)\Omega
M=M^{\prime}\Omega\left(K_{n}+\Upsilon_{G}\right)\Omega M,$

as stated in (21). This completes the proof. $\square$

###  A.2 Block Correlation Matrix with Multivariate $t$-Distribution

Next, we prove the results in Section 4.2. For latter use, we define the
following variables:

$Q_{Y}=Y_{0}^{\prime}A^{-1}Y_{0}+\sum_{k=1}^{K}\lambda_{k}^{-1}Y_{k}^{\prime}Y_{k},\quad
W=\frac{\nu+n}{\nu-2+Q_{Y}},\quad\nabla_{A}=\frac{\partial\ell}{\partial{\rm
vec}\left(A\right)},\quad\Pi_{A}=\frac{\partial{\rm
vec}\left(A\right)}{\partial{\rm vec}\left(W\right)^{\prime}}.$

By (6) we have following form of log-likelihood function

$\ell(Z)=c-\tfrac{1}{2}\log|A|-\frac{1}{2}\sum_{k=1}^{K}\left(n_{k}-1\right)\log\lambda_{k}-\tfrac{\nu+n_{k}}{2}\log\left(1+\tfrac{1}{\nu-2}Q_{Y}\right).$

Because $\tilde{C}=\Lambda_{n}^{-1}W\Lambda_{n}^{-1}$, we have

$\eta={\rm vech}(\tilde{C})=L_{K}\Lambda_{n\otimes}^{-1}{\rm
vec}\left(W\right),$

and the score is given by

$\displaystyle\nabla^{\prime}=\frac{\partial\ell}{\partial\eta^{\prime}}$
$\displaystyle=\underset{=\nabla_{A}^{\prime}}{\underbrace{\frac{\partial\ell}{\partial{\rm
vec}\left(A\right)^{\prime}}}}\underset{=\Pi_{A}}{\underbrace{\frac{\partial{\rm
vec}\left(A\right)}{\partial{\rm
vec}\left(W\right)^{\prime}}\frac{\partial{\rm
vec}\left(W\right)}{\partial\eta^{\prime}}}}.$

Proof of Lemma 1. We have $\Pi_{A}=\frac{\partial{\rm vec}(A)}{\partial{\rm
vec}(W)^{\prime}}\frac{\partial{\rm
vec}\left(W\right)}{\partial\eta^{\prime}}$, where $\frac{\partial{\rm
vec}\left(W\right)}{\partial\eta^{\prime}}=\Lambda_{n\otimes}D_{K}$ and

$\frac{\partial{\rm vec}\left(W\right)}{\partial{\rm
vec}\left(A\right)^{\prime}}=\frac{\partial{\rm vec}\left(\log
A\right)}{\partial{\rm vec}\left(A\right)^{\prime}}-\frac{\partial{\rm
vec}\left(\log\Lambda_{\lambda}\right)}{\partial{\rm
vec}\left(A\right)^{\prime}}=\frac{\partial{\rm vec}\left(\log
A\right)}{\partial{\rm
vec}\left(A\right)^{\prime}}-E_{d}^{\prime}\frac{\partial{\rm
diag}\left(\log\Lambda_{\lambda}\right)}{\partial{\rm
diag}\left(A\right)^{\prime}}E_{d},$

where the matrix $\tilde{\Phi}=\partial{\rm
diag}\left(\log\Lambda_{\lambda}\right)/\partial{\rm
diag}\left(A\right)^{\prime}$ is a diagonal matrix with diagonal elements
$\left(A_{kk}-n_{k}\right)^{-1}=\lambda_{k}^{-1}\left(1-n_{k}\right)^{-1}$ for
$k=1,\ldots,K$. The formula for $d{\rm vec}(\log A)/d{\rm vec}(A)$ is given by
$\Gamma_{A}^{-1}$ in (A.2). So, we have

$\Pi_{A}=\frac{\partial{\rm vec}\left(A\right)}{\partial{\rm
vec}\left(W\right)^{\prime}}\Lambda_{n\otimes}D_{K}=\left(\frac{\partial{\rm
vec}\left(W\right)}{\partial{\rm
vec}\left(A\right)^{\prime}}\right)^{-1}\Lambda_{n\otimes}D_{K}.$

Using the Woodbury formula, we simplify the inverse of the $K^{2}\times K^{2}$
matrix,

$\left(\frac{\partial{\rm vec}\left(W\right)}{\partial{\rm
vec}\left(A\right)^{\prime}}\right)^{-1}=\left(\Gamma_{A}^{-1}+E_{d}^{\prime}\tilde{\Phi}E_{d}\right)^{-1}=\Gamma_{A}-\Gamma_{A}E_{d}^{\prime}\left(\tilde{\Phi}^{-1}+E_{d}\Gamma_{A}E_{d}^{\prime}\right)^{-1}E_{d}\Gamma_{A},$

which only requires the inverse of the low dimension, $K\times K$ matrix,
$\tilde{\Phi}^{-1}+E_{d}\Gamma_{A}E_{d}^{\prime}$ to be evaluated. Moreover,
because $\tilde{\Phi}$ is a diagonal matrix with elements
$\tilde{\Phi}_{kk}=\lambda_{k}^{-1}\left(n_{k}-1\right)^{-1}$, we define the
diagonal matrix $\Phi$ with diagonal elements
$\Phi_{kk}=\lambda_{k}\left(n_{k}-1\right)$, such that
$\Phi=\tilde{\Phi}^{-1}$. This proves (22) and completes the proof of Lemma 1.
$\square$

Proof of Theorem 3. The expression for $\nabla_{A}$ is given by,

$\nabla_{A}^{\prime}=-\tfrac{1}{2}{\rm
vec}\left(A^{-1}\right)^{\prime}-\tfrac{1}{2}\sum_{k=1}^{K}\frac{n_{k}-1}{\lambda_{k}}\frac{\partial\lambda_{k}}{\partial{\rm
vec}\left(A\right)^{\prime}}-\tfrac{1}{2}W\frac{\partial Q_{Y}}{\partial{\rm
vec}(A)^{\prime}},$

with

$\frac{\partial\lambda_{k}}{\partial{\rm
vec}\left(A\right)^{\prime}}=\frac{\partial\lambda_{k}}{\partial{\rm
diag}\left(A\right)^{\prime}}\frac{\partial{\rm
diag}\left(A\right)}{\partial{\rm
vec}\left(A\right)^{\prime}}=\left(1-n_{k}\right)^{-1}e_{k,K}^{\prime}E_{d},$

where $e_{k,K}$ is the $k$-th column of the $K\times K$ identity matrix
$I_{K}$. Then we obtain

$\frac{\partial Q_{Y}}{\partial{\rm vec}(A)^{\prime}}=-{\rm
vec}\left(Y_{0}Y_{0}^{\prime}\right)^{\prime}A_{\otimes}^{-1}-\sum_{k=1}^{K}\lambda_{k}^{-2}\left(1-n_{k}\right)^{-1}\left(Y_{k}^{\prime}Y_{k}\right)e_{k,K}^{\prime}E_{d},$

which leads to

$\displaystyle\nabla_{A}^{\prime}$ $\displaystyle=\tfrac{1}{2}\left[W{\rm
vec}\left(Y_{0}Y_{0}^{\prime}\right)^{\prime}-{\rm
vec}\left(A\right)^{\prime}\right]A_{\otimes}^{-1}+\tfrac{1}{2}S^{\prime}E_{d}$
$\displaystyle=\tfrac{1}{2}\left[W{\rm
vec}\left(X_{0}X_{0}^{\prime}\right)^{\prime}-{\rm
vec}\left(I_{K}\right)^{\prime}\right]A_{\otimes}^{-\frac{1}{2}}+\tfrac{1}{2}S^{\prime}E_{d},$
(A.5)

where $S$ is a $K\times 1$ vector defined by

$S=\sum_{k=1}^{K}\left(\lambda_{k}^{-1}-W\lambda_{k}^{-1}\left(n_{k}-1\right)^{-1}X_{k}^{\prime}X_{k}\right)e_{k}^{\prime},\quad{\rm
with}\quad
S_{k}=\frac{\left(n_{k}-1\right)-WX_{k}^{\prime}X_{k}}{\lambda_{k}\left(n_{k}-1\right)}.$

#### The Information Matrix

First, from the formula of score, we have following form of information
matrix,

$\mathcal{I}=\Pi_{A}^{\prime}\mathcal{I}_{A}\Pi_{A}.$

So we need to compute the matrix
$\mathcal{I}_{A}=\mathbb{E}\left(\nabla_{A}\nabla_{A}^{\prime}\right)$. From,
(A.5), we could find its first term is a function of $X_{0}$ and the second
term is a function of $X_{k}^{\prime}X_{k},k=1,2,..K$. So, for the first term,
we have

$\displaystyle\nabla_{A}^{(1)}$ $\displaystyle\equiv$
$\displaystyle\tfrac{1}{2}A_{\otimes}^{-\frac{1}{2}}\left[W{\rm
vec}\left(X_{0}X_{0}^{\prime}\right)-{\rm vec}\left(I_{K}\right)\right]$
$\displaystyle\mathcal{I}_{A}^{(1)}$ $\displaystyle=$
$\displaystyle\tfrac{1}{4}A_{\otimes}^{-1/2}\left(\mathbb{E}\left[W^{2}{\rm
vec}\left(X_{0}X_{0}^{\prime}\right){\rm
vec}\left(X_{0}X_{0}^{\prime}\right)^{\prime}\right]-{\rm
vec}\left(I_{K}\right){\rm
vec}\left(I_{K}\right)^{\prime}\right)A_{\otimes}^{-1/2}.$

Similar to (A.4), we have

$\displaystyle\mathcal{I}_{A}^{(1)}$ $\displaystyle=\tfrac{1}{4}\left[\phi
A_{\otimes}^{-1}H_{K}+\left(\phi-1\right){\rm vec}\left(A^{-1}\right){\rm
vec}\left(A^{-1}\right)^{\prime}\right].$

For the second term, we first define

$\displaystyle\nabla_{A}^{(2)}$
$\displaystyle\equiv\tfrac{1}{2}E_{d}^{\prime}S=\tfrac{1}{2}E_{d}^{\prime}\Lambda_{\lambda}\bar{S},\quad\mathcal{I}_{A}^{(2)}=\tfrac{1}{4}E_{d}^{\prime}\Lambda_{\lambda}\mathbb{E}\left(\bar{S}\bar{S}^{\prime}\right)\Lambda_{\lambda}E_{d},$

where the element in vector $\bar{S}$ and diagonal matrix $\Lambda_{\lambda}$
are define by $\bar{S}_{k}=WX_{k}^{\prime}X_{k}-\left(n_{k}-1\right)$ and
$\left[\Lambda_{\lambda}\right]_{kk}=\left[-\lambda_{k}\left(n_{k}-1\right)\right]^{-1}$.
We know that

$\displaystyle\mathbb{E}\left[W^{2}\left(X_{k}^{\prime}X_{k}\right)\left(X_{l}^{\prime}X_{l}\right)\right]$
$\displaystyle=\phi\mathbb{E}\left[\left(\tilde{Z}_{k}^{\prime}\tilde{Z}_{k}\right)\left(\tilde{Z}_{l}^{\prime}\tilde{Z}_{l}\right)\right]=\begin{cases}\phi\left(n_{k}-1\right)\left(n_{l}-1\right)&k\neq
l,\\\
\phi\left[\left(n_{k}-1\right)^{2}+2\left(n_{k}-1\right)\right]&k=l,\end{cases}$

where $\tilde{Z}_{k}\sim N\left(0,I_{n_{k}}\right)$. So
$\mathbb{E}\left(\bar{S}_{k}\bar{S}_{l}^{\prime}\right)=\mathbb{E}\left[W^{2}\left(X_{k}^{\prime}X_{k}\right)\left(X_{l}^{\prime}X_{l}\right)\right]-\left(n_{k}-1\right)\left(n_{l}-1\right)$
is given by

$\mathbb{E}\left(\bar{S}_{k}\bar{S}_{l}^{\prime}\right)=\begin{cases}\left(\phi-1\right)\left(n_{k}-1\right)\left(n_{l}-1\right)&k\neq
l,\\\
\left(\phi-1\right)\left(n_{k}-1\right)^{2}+2\phi\left(n_{k}-1\right)&k=l,\end{cases}$

and along with the following $K\times 1$ vector $\xi$ and diagonal matrix
$\Xi$, we have

$\Lambda_{\lambda}\mathbb{E}\left(\bar{S}\bar{S}^{\prime}\right)\Lambda_{\lambda}=\left(\phi-1\right)\xi\xi^{\prime}+2\phi\Xi,\quad\xi_{k}=\lambda_{k}^{-1},\quad\Xi_{kk}=\lambda_{k}^{-2}\left(n_{k}-1\right)^{-1}.$

So,

$\displaystyle\mathcal{I}_{A}^{(2)}$
$\displaystyle=\tfrac{1}{4}E_{d}^{\prime}\left[\left(\phi-1\right)\xi\xi^{\prime}+2\phi\Xi\right]E_{d}=\tfrac{\phi-1}{4}E_{d}^{\prime}\xi\xi^{\prime}E_{d}+\tfrac{\phi}{2}E_{d}^{\prime}\Xi
E_{d},$

and we also have

$\displaystyle\mathbb{E}\left(W{\rm vec}\left(X_{0}X_{0}^{\prime}\right)-{\rm
vec}(I_{K})\right)\bar{S}_{k}$ $\displaystyle=\mathbb{E}\left[W^{2}{\rm
vec}\left(X_{0}X_{0}^{\prime}\right)\left(X_{k}^{\prime}X_{k}\right)\right]-\left(n_{k}-1\right){\rm
vec}(I_{K})$ $\displaystyle=\left(\phi-1\right)\left(n_{k}-1\right){\rm
vec}\left(I_{K}\right).$

Hence,

$\displaystyle\mathcal{I}_{A}^{(12)}=\mathbb{E}\left(\nabla_{A}^{(1)}\nabla_{A}^{(2)\prime}\right)=$
$\displaystyle-\tfrac{1}{4}A_{\otimes}^{-1/2}\left(\phi-1\right){\rm
vec}\left(I_{K}\right)\xi^{\prime}E_{d}=\tfrac{1-\phi}{4}{\rm
vec}\left(A_{\otimes}^{-1}\right)\xi^{\prime}E_{d}.$

Finally, we have

$\mathcal{I}_{A}=\mathcal{I}_{A}^{(1)}+\mathcal{I}_{A}^{(2)}+\mathcal{I}_{A}^{(12)}+\mathcal{I}_{A}^{(21)},$

which gives the expression in Theorem 3. Finally, in the limited case,
$\nu\rightarrow\infty$, which corresponds to the multivariate normal
distribution, we have $\phi\rightarrow 1$, and the information matrix
simplifies to
$\mathcal{I}_{A}=\tfrac{1}{4}A_{\otimes}^{-1}H_{K}+\tfrac{1}{2}E_{d}^{\prime}\Xi
E_{d}.$ This completes the proof. $\square$

Proof of Theorem 4. For this case we have

$\nabla_{A}^{\prime}=\tfrac{1}{2}\left[W_{0}{\rm
vec}\left(X_{0}X_{0}^{\prime}\right)^{\prime}-{\rm
vec}\left(I_{K}\right)^{\prime}\right]A_{\otimes}^{-\frac{1}{2}}+\tfrac{1}{2}S^{\prime}E_{d},$

where $W_{0}$ and $S\in\mathbb{R}^{K}$ are given by

$W_{0}=\frac{\nu_{0}+K}{\nu_{0}-2+X_{0}^{\prime}X_{0}},\quad
S_{k}=\frac{(n_{k}-1)-W_{k}X_{k}^{\prime}X_{k}}{\lambda_{k}(n_{k}-1)},\quad\text{with}\quad
W_{k}=\frac{\nu_{k}+n_{k}-1}{\nu_{k}-2+X_{k}^{\prime}X_{k}}.$

The covariance of the first part was derived in Theorem 3 and is given by

$\mathcal{I}_{A}^{(1)}=\tfrac{1}{4}\left(\phi_{0}A_{\otimes}^{-1}H_{K}+\left(\phi-1\right){\rm
vec}\left(A^{-1}\right){\rm
vec}\left(A^{-1}\right)^{\prime}\right),\quad\phi_{0}=\frac{\nu_{0}+K}{\nu_{0}+K+2}.$

For the second part, we have $\mathbb{E}\left(\bar{S}_{k}\bar{S}_{l}\right)=0$
for $k\neq l$, and

$\mathbb{E}\left(\bar{S}_{k}^{2}\right)=\left(\phi_{k}-1\right)\left(n_{k}-1\right)^{2}+2\phi_{k}\left(n_{k}-1\right),$

with $\phi_{k}=\left(\nu_{k}+n_{k}-1\right)/\left(\nu_{k}+n_{k}+1\right)$.
Therefore, we have

$\mathcal{I}_{A}^{(2)}=E_{d}^{\prime}\Lambda_{\lambda}\mathbb{E}\left(\bar{S}\bar{S}^{\prime}\right)\Lambda_{\lambda}E_{d}=\Xi,\quad\Xi_{kk}=\frac{2\phi_{k}}{\lambda_{k}^{2}\left(n_{k}-1\right)}+\frac{\phi_{k}-1}{\lambda_{k}^{2}},$

and
$\mathcal{I}_{A}^{(12)}=\mathbb{E}\left(\nabla_{A}^{(1)}\nabla_{A}^{(2)\prime}\right)=0$.
Finally, we obtain

$\mathcal{I}_{A}=\tfrac{1}{4}\left[\phi_{0}A_{\otimes}^{-1}H_{K}+(\phi-1){\rm
vec}\left(A^{-1}\right){\rm vec}\left(A^{-1}\right)^{\prime}+E_{d}^{\prime}\Xi
E_{d}\right].$

##  Appendix B Block Correlation Matrix with Cluster-$t$ Distribution

Because $P=I_{n}$, and $\bm{n}=\bm{m}$, we have $V=U$, and $V_{k}=U_{k}$. Then
the log-likelihood function is given by

$\displaystyle\ell\left(Z\right)$
$\displaystyle=-\tfrac{1}{2}\log|C|+\sum_{k=1}^{K}c_{k}-\tfrac{\nu_{k}+n_{k}}{2}\log\left(1+\tfrac{1}{\nu_{k}-2}U_{k}^{\prime}U_{k}\right),$

by using the canonical representation of block correlation matrix
$C=QDQ^{\prime}$, we define the vectors $X$ and $Y$ as $Y=Q^{\prime}Z$ and
$X=Q^{\prime}U$, so we have $X_{0}=A^{-\frac{1}{2}}Y_{0}$,
$X_{k}=\lambda_{k}^{-\frac{1}{2}}Y_{k}$. From $U=QX$ and the structure of $Q$,
we have

$\displaystyle U_{k,i}$
$\displaystyle=n_{k}^{-1/2}X_{0,k}+\left(\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}\right)X_{k}$
$\displaystyle U_{k,i}^{2}$
$\displaystyle=X_{0,k}^{2}n_{k}^{-1}+X_{k}^{\prime}\left(v_{n_{k}}^{\perp\prime}\tilde{e}_{i}\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}\right)X_{k}+2X_{0,k}\left(v_{n_{k}}^{\prime}\tilde{e}_{i}\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}\right)X_{k},$

for $i=1,\ldots,n_{k}$, and $\tilde{e}_{i}\in\mathbb{R}^{n_{k}\times 1}$
denote the $i$-th column of $I_{n_{k}}$. So we obtain

$U_{k}^{\prime}U_{k}=\sum_{i=1}^{n_{k}}U_{k,i}^{2}=X_{0,k}^{2}+X_{k}^{\prime}X_{k}=Y_{0}^{\prime}\tilde{A}_{k}^{-1}Y_{0}+\lambda_{k}^{-1}Y_{k}^{\prime}Y_{k},$

where $\tilde{A}_{k}^{-1}=A^{-\tfrac{1}{2}}J_{k}^{e}A^{-\tfrac{1}{2}}$ , and
$J_{k}^{e}=e_{k}e_{k}^{\prime}$ with $e_{k}$ the $k$-th column of the $K\times
K$ identity matrix $I_{K}$. This leads to the the simplified expression for
log-likelihood function

$\displaystyle\ell\left(Z\right)$
$\displaystyle=-\tfrac{1}{2}\log|A|+\sum_{k=1}^{K}\left[c_{v_{k},n_{k}}-\tfrac{n_{k}-1}{2}\log\lambda_{k}-\tfrac{\nu_{k}+n_{k}}{2}\log\left(1+\tfrac{1}{\nu_{k}-2}\left(X_{0,k}^{2}+X_{k}^{\prime}X_{k}\right)\right)\right].$

Note that we have

$X_{0,k}=v_{n_{k}}^{\prime}U_{k},\quad X_{k}=v_{n_{k}}^{\perp\
\prime}U_{k}\qquad\text{for}\quad k=1,\ldots,K,$

such that $X_{0,k}$ and $X_{k}$ are simply linear combinations of $U_{k}$.
From the structure of $Q$ it follows that $X_{0,k}$ and $X_{0,l}$ are
independent for $k\neq l$, just as it the case for $X_{k}$ and $X_{l}$ (by
their definition). We also have that $X_{0,k}$ and $X_{k}$ are uncorrelated,
but they are not independent, because they have $t$-distributed shocks in
common.

###  B.1 The Form of the Score

By using $X_{0,k}=e_{k,K}^{\prime}A^{-\frac{1}{2}}Y_{0}$, we first have

$\displaystyle\frac{\partial X_{0,k}^{2}}{\partial{\rm
vec}\left(A\right)^{\prime}}$ $\displaystyle=\frac{\partial
X_{0,k}^{2}}{\partial
X_{0,k}}\frac{\partial\left(e_{k}^{\prime}A^{-\frac{1}{2}}Y_{0}\right)}{\partial{\rm
vec}\left(A^{-\frac{1}{2}}\right)^{\prime}}\frac{\partial{\rm
vec}\left(A^{-\frac{1}{2}}\right)^{\prime}}{\partial{\rm
vec}\left(A^{\frac{1}{2}}\right)^{\prime}}\frac{\partial{\rm
vec}\left(A^{\frac{1}{2}}\right)^{\prime}}{\partial{\rm
vec}\left(A\right)^{\prime}}$
$\displaystyle=-2X_{0,k}\left(Y_{0}^{\prime}\otimes
e_{k}^{\prime}\right)A_{\otimes}^{-\frac{1}{2}}\left(A^{\frac{1}{2}}\oplus
I\right)^{-1}$ $\displaystyle=-2X_{0,k}\left(X_{0}^{\prime}\otimes
e_{k}^{\prime}A^{-\frac{1}{2}}\right)\left(A^{\frac{1}{2}}\oplus
I\right)^{-1}$ $\displaystyle=-2X_{0,k}{\rm
vec}\left(A^{-\frac{1}{2}}e_{k}X_{0}^{\prime}\right)^{\prime}\left(A^{\frac{1}{2}}\oplus
I\right)^{-1}$ $\displaystyle=-2X_{0,k}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}\Omega,$

where $\Omega=\left(I\otimes
A^{-\frac{1}{2}}\right)\left(A^{\frac{1}{2}}\oplus I\right)^{-1}$. It follows
that

$\nabla_{A}^{\prime}=\frac{\partial\ell}{\partial{\rm
vec}\left(A\right)^{\prime}}=\left[\sum_{k=1}^{K}W_{k}X_{0,k}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}-{\rm
vec}\left(I_{K}\right)^{\prime}\right]\Omega+\tfrac{1}{2}E_{d}^{\prime}S$

where $S$ is a $K\times 1$ vector defined by

$S_{k}=\frac{\left(n_{k}-1\right)-W_{k}X_{k}^{\prime}X_{k}}{\left(n_{k}-1\right)\lambda_{k}},\quad
W_{k}=\frac{\nu_{k}+n_{k}}{\nu_{k}-2+X_{0,k}^{2}+X_{k}^{\prime}X_{k}}.$

###  B.2 The Form of the Information Matrix

The information matrix of $\nabla_{A}$ can be decompose into four components,
given by

$\mathcal{I}_{A}=\mathcal{I}_{A}^{(1)}+\mathcal{I}_{A}^{(2)}+\mathcal{I}_{A}^{(21)}+\mathcal{I}_{A}^{(12)}.$

####  B.2.1 The Form of Matrix $\mathcal{I}_{A}^{(1)}$

Similar to previous proof, the covariance of the first part of
$\partial\ell/\partial{\rm vec}\left(A\right)$ is given by

$\displaystyle\mathcal{I}_{A}^{(1)}$
$\displaystyle=\Omega\mathbb{E}\left[\sum_{k=1}^{K}\sum_{l=1}^{K}W_{k}W_{l}X_{0,k}X_{0,l}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}-{\rm vec}\left(I_{K}\right){\rm
vec}\left(I_{K}\right)^{\prime}\right]\Omega,$
$\displaystyle=\Omega\left(K_{K}+\Psi^{e}\right)\Omega$

where $\Psi^{e}=\sum_{k=1}^{K}\Psi_{k}^{e}$ with

$\displaystyle\Psi_{k}^{e}$
$\displaystyle=\psi_{k}\left(I-J_{k}^{e}\right)\otimes
J_{k}^{e}+\phi_{k}J_{k\otimes}^{e}+\left(\phi_{k}-1\right)\left[J_{k\otimes}^{e}K_{K}+{\rm
vec}\left(J_{k}^{e}\right){\rm vec}\left(J_{k}^{e}\right)^{\prime}\right],$

with $\phi_{k}=(v_{k}+n_{k})/(v_{k}+n_{k}+2)$.

####  B.2.2 The Form of Matrix $\mathcal{I}_{A}^{(2)}$

As for the second part, we have

$\mathcal{I}_{A}^{(2)}=\tfrac{1}{4}E_{d}^{\prime}\mathbb{E}\left(SS^{\prime}\right)E_{d}=\tfrac{1}{4}E_{d}^{\prime}\Xi
E_{d},$

where $\Xi_{kk}=\mathbb{E}\left(S_{k}^{2}\right)$ given by

$\displaystyle\mathbb{E}\left(S_{k}^{2}\right)$
$\displaystyle=\left[\mathbb{E}\left(W_{k}^{2}\left(X_{k}^{\prime}X_{k}\right)^{2}\right)-\left(n_{k}-1\right)^{2}\right]/\left[\lambda_{k}^{2}\left(n_{k}-1\right)^{2}\right]$
$\displaystyle=\left[\phi_{k}\left[\left(n_{k}-1\right)^{2}+2\left(n_{k}-1\right)\right]-\left(n_{k}-1\right)^{2}\right]/\left[\lambda_{k}^{2}\left(n_{k}-1\right)^{2}\right]$
$\displaystyle=\left(\phi_{k}-1\right)\lambda_{k}^{-2}+2\phi_{k}\lambda_{k}^{-2}\left(n_{k}-1\right)^{-1},$

and $\Xi_{kl}=\mathbb{E}\left(S_{k}S_{l}\right)=0$ for $k\neq l$, so $\Xi$ is
a $K\times K$ diagonal matrix.

####  B.2.3 The Form of Matrix $\mathcal{I}_{A}^{(12)}$

As for the interaction term, we have
$\mathcal{I}_{A}^{(21)}=\left[\mathcal{I}_{A}^{(12)}\right]^{\prime}$, and

$\mathcal{I}_{A}^{(12)}=\tfrac{1}{2}E_{d}^{\prime}\mathbb{E}\left[S\left(\sum_{k=1}^{K}W_{k}X_{0,k}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}-{\rm
vec}\left(I\right)^{\prime}\right)\right]\Omega=\tfrac{1}{2}E_{d}^{\prime}\Theta\Omega,$

where the $k$-th row of the $K\times K^{2}$ matrix $\Theta$ is given by

$\displaystyle e_{k}^{\prime}\Theta$
$\displaystyle=\mathbb{E}\left[S_{k}\left(\sum_{l=1}^{K}W_{l}X_{0,l}{\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}-{\rm
vec}\left(I_{K}\right)^{\prime}\right)\right]$
$\displaystyle=\left(n_{k}-1\right)^{-1}\lambda_{k}^{-1}\left[\mathbb{E}\left(\sum_{l=1}^{K}W_{k}W_{l}\left(X_{k}^{\prime}X_{k}\right)X_{0,l}{\rm
vec}\left(e_{l}X_{0}^{\prime}\right)\right)-\left(n_{k}-1\right){\rm
vec}\left(I_{K}\right)\right],$

when $k=l$, we have

$\displaystyle\mathbb{E}\left[W_{k}^{2}\left(X_{k}^{\prime}X_{k}\right)X_{0,k}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)\right]=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{K}W_{k}^{2}\left(X_{k}^{\prime}X_{k}\right)X_{0,k}{\rm
vec}\left(e_{k}X_{0,p}e_{p}^{\prime}\right)\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k}^{2}\left(X_{k}^{\prime}X_{k}\right)X_{0,k}^{2}{\rm
vec}\left(J_{k}^{e}\right)\right]$ $\displaystyle=$
$\displaystyle\phi_{k}\left(n_{k}-1\right){\rm vec}\left(J_{k}^{e}\right),$

when $k\neq l$, we have

$\displaystyle\mathbb{E}\left[W_{k}W_{l}\left(X_{k}^{\prime}X_{k}\right)X_{0,l}{\rm
vec}\left(e_{l}X_{0}^{\prime}\right)\right]=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{K}W_{k}W_{l}\left(X_{k}^{\prime}X_{k}\right)X_{0,l}{\rm
vec}\left(e_{l}X_{0,p}e_{l}^{\prime}\right)\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k}W_{l}\left(X_{k}^{\prime}X_{k}\right)X_{0,l}^{2}\right]{\rm
vec}\left(J_{l}^{e}\right)$ $\displaystyle=$
$\displaystyle\left(n_{k}-1\right){\rm vec}\left(J_{l}^{e}\right).$

Thus, we have

$\displaystyle\mathbb{E}\left[\sum_{l=1}^{K}W_{k}W_{l}\left(X_{k}^{\prime}X_{k}\right)X_{0,l}{\rm
vec}\left(e_{l}X_{0}^{\prime}\right)\right]-\left(n_{k}-1\right){\rm
vec}\left(I_{K}\right)$ $\displaystyle=$
$\displaystyle\sum_{l}\left(n_{k}-1\right){\rm
vec}\left(J_{l}^{e}\right)-\left(n_{k}-1\right){\rm
vec}\left(J_{k}^{e}\right)+\phi_{k}\left(n_{k}-1\right){\rm
vec}\left(J_{k}^{e}\right)-\left(n_{k}-1\right){\rm vec}\left(I_{K}\right)$
$\displaystyle=$ $\displaystyle\left(n_{k}-1\right){\rm
vec}\left(I_{K}\right)-\left(n_{k}-1\right){\rm
vec}\left(J_{k}^{e}\right)+\phi_{k}\left(n_{k}-1\right){\rm
vec}\left(J_{k}^{e}\right)-\left(n_{k}-1\right){\rm vec}\left(I_{K}\right)$
$\displaystyle=$ $\displaystyle\left(n_{k}-1\right)\left(\phi_{k}-1\right){\rm
vec}\left(J_{k}^{e}\right).$

Finally, the $k$-th row of matrix $M$ is
$e_{k}^{\prime}\Theta=-\lambda_{k}^{-1}\left(\phi_{k}-1\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}$, so we have

$\Theta=\sum_{k=1}^{K}e_{k}\left(e_{k}^{\prime}\Theta\right)=\sum_{k=1}^{K}\lambda_{k}^{-1}\left(1-\phi_{k}\right)e_{k}{\rm
vec}\left(J_{k}^{e}\right)^{\prime}.$

##  Appendix C Block Correlation Matrix with Hetero-$t$ Distribution

Because $P=I$, we have $U=PV=V$. By using $C=QDQ^{\prime}$, the log-likelihood
function is now given by

$\ell(Z)=c-\tfrac{1}{2}\log|A|-\tfrac{1}{2}\sum_{k=1}^{K}\left(n_{k}-1\right)\log\lambda_{k}-\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}\tfrac{\nu_{k,i}+1}{2}\log\left(1+\tfrac{1}{\nu_{k,i}-2}U_{k,i}^{2}\right),$

where $c=\sum_{i=1}^{n}c\left(\nu_{i},1\right)$, and $U_{k,i}$ is the $i$-th
innovation of $U_{k}$. The cluster structure is implied by the block
correlation matrix. To simplify the notation we let
$\tilde{e}_{i}\in\mathbb{R}^{n_{k}\times 1}$ denote the $i$-th column of
$I_{n_{k}}$. The identity
$U_{k,i}=n_{k}^{-1/2}X_{0,k}+\left(e_{i}^{\prime}v_{n_{k}}^{\perp}\right)X_{k}$,
which means that

$\displaystyle\frac{\partial\left(U_{k,i}^{2}\right)}{\partial{\rm
vec}\left(A\right)}$
$\displaystyle=\frac{\partial\left(U_{k,i}^{2}\right)}{\partial
U_{k,i}}\frac{\partial\left(n_{k}^{-1/2}X_{0,k}+\left(\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}\right)X_{k}\right)}{\partial{\rm
vec}\left(A\right)}$ $\displaystyle=2U_{k,i}\left[n_{k}^{-1/2}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}\Omega-\frac{1}{2}\left(e_{i}^{\prime}v_{n_{k}}^{\perp}\right)X_{k}\lambda_{k}^{-1}\left(1-n_{k}\right)^{-1}e_{k}^{\prime}E_{d}\right]$
$\displaystyle=2U_{k,i}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-1/2}\Omega-
U_{k,i}F_{k,i}U_{k}e_{k}^{\prime}E_{d}/\left[\lambda_{k}\left(1-n_{k}\right)\right],$

where $e_{k}$ is $k$-th column of the $K\times K$ identity matrix $I_{K}$,
$\tilde{e}_{i}$ is the $i$-th column of the $n_{k}\times n_{k}$ identity
matrix $I_{n_{k}}$, $X_{k}=v_{n_{k}}^{\perp\ \prime}U_{k}$ and
$F_{k,i}=\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}v_{n_{k}}^{\perp\prime}$. So
we have

$\displaystyle\nabla_{A}^{\prime}=\frac{\partial\ell}{\partial{\rm
vec}\left(A\right)^{\prime}}$
$\displaystyle=\left[\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}W_{k,i}U_{k,i}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}-{\rm
vec}(I_{K})^{\prime}\right]\Omega+\tfrac{1}{2}S^{\prime}E_{d},$

with the $W_{k,i}$ and $S_{k}$ defined by

$W_{k,i}=\frac{\nu_{k,i}+1}{\nu_{k,i}-2+U_{k,i}^{\prime}U_{k,i}},\quad
S_{k}=\frac{\left(n_{k}-1\right)-\sum_{i=1}^{n_{k}}W_{k,i}U_{k,i}F_{k,i}U_{k}}{\left(n_{k}-1\right)\lambda_{k}}.$

The information matrix of $\nabla_{A}$ can be expressed by
$\mathcal{I}_{A}=\mathcal{I}_{A}^{(1)}+\mathcal{I}_{A}^{(2)}+\mathcal{I}_{A}^{(21)}+\mathcal{I}_{A}^{(12)}$.

###  C.1 The Form of Matrix $\mathcal{I}_{A}^{(1)}$

$\displaystyle\mathcal{I}_{A}^{(1)}=$
$\displaystyle\mathbb{E}\left[\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}W_{k,i}U_{k,i}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]\left[\sum_{l=1}^{K}\sum_{j=1}^{n_{k}}W_{l,j}U_{l,j}{\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}\right]-{\rm
vec}(I_{K}){\rm vec}(I_{K})^{\prime}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{k=1}^{K}\sum_{l=1}^{K}\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{l}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]-{\rm
vec}(I_{K}){\rm vec}(I_{K})^{\prime}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-1}\right]-{\rm
vec}(I_{K}){\rm vec}(I_{K})^{\prime}$
$\displaystyle+\mathbb{E}\left[\sum_{k=1}^{K}\sum_{l\neq
k}^{K}\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{l}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle+\mathbb{E}\left[\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}\sum_{j\neq
i}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right],$

and for later use, we have
$X_{0}=\sum_{k}X_{0,k}e_{k}=\sum_{p}U_{p}^{\prime}v_{n_{p}}e_{p}$.

####  C.1.1 The First Term

We have

$\displaystyle\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{K}\sum_{q=1}^{K}W_{k,i}^{2}U_{k,i}^{2}{\rm
vec}\left(e_{k}U_{p}^{\prime}v_{n_{p}}e_{p}^{\prime}\right){\rm
vec}\left(e_{k}U_{q}^{\prime}v_{n_{q}}e_{q}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$ $\displaystyle\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}{\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right){\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle+\mathbb{E}\left[\sum_{p\neq k}^{K}W_{k,i}^{2}U_{k,i}^{2}{\rm
vec}\left(e_{k}U_{p}^{\prime}v_{n_{p}}e_{p}^{\prime}\right){\rm
vec}\left(e_{k}U_{p}^{\prime}v_{n_{p}}e_{p}^{\prime}\right)^{\prime}n_{k}^{-1}\right],$

and with $p=k=q$, we have

$\displaystyle\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}{\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right){\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{n_{k}}\sum_{q=1}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}U_{k,p}U_{k,q}{\rm
vec}\left(e_{k}e_{k}^{\prime}\right){\rm
vec}\left(e_{k}e_{k}^{\prime}\right)^{\prime}n_{k}^{-2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{n_{k}}\sum_{q=1}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}U_{k,p}U_{k,q}J_{k\otimes}^{e}n_{k}^{-2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{4}\right]J_{k\otimes}^{e}n_{k}^{-2}+\sum_{p\neq
i}^{n_{k}}\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}U_{k,p}^{2}\right]J_{k\otimes}^{e}n_{k}^{-2}$
$\displaystyle=$
$\displaystyle\left[3\phi_{k,i}+\psi_{k,i}\left(n_{k}-1\right)\right]J_{k\otimes}^{e}n_{k}^{-2},$

where we use that ${\rm vec}\left(e_{k}e_{k}^{\prime}\right)=e_{k}\otimes
e_{k}$ because $e_{k}$ is a $K\times 1$ vector. Next, when $p\neq k$, we have

$\displaystyle\sum_{p\neq k}^{K}\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}{\rm
vec}\left(e_{k}U_{p}^{\prime}v_{n_{p}}e_{p}^{\prime}\right){\rm
vec}\left(e_{k}U_{p}^{\prime}v_{n_{p}}e_{p}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$ $\displaystyle\sum_{p\neq
k}^{K}\mathbb{E}\left[\sum_{r=1}^{n_{p}}\sum_{m=1}^{n_{p}}W_{k,i}^{2}U_{k,i}^{2}U_{p,r}U_{p,m}{\rm
vec}\left(e_{k}e_{p}^{\prime}\right){\rm
vec}\left(e_{k}e_{p}^{\prime}\right)^{\prime}n_{k}^{-1}n_{p}^{-1}\right]$
$\displaystyle=$ $\displaystyle\sum_{p\neq
k}^{K}\mathbb{E}\left[\sum_{r=1}^{n_{p}}W_{k,i}^{2}U_{k,i}^{2}U_{p,r}^{2}\left(e_{p}\otimes
e_{k}\right)\left(e_{p}^{\prime}\otimes
e_{k}^{\prime}\right)n_{k}^{-1}n_{p}^{-1}\right]$ $\displaystyle=$
$\displaystyle\sum_{p\neq k}^{K}\psi_{k,i}n_{p}\left(e_{p}\otimes
e_{k}\right)\left(e_{p}^{\prime}\otimes
e_{k}^{\prime}\right)n_{k}^{-1}n_{p}^{-1}$ $\displaystyle=$
$\displaystyle\sum_{p\neq k}^{K}\psi_{k,i}\left(J_{p}^{e}\otimes
J_{k}^{e}\right)n_{k}^{-1}$ $\displaystyle=$
$\displaystyle\psi_{k,i}\left[\left(I_{K}-J_{k}^{e}\right)\otimes
J_{k}\right]n_{k}^{-1}.$

So, we have

$\displaystyle\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}\left[\left[3\phi_{k,i}+\psi_{k,i}\left(n_{k}-1\right)\right]n_{k}^{-2}J_{k\otimes}+\psi_{k,i}n_{k}^{-1}\left[\left(I_{K}-J_{k}\right)\otimes
J_{k}\right]\right]$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\left[n_{k}^{-1}\left(3\bar{\phi}_{k}-\bar{\psi}_{k}\right)J_{k\otimes}^{e}+\bar{\psi}_{k}\left(I_{K}\otimes
J_{k}^{e}\right)\right],$

where $\bar{\phi}_{k}$ and $\bar{\psi}_{k}$, $k=1,\ldots,K$ are defined as

$\bar{\phi}_{k}=\frac{1}{n_{k}}\sum_{k=1}^{n_{k}}\phi_{k,i},\qquad\text{and}\qquad\bar{\psi}_{k}=\frac{1}{n_{k}}\sum_{k=1}^{n_{k}}\psi_{k,i},$

respectively.

####  C.1.2 The Second Term

For $l\neq k$, we have

$\displaystyle\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{n_{k}}\sum_{q=1}^{n_{k}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}U_{p}^{\prime}v_{n_{p}}e_{p}^{\prime}\right){\rm
vec}\left(e_{l}U_{q}^{\prime}v_{n_{q}}e_{q}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right){\rm
vec}\left(e_{l}U_{l}^{\prime}v_{n_{l}}e_{l}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle+\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}U_{l}^{\prime}v_{n_{l}}e_{l}^{\prime}\right){\rm
vec}\left(e_{l}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right].$

The first term is given by

$\displaystyle\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right){\rm
vec}\left(e_{l}U_{l}^{\prime}v_{n_{l}}e_{l}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{r}^{n_{k}}\sum_{m}^{n_{l}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}U_{k,r}U_{l,m}{\rm
vec}\left(e_{k}e_{k}^{\prime}\right){\rm
vec}\left(e_{l}e_{l}^{\prime}\right)^{\prime}n_{l}^{-1}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{r}^{n_{k}}\sum_{m}^{n_{l}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}U_{k,r}U_{l,m}{\rm
vec}\left(J_{k}^{e}\right){\rm
vec}\left(J_{l}^{e}\right)^{\prime}n_{l}^{-1}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}^{2}U_{l,j}^{2}{\rm
vec}\left(J_{k}^{e}\right){\rm
vec}\left(J_{l}^{e}\right)^{\prime}n_{l}^{-1}n_{k}^{-1}\right]$
$\displaystyle=$ $\displaystyle{\rm vec}\left(J_{k}^{e}\right){\rm
vec}\left(J_{l}^{e}\right)^{\prime}n_{l}^{-1}n_{k}^{-1}.$

The second term is given by

$\displaystyle\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}U_{l}^{\prime}v_{n_{l}}e_{l}^{\prime}\right){\rm
vec}\left(e_{l}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{r=1}^{n_{l}}\sum_{m=1}^{n_{k}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}U_{l,r}U_{k,m}{\rm
vec}\left(e_{k}e_{l}^{\prime}\right){\rm
vec}\left(e_{l}e_{k}^{\prime}\right)^{\prime}n_{l}^{-1}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}^{2}U_{l,j}^{2}{\rm
vec}\left(e_{k}e_{l}^{\prime}\right){\rm
vec}\left(e_{l}e_{k}^{\prime}\right)^{\prime}n_{l}^{-1}n_{k}^{-1}\right]$
$\displaystyle=$ $\displaystyle{\rm vec}\left(e_{k}e_{l}^{\prime}\right){\rm
vec}\left(e_{l}e_{k}^{\prime}\right)^{\prime}n_{l}^{-1}n_{k}^{-1}$
$\displaystyle=$ $\displaystyle\left(J_{l}^{e}\otimes
J_{k}^{e}\right)K_{K}n_{l}^{-1}n_{k}^{-1}.$

So, we have

$\displaystyle\sum_{k=1}^{K}\sum_{l\neq
k}^{K}\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{l}}\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}U_{l,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{K}\sum_{l\neq
k}^{K}\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{l}}\left[{\rm
vec}\left(J_{k}^{e}\right){\rm
vec}\left(J_{l}^{e}\right)^{\prime}n_{l}^{-1}n_{k}^{-1}+\left(J_{l}^{e}\otimes
J_{k}^{e}\right)K_{K}n_{l}^{-1}n_{k}^{-1}\right]$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\sum_{l\neq k}^{K}\left[{\rm
vec}\left(J_{k}^{e}\right){\rm
vec}\left(J_{l}^{e}\right)^{\prime}+\left(J_{l}^{e}\otimes
J_{k}^{e}\right)K_{K}\right]$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{K}\left[{\rm vec}\left(J_{k}^{e}\right){\rm
vec}\left(I_{K}\right)^{\prime}+\left(I_{K}\otimes
J_{k}^{e}\right)K_{K}-J_{k\otimes}^{e}\left(K_{K}+I_{K^{2}}\right)\right]$
$\displaystyle=$ $\displaystyle{\rm vec}\left(I_{K}\right){\rm
vec}\left(I_{K}\right)^{\prime}+K_{K}-2\sum_{k=1}^{K}J_{k\otimes}^{e},$

the last equality use the fact that $J_{k\otimes}^{e}={\rm
vec}\left(J_{k}^{e}\right){\rm vec}\left(J_{k}^{e}\right)^{\prime}$ as
$J_{k}^{e}=e_{k}e_{k}^{\prime}$, and $J_{k\otimes}^{e}K_{K}=J_{k\otimes}^{e}$.

####  C.1.3 The Third Term

We have $i\neq j$, then

$\displaystyle\mathbb{E}\left[W_{k,i}W_{k,j}U_{k,i}U_{k,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p}^{n_{k}}\sum_{q}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}{\rm
vec}\left(e_{k}U_{p}^{\prime}v_{n_{p}}e_{p}^{\prime}\right){\rm
vec}\left(e_{k}U_{q}^{\prime}v_{n_{q}}e_{q}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}W_{k,j}U_{k,i}U_{k,j}{\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right){\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\sum_{r}^{n_{k}}\sum_{m}^{n_{k}}\mathbb{E}\left[W_{k,i}W_{k,j}U_{k,i}U_{k,j}U_{k,r}U_{k,m}{\rm
vec}\left(e_{k}e_{k}^{\prime}\right){\rm
vec}\left(e_{k}e_{k}^{\prime}\right)^{\prime}n_{k}^{-2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}W_{k,j}U_{k,i}U_{k,j}U_{k,i}U_{k,j}+W_{k,i}W_{k,j}U_{k,i}U_{k,j}U_{k,j}U_{k,i}\right]J_{k\otimes}^{e}n_{k}^{-2}$
$\displaystyle=$ $\displaystyle
2\mathbb{E}\left[W_{k,i}W_{k,j}U_{k,i}^{2}U_{k,j}^{2}\right]J_{k\otimes}^{e}n_{k}^{-2}$
$\displaystyle=$ $\displaystyle 2J_{k\otimes}^{e}n_{k}^{-2}.$

So we have

$\displaystyle\mathbb{E}\left[\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}\sum_{j\neq
i}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}\sum_{j\neq
i}^{n_{k}}2J_{k\otimes}^{e}n_{k}^{-2}\right]=\sum_{k=1}^{K}2J_{k\otimes}^{e}\left(1-n_{k}^{-1}\right).$

####  C.1.4 Combine

Now we have

$\displaystyle\mathbb{E}\left[\sum_{k=1}^{K}\sum_{i=1}^{n_{k}}W_{k,i}U_{k,i}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]\left[\sum_{l=1}^{K}\sum_{j=1}^{n_{k}}W_{l,j}U_{l,j}{\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}\right]-{\rm
vec}\left(I_{K}\right){\rm vec}\left(I_{K}\right)^{\prime}$ $\displaystyle=$
$\displaystyle
K_{K}+\sum_{k=1}^{K}\left[n_{k}^{-1}\left(3\bar{\phi}_{k}-\bar{\psi}_{k}\right)J_{k\otimes}^{e}+\bar{\psi}_{k}\left(I_{K}\otimes
J_{k}^{e}\right)-2J_{k\otimes}^{e}+2J_{k\otimes}^{e}\left(1-n_{k}^{-1}\right)\right]$
$\displaystyle=$ $\displaystyle K_{K}+\Upsilon_{K}^{e},$

where $\Upsilon_{K}^{e}=\sum_{k=1}^{K}\Psi_{k}^{e}$ with

$\displaystyle\Psi_{k}^{e}$
$\displaystyle=n_{k}^{-1}\left(3\bar{\phi}_{k}-\bar{\psi}_{k}\right)J_{k\otimes}^{e}+\bar{\psi}_{k}\left(I_{K}\otimes
J_{k}^{e}\right)-2J_{k\otimes}^{e}+2J_{k\otimes}^{e}\left(1-n_{k}^{-1}\right)$
$\displaystyle=n_{k}^{-1}\left(3\bar{\phi}_{k}-2-\bar{\psi}_{k}\right)J_{k\otimes}^{e}+\bar{\psi}_{k}\left(I_{K}\otimes
J_{k}^{e}\right).$

###  C.2 The Form of Matrix $\mathcal{I}_{A}^{(2)}$

We first define the $K\times 1$ vector $\bar{S}$ with elements

$\bar{S}_{k}=\sum_{i=1}^{n_{k}}W_{k,i}U_{k,i}F{}_{k,i}U_{k}-\left(n_{k}-1\right),$

and obviously we have $\mathbb{E}\left(S_{k}S_{l}\right)=0$ for $k\neq l$. And
we need to compute

$\displaystyle\mathbb{E}\left(\bar{S}_{k}^{2}\right)$
$\displaystyle=\mathbb{E}\left(\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}F_{k,i}U_{k}U_{k}^{\prime}F_{k,j}^{\prime}\right)-\left(n_{k}-1\right)^{2}$
$\displaystyle=\mathbb{E}\left(\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{k}}W_{k,i}W_{k,j}\left(U_{k,i}U_{k,j}\right)F_{k,i}\left(U_{k}U_{k}^{\prime}\right)F_{k,j}^{\prime}\right)-\left(n_{k}-1\right)^{2}$
$\displaystyle=\mathbb{E}\left(\sum_{i=1}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}F_{k,i}\left(U_{k}U_{k}^{\prime}\right)F_{k,i}^{\prime}\right)-\left(n_{k}-1\right)^{2}$
$\displaystyle\quad+\mathbb{E}\left(\sum_{i=1}^{n_{k}}\sum_{j\neq
i}^{n_{k}}W_{k,i}W_{k,j}\left(U_{k,i}U_{k,j}\right)F_{k,i}\left(U_{k}U_{k}^{\prime}\right)F_{k,j}^{\prime}\right).$

Based on the following results on
$F_{k,i}=\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}v_{n_{k}}^{\perp\ \prime}$,
for $i\neq j$ we have

$\displaystyle F_{k,i}F_{k,i}^{\prime}$
$\displaystyle=\tilde{e}_{i}^{\prime}\left(v_{n_{k}}^{\perp}v_{n_{k}}^{\perp\
\prime}\right)\tilde{e}_{i}=\tilde{e}_{i}^{\prime}\left(I_{n_{k}}-v_{n_{k}}v_{n_{k}}^{\prime}\right)\tilde{e}_{i}=1-n_{k}^{-1}$
$\displaystyle F_{k,i}\tilde{e}_{i}\tilde{e}_{i}^{\prime}F_{k,i}^{\prime}$
$\displaystyle=\left(\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}v_{n_{k}}^{\perp\
\prime}\tilde{e}_{i}\right)\left(\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}v_{n_{k}}^{\perp\
\prime}\tilde{e}_{i}\right)=\left(1-n_{k}^{-1}\right)^{2}$ $\displaystyle
F_{k,i}\tilde{e}_{j}\tilde{e}_{i}^{\prime}F_{k,j}^{\prime}$
$\displaystyle=\left(\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}v_{n_{k}}^{\perp\
\prime}\tilde{e}_{j}\right)\left(\tilde{e}_{i}^{\prime}v_{n_{k}}^{\perp}v_{n_{k}}^{\perp\
\prime}\tilde{e}_{j}\right)=\left(-n_{k}^{-1}\right)^{2}=n_{k}^{-2},$

we have

$\displaystyle\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}F_{k,i}\left(U_{k}U_{k}^{\prime}\right)F_{k,i}^{\prime}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p}^{n_{k}}\sum_{q}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}\left(U_{k,p}U_{k,q}\right)F_{k,i}\left(\tilde{e}_{p}\tilde{e}_{q}^{\prime}\right)F_{k,i}^{\prime}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{4}F_{k,i}\left(\tilde{e}_{i}\tilde{e}_{i}^{\prime}\right)F_{k,i}^{\prime}\right]+\sum_{p\neq
i}^{n_{k}}\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}U_{k,p}^{2}F_{k,i}\left(\tilde{e}_{p}\tilde{e}_{p}^{\prime}\right)F_{k,i}^{\prime}\right]$
$\displaystyle=$ $\displaystyle
3\phi_{k,i}F_{k,i}\left(\tilde{e}_{i}\tilde{e}_{i}^{\prime}\right)F_{k,i}^{\prime}+\sum_{p\neq
i}^{n_{k}}\psi_{k,i}F_{k,i}\left(\tilde{e}_{p}\tilde{e}_{p}^{\prime}\right)F_{k,i}^{\prime}$
$\displaystyle=$ $\displaystyle
3\phi_{k,i}F_{k,i}\left(\tilde{e}_{i}\tilde{e}_{i}^{\prime}\right)F_{k,i}^{\prime}+\psi_{k,i}n_{k}^{-2}\left(n_{k}-1\right)$
$\displaystyle=$ $\displaystyle
3\phi_{k,i}\left(1-n_{k}^{-1}\right)^{2}+\psi_{k,i}n_{k}^{-2}\left(n_{k}-1\right).$

So

$\sum_{i=1}^{n_{k}}\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}F_{k,i}\left(U_{k}U_{k}^{\prime}\right)F_{k,i}^{\prime}\right]=3n_{k}\bar{\phi}_{k}\left(1-n_{k}^{-1}\right)^{2}+\bar{\psi}_{k}\left(1-n_{k}^{-1}\right),$

and for $i\neq j$, we have

$\displaystyle\mathbb{E}\left[W_{k,i}W_{k,j}\left(U_{k,i}U_{k,j}\right)F_{k,i}\left(U_{k}U_{k}^{\prime}\right)F_{k,j}^{\prime}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p}^{n_{k}}\sum_{q}^{n_{k}}W_{k,i}W_{k,j}\left(U_{k,i}U_{k,j}\right)\left(U_{k,p}U_{k,q}\right)F_{k,i}\left(\tilde{e}_{p}\tilde{e}_{q}^{\prime}\right)F_{k,j}^{\prime}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}W_{k,j}\left(U_{k,i}^{2}U_{k,j}^{2}\right)F_{k,i}\left(\tilde{e}_{i}\tilde{e}_{j}^{\prime}\right)F_{k,j}^{\prime}\right]+\mathbb{E}\left[W_{k,i}W_{k,j}\left(U_{k,i}^{2}U_{k,j}^{2}\right)F_{k,i}\left(\tilde{e}_{j}\tilde{e}_{i}^{\prime}\right)F_{k,j}^{\prime}\right]$
$\displaystyle=$ $\displaystyle
F_{k,i}\left(\tilde{e}_{i}\tilde{e}_{j}^{\prime}+\tilde{e}_{j}\tilde{e}_{i}^{\prime}\right)F_{k,j}^{\prime}=\left(1-n_{k}^{-1}\right)^{2}+n_{k}^{-2}.$

So,

$\mathbb{E}\left[\sum_{j\neq
i}^{n_{k}}W_{k,i}W_{k,j}\left(U_{k,i}U_{k,j}\right)F_{k,i}\left(U_{k}U_{k}^{\prime}\right)F_{k,j}^{\prime}\right]=\left(n_{k}-1\right)\left[\left(1-n_{k}^{-1}\right)^{2}+n_{k}^{-2}\right],$

and this leads to

$\displaystyle\mathbb{E}\left(\bar{S}_{k}^{2}\right)$
$\displaystyle=3n_{k}\bar{\phi}_{k}\left(1-n_{k}^{-1}\right)^{2}+\bar{\psi}_{k}\left(1-n_{k}^{-1}\right)+n_{k}\left(n_{k}-1\right)\left[\left(1-n_{k}^{-1}\right)^{2}+n_{k}^{-2}\right]-\left(n_{k}-1\right)^{2},$
$\displaystyle\mathbb{E}\left(S_{k}^{2}\right)$
$\displaystyle=\lambda_{k}^{-2}\left(n_{k}-1\right)^{-2}\mathbb{E}\left(\bar{S}_{k}^{2}\right)=\lambda_{k}^{-2}n_{k}^{-1}\left[3\bar{\phi}_{k}-1+\left(\bar{\psi}_{k}+1\right)\left(n_{k}-1\right)^{-1}\right],$

and define the matrix $\Xi$ as $\Xi_{kk}=\mathbb{E}\left(S_{k}^{2}\right)$ and
$\Xi_{kl}=\mathbb{E}\left(S_{k}S_{j}\right)=0$ for $k\neq l$, we have
$\mathcal{I}_{A}^{(2)}=\frac{1}{4}E_{d}^{\prime}\Xi E_{d}$.

###  C.3 The Form of Matrix $\mathcal{I}_{A}^{(12)}$

We first need to compute

$\displaystyle\mathbb{E}\left[\sum_{i=1}^{n_{k}}W_{k,i}U_{k,i}F_{k,i}U_{k}-\left(n_{k}-1\right)\right]\left[\sum_{l=1}^{K}\sum_{j=1}^{n_{l}}W_{l,j}U_{l,j}{\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}-{\rm
vec}(I_{K})^{\prime}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{l=1}^{K}\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{l}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}\right]-\left(n_{k}-1\right){\rm
vec}(I_{K})^{\prime}$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{i=1}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]-\left(n_{k}-1\right){\rm
vec}(I_{K})^{\prime}$
$\displaystyle+\mathbb{E}\left[\sum_{i=1}^{n_{k}}\sum_{j\neq
i}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle+\mathbb{E}\left[\sum_{l\neq
k}^{K}\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{l}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}\right].$

For the first term, we have

$\displaystyle\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{2}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{q}^{K}\sum_{p}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}\left(F_{k,i}U_{k,p}\tilde{e}_{p}\right){\rm
vec}\left(e_{k}U_{q}^{\prime}v_{n_{q}}e_{q}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}U_{k,p}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p}^{n_{k}}\sum_{r}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}U_{k,p}U_{k,r}\left(F_{k,i}\tilde{e}_{r}\right){\rm
vec}\left(e_{k}e_{k}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}^{2}U_{k,i}^{4}\left(F_{k,i}\tilde{e}_{i}\right){\rm
vec}\left(e_{k}e_{k}^{\prime}\right)^{\prime}n_{k}^{-1}\right]+\mathbb{E}\left[\sum_{p\neq
i}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}U_{k,p}^{2}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(e_{k}e_{k}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$ $\displaystyle
3\phi_{k,i}\left[\left(F_{k,i}\tilde{e}_{i}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}\right]+\sum_{p\neq
i}^{n_{k}}\psi_{k,i}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}$ $\displaystyle=$ $\displaystyle
3\phi_{k,i}\left(1-n_{k}^{-1}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}-\psi_{k,i}\left(n_{k}-1\right)n_{k}^{-1}{\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}$ $\displaystyle=$
$\displaystyle\left(3\phi_{k,i}-\psi_{k,i}\right)\left(1-n_{k}^{-1}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}.$

So

$\displaystyle\mathbb{E}\left[\sum_{i}^{n_{k}}W_{k,i}^{2}U_{k,i}^{2}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=\left(3\bar{\phi}_{k}-\bar{\psi}_{k}\right)\left(1-n_{k}^{-1}\right){\rm
vec}\left(J_{k}\right)^{\prime},$

and for the second term, we have $i\neq j$ and

$\displaystyle\mathbb{E}\left[W_{k,i}W_{k,j}U_{k,i}U_{k,j}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{q=1}^{K}\sum_{p=1}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}U_{k,p}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(e_{k}U_{q}^{\prime}v_{n_{q}}e_{q}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}U_{k,p}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(e_{k}U_{k}^{\prime}v_{n_{k}}e_{k}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{n_{k}}\sum_{r=1}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}U_{k,p}U_{k,r}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(e_{k}e_{k}^{\prime}\right)^{\prime}n_{k}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}W_{k,j}U_{k,i}^{2}U_{k,j}^{2}\left(F_{k,i}\tilde{e}_{i}\right){\rm
vec}\left(J_{k}\right)^{\prime}n_{k}^{-1}\right]+\mathbb{E}\left[W_{k,i}W_{k,j}U_{k,i}^{2}U_{k,j}^{2}\left(F_{k,i}\tilde{e}_{j}\right){\rm
vec}\left(J_{k}\right)^{\prime}n_{k}^{-1}\right]$ $\displaystyle=$
$\displaystyle\left(F_{k,i}\tilde{e}_{i}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}+\left(F_{k,i}\tilde{e}_{j}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}$ $\displaystyle=$
$\displaystyle\left(1-n_{k}^{-1}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}-n_{k}^{-1}{\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}$ $\displaystyle=$
$\displaystyle\left(1-2n_{k}^{-1}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}n_{k}^{-1}.$

So

$\mathbb{E}\left[\sum_{i=1}^{n_{k}}\sum_{j\neq
i}^{n_{k}}W_{k,i}W_{k,j}U_{k,i}U_{k,j}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}\right]=\left(n_{k}-1\right)\left(1-2n_{k}^{-1}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime},$

as for the third term, we have $k\neq l$, and

$\displaystyle\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}U_{l,j}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{q=1}^{K}\sum_{p=1}^{n_{k}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}U_{k,p}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(e_{l}U_{q}^{\prime}v_{n_{q}}e_{q}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{n_{k}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}U_{k,p}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(e_{l}U_{l}^{\prime}v_{n_{l}}e_{l}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\sum_{p=1}^{n_{k}}\sum_{r=1}^{n_{l}}W_{k,i}W_{l,j}U_{k,i}U_{k,p}U_{l,j}U_{l,r}\left(F_{k,i}\tilde{e}_{p}\right){\rm
vec}\left(e_{l}e_{l}^{\prime}\right)^{\prime}n_{l}^{-1}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[W_{k,i}W_{l,j}U_{k,i}^{2}U_{l,j}^{2}\left(F_{k,i}\tilde{e}_{i}\right){\rm
vec}\left(e_{l}e_{l}^{\prime}\right)^{\prime}n_{l}^{-1}\right]$
$\displaystyle=$ $\displaystyle\left(F_{k,i}\tilde{e}_{i}\right){\rm
vec}\left(J_{l}^{e}\right)^{\prime}n_{l}^{-1}$ $\displaystyle=$
$\displaystyle\left(1-n_{k}^{-1}\right){\rm
vec}\left(J_{l}^{e}\right)^{\prime}n_{l}^{-1}.$

So, we have

$\mathbb{E}\left[\sum_{l\neq
k}^{K}\sum_{i=1}^{n_{k}}\sum_{j=1}^{n_{l}}W_{k,i}W_{l,j}U_{k,i}U_{l,j}\left(F_{k,i}U_{k}\right){\rm
vec}\left(e_{l}X_{0}^{\prime}\right)^{\prime}n_{l}^{-\frac{1}{2}}\right]=\left(n_{k}-1\right){\rm
vec}\left(I_{K}-J_{k}^{e}\right)^{\prime},$

and

$\displaystyle\mathbb{E}\left[\sum_{i=1}^{n_{k}}W_{k,i}U_{k,i}F_{k,i}U_{k}-\left(n_{k}-1\right)\right]\left[\sum_{l=1}^{K}\sum_{j=1}^{n_{k}}W_{k,j}U_{k,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}-{\rm
vec}(I_{K})^{\prime}\right]$ $\displaystyle=$
$\displaystyle\left(3\bar{\phi}_{k}-\bar{\psi}_{k}\right)\left(1-n_{k}^{-1}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}+\left(n_{k}-1\right)\left(1-2n_{k}^{-1}\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}$ $\displaystyle+\left(n_{k}-1\right){\rm
vec}\left(I_{K}-J_{k}^{e}\right)^{\prime}-\left(n_{k}-1\right){\rm
vec}(I_{K})^{\prime}$ $\displaystyle=$
$\displaystyle\left[\left(3\bar{\phi}_{k}-\bar{\psi}_{k}-2\right)\left(1-n_{k}^{-1}\right)\right]{\rm
vec}\left(J_{k}^{e}\right)^{\prime}.$

We finally arrive at the following expression for $\Theta$, with
$\mathcal{I}_{A}^{(21)}=\frac{1}{2}E_{d}^{\prime}\Theta E_{d}$,

$\Theta=\mathbb{E}\left[S\left(\sum_{k=1}^{K}\sum_{j=1}^{n_{k}}W_{k,j}U_{k,j}{\rm
vec}\left(e_{k}X_{0}^{\prime}\right)^{\prime}n_{k}^{-\frac{1}{2}}-{\rm
vec}(I_{K})^{\prime}\right)\right]=\sum_{k=1}^{K}e_{k}\left(e_{k}^{\prime}\Theta\right),$

where its $k$-th row $e_{k}^{\prime}\Theta$ is given by

$\displaystyle e_{k}^{\prime}\Theta$
$\displaystyle=-\left(n_{k}-1\right)^{-1}\lambda_{k}^{-1}\left[\left(3\bar{\phi}_{k}-\bar{\psi}_{k}-2\right)\left(1-n_{k}^{-1}\right)\right]{\rm
vec}\left(J_{k}^{e}\right)^{\prime}$
$\displaystyle=-\lambda_{k}^{-1}n_{k}^{-1}\left(3\bar{\phi}_{k}-\bar{\psi}_{k}-2\right){\rm
vec}\left(J_{k}^{e}\right)^{\prime}.$


[a,b]Sarah. M. Wagner,

# Statistical properties of flux variations in blazar light curves at GeV and
TeV energies

P. R. Burd  D. Dorner  K. Mannheim  S. Buson  A. Gokus  G. Madejski  J. D.
Scargle  on behalf of the Fermi-LAT Collaboration  A. Arbet-Engels  D. Baack
M. Balbo  A. Biland  T. Bretz  J. Buss  D. Elsaesser  L. Eisenberger  D.
Hildebrand  R. Iotov  A. Kalenski  D. Neise  M. Noethe  A. Paravac  W. Rhode
B. Schleicher  V. Sliusar  R. Walter  on behalf of the FACT Collaboration

###### Abstract

Despite numerous detections of individual flares, the physical origin of the
rapid variability observed from blazars remains uncertain. Using Bayesian
blocks and the Eisenstein-Hut HOP algorithm, we characterize flux variations
of high significance in the $\gamma$-ray light curves of two samples of
blazars. Daily binned long-term light curves of TeV-bright blazars observed
with the First G-APD Cherenkov Telescope (FACT) are compared to those of GeV-
bright blazars observed with the Large Area Telescope on board the Fermi
Gamma-ray Space Telescope (Fermi-LAT). We find no evidence for systematic
asymmetry of the flux variations based on the derived rise and decay time
scales. Additionally, we show that the daily-binned blazar light curves can be
described by an exponential stochastic Ornstein-Uhlenbeck process with
parameters depending on energy. Our analysis suggests that the flux
variability in both samples is a superposition of faster fluctuations. This
is, for instance, challenging to explain by shock-acceleration but expected
for magnetic reconnection.

##  1 Introduction

Blazars are active galactic nuclei (AGN) with highly relativistic outflows
that are directed closely towards our line of sight. They can be sub-
classified based on their spectral energy distribution (SED), which
characteristically shows a double-hump structure. The low-energy hump is well
described by synchrotron emission while the high-energy hump is likely due to
inverse Compton scattering and/or hadronic processes. Based on the position of
the peak of the former, blazars can be divided in low- and high-synchrotron-
peaked objects (LSP and HSP) [1]. There have been many attempts to link these
observational properties to the physical nature of blazars [2]. However, it
remains uncertain which intrinsic differences there are between the blazar
classes and what physical mechanisms accelerate the particles necessary to
produce the observed $\gamma$-radiation in the first place [3]. We study the
flux variability in the range of the high-energy hump in the brightest LSP and
HSP objects in order to compare the blazar classes and constrain possible
emission models.

##  2 Data and instruments

The high-energy hump of LSP and HSP objects falls in different energy ranges.
Currently, no instrument covers the wide $\gamma$-ray range with continuous
sensitivity. Hence, we utilize the two instruments described in the following
to measure flux variations of blazars in daily binning at both lower as well
as higher $\gamma$-ray energies.

###  Fermi-LAT

The Large Area Telescope (LAT) on board the Fermi satellite operates from 20
MeV up to 1 TeV. Due to its capability of monitoring the entire sky within a
day, it is possible to gain continuous information about each $\gamma$-ray
source over the whole length of the Fermi mission. Hence, we computed light
curves with daily binning from 2008-08-05 to 2020-12-31 (MET 239587201 -
631065605) for an energy range between 100 MeV and 300 GeV. Our source sample
for Fermi-LAT analysis consists of the brightest blazars in the Fourth Fermi-
LAT source catalog (4FGL)[4], as listed in Table 1 on the left.

We performed the standard data reduction process111
https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/ with Science
Tools 1.2.23,fermipy 0.20.0. We model all sources included in the 4FGL catalog
that are within a region of interest (ROI) of $15^{\circ}$ around each source,
and model the isotropic diffusion emission with iso_P8R3_SOURCE_V2_v1 and the
Galactic diffuse emission with gll_iem_v07. For further analysis of the light
curves, we only consider flux bins with test statistic (TS)$>4$ and a flux
uncertainty equal to or smaller than its flux, analogous to [5].

### FACT

Since October 2011, the First G-APD Cherenkov Telescope (FACT) has been
measuring $\gamma$-rays from hundreds of GeV up to several TeV. The three
brightest blazars with continuous light curves observed by FACT are listed in
Table 1. The data were taken from 2011-11-15 to 2020-01-23 and are analyzed
using the Modular Analysis and Reconstruction Software (MARS) [6]. For the
background suppression, the ’light curve cuts’ as described in [7] were used.
A data quality selection based on the artificial trigger rate [7, 8] was
applied. More details on the analysis can be found in [9]. Since the
observations are ground-based, there are seasonal gaps of $\sim 120$ days each
year. We divide each FACT light curve into seven stretches where the source
could be observed continuously222 Further short gaps on the order of days can
be caused by bad weather and the break in observing around full moon. and
refer to these as (FACT light curve) chunks.

|  Fermi-LAT (all FSRQ and LSP)
---
Name | $\mathrm{log}(\nu_{\text{syn}}/\mathrm{Hz})$ | $z$
3C 454.3 | 13.6 | 0.859
PKS 1510-089 | 13.1 | 0.36
3C 279 | 12.6 | 0.536
PKS 1424-41 | 13.1 | 1.522
PKS 1830-211 | 12.2 | 2.5
4C +21.35 | 13.5 | 0.433
3C 273 | 13.5 | 0.158
| FACT (all BLL and HSP)
---
Name | $\mathrm{log}(\nu_{\text{syn}}/\mathrm{Hz})$ | $z$
Mrk 421 | 16.6 | 0.03
Mrk 501 | 17.1 | 0.033
1ES 1959+650 | 16.6 | 0.047
Table 1: Sources considered for Fermi-LAT (left) and FACT (right) with the
corresponding logarithm of the position of the synchrotron peak frequency
$\nu_{\text{syn}}$ and the redshift $z$ taken from 4LAC [10] and [1].

##  3 Definition of flares and their asymmetry

We derive the best fit piece wise constant representation for all Fermi-LAT
light curves and FACT chunks by applying the Bayesian block algorithm333
https://docs.astropy.org/en/stable/api/astropy.stats.bayesian_blocks.html444
Assuming that the flux errors are Gaussian, we set fitness to ’measures’.
[11]. For the purpose of this work, we define flares as groups of blocks,
following [5]. The algorithms implemented for this procedure are available on
GitHub555 Code for flare analysis: https://github.com/swagner-
astro/lightcurves.

We determine the peak times to be at the center of local maxima in the block
representation and every block that is subsequently lower to the left and
right belongs to that peak (“hop666 Thus, referred to as HOP algorithm, which
is not an acronym.” downwards). For the resulting “valley” blocks, we
implemented four methods to define where one flare ends and another one
starts. We illustrate those with an exemplary part of the Fermi-LAT light
curve of 3C 279 in Fig. 1.

Figure 1: All panels show an identical part of the daily-binned Fermi-LAT
light curve of 3C 279 (black bins) and its Bayesian block representation (blue
line). Flares defined with the corresponding method (top left of each panel)
are shaded orange and purple. The green baseline in the top right panel shows
the quiescent background introduced in [5].

The half and sharp methods simply divide the valley block in half, or neglect
it completely, respectively. The flip method extrapolates the slope of the
flare by flipping the length of the adjacent block onto the valley block. The
start/end then sits at the bordering edge of the valley block minus/plus the
length of the adjacent block, as shown in Fig. 1 on the left. In order to
avoid overlap, we set the start/end of a flare to the center of the block
(half) if flipping the block would exceed the center of the valley block.
Originally, the start/end of the so-called HOP groups is determined by the
flux exceeding/dropping under a predefined flux level (baseline, e.g. the
quiescent level [5], see top panel of the right plot in Fig. 1). We refer to
this as baseline method. In case of multiple flares above the baseline, we
again divide the valley block in half (half). While all of the methods have
advantages and disadvantages, it becomes clear that an objective definition of
a flare or flux variation in this context always needs some kind of
assumption. For this work, we choose the flip method since it allows to factor
in as much data as possible without stretching the flares to be unnaturally
long. The results do not change significantly, for either of the methods
illustrated here. Based on the determined start, peak and end time of each
flare, we compute the asymmetry measure

$A=\frac{t_{\text{rise}}-t_{\text{decay}}}{t_{\text{rise}}+t_{\text{decay}}}$
(1)

which can shed light on the physical mechanisms that cause the high-energy
variability of blazars. A negative asymmetry indicates
$t_{\text{rise}}<t_{\text{decay}}$ and can for instance be explained by a
moving shock colliding with a stationary feature [12]. Symmetric flares can be
explained by mini-jets sweeping across the line of sight with a constant
angular velocity [13]. Physical emission models producing flares with positive
asymmetry ($t_{\text{rise}}>t_{\text{decay}}$) are less common.

A large portion of flares in the Fermi-LAT light curves as well as the FACT
chunks are found to consist of just a single Bayesian block (47% and 48%,
respectively). Such flares often result in a trivial asymmetry of exactly zero
(flip to both sides). Utilizing Bayesian blocks for binning777 Due to the
variable bin width we divide the number of counts in each bin by its width,
resulting in the probability density. This holds for all histograms in this
work. the distribution of asymmetry measures, as shown in Fig. 2 on the left
with thick solid lines, illustrates how prominent this feature is.
Interestingly, this artefact is not at all captured with constant binning
(thin dashed lines). Since single block flares do not allow the resolution of
the true asymmetry of the flux variation, we exclude them for both samples.
The asymmetry measures of all remaining flares are shown in the same figure on
the right for Fermi-LAT in red and FACT in blue. In this case, the Bayesian
binning (thick solid lines) results in one block confirming the visual
impression of the constant binning (thin dashed lines) that every kind of
asymmetry occurs to a comparable degree. Based on a two-sided Kolmogorov-
Smirnov (KS) test888
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.ks_2samp.html,
it can not be rejected that the asymmetry measures from Fermi-LAT and FACT are
drawn from the same distribution. This holds for all flares (KS: p $\sim
0.42$) as well as the sub-sample with at least two blocks per flare (KS: p
$\sim 0.97$). The distribution of asymmetry measures is not correlated with
the (redshift corrected) duration of the flares nor their flux amplitude999
The Pearson and Spearman correlation coefficients show a range of
$-0.17<r_{\mathrm{Pearson/Spearman}}<0.02$. Based on the method utilized here,
we can not distinguish the asymmetry of the TeV (Fermi-LAT) and GeV (FACT)
flux variations with daily binning.

Figure 2: Distribution of asymmetry measures for the Fermi-LAT (red) and FACT
(blue) flares in Bayesian (solid thick line) and constant binning (thin dashed
line); Left: all flares; Right: flares with at least two blocks.

##  4 Quantization of the variability using Ornstein-Uhlenbeck (OU) parameters

In an additional approach, we interpret the flux variations as correlated
noise parametrized by a first order auto-regressive process. We use the
discrete OU process (see Eq. 5e in [14])

$u_{T+1}=u_{T}+\theta\Delta t(\mu-u_{T})+\sigma\sqrt{\Delta t}\mathcal{N}_{T}$
(2)

with mean revision level $\mu$, mean revision rate $\theta$, and white noise
constituted by independent draws from a normal distribution
$\mathcal{N}(m=0,\sigma^{2})$. We interpret the value of the time series
$u_{T}$ to be the logarithm of flux and extract the OU parameters
$(\mu,\sigma,\theta)$ of each light curve101010 The fact that we are able to
identify HOP groups indicates stationarity of the analyzed light curves on
relevant time-scales which is necessary for this procedure. following
[14]111111 Code for OU paramter extraction: https://github.com/PRBurd/astro-
wue.

Figure 3 (top-left) shows the resulting distribution of $\sigma$-parameters
for the Fermi-LAT light curves (red), the complete FACT light curves (green)
and the FACT chunks (blue) in Bayesian binning. The $\sigma$-value can be
interpreted as the strength of the innovation, which corresponds to the
maximum random amplitude. There are significant differences for this parameter
among the three classes. A two-sided KS test yields a high probability that
the Fermi-LAT sample is not drawn from the same distribution as the FACT
samples (KS complete:$p\sim 0.017$, KS chunks: $p\sim 1.4\times 10^{-5}$).
This indicates that the amplitude of random variability is smaller in the
Fermi-LAT sample as compared to the FACT chunks and complete light curves.
Figure 3 (top-right) shows the $\theta$-distribution for the Fermi-LAT light
curves (red), the complete FACT light curves (green) and the FACT chunks
(blue). This parameter denotes the reversion rate drawing back the flux
variation to a mean reversion level. Smaller $\theta$-values indicate light
curves (or segments in light curves) where variability takes place on time
scales which are large with respect to the time binning, while larger
$theta$-values indicate flux variations which quickly (with respect to the
time-sampling) level back out to the reversion level. Also in this case,
though on a smaller significance level, the Fermi-LAT light curves and the
FACT light curve chunks are not drawn from the same distribution (KS chunks:
$p\sim 0.05$). This does, however, not hold for the complete FACT light curves
(KS complete:$p\sim 0.85$).

In Fig. 3 (bottom-left) the number of Bayesian blocks in each light curve
(chunk) is plotted against its $\theta$-parameter resulting in an anti-
correlation121212 $R_{\mathrm{Pearson}}=-0.54/-0.58$ and
$R_{\mathrm{Spearman}}=-0.57/-0.61$ for FACT chunks/Fermi-LAT respectively..
As expected, the overall longer Fermi-LAT and complete FACT light curves
consist of more blocks than the short FACT chunks. A power law of the form
$\text{Number of block}\propto\theta^{s}$ is fit with the resulting slopes
$s_{\mathrm{FACTchunks}}=-0.458\pm 0.093$ and $s_{\mathrm{Fermi-LAT}}=-0.41\pm
0.24$. Three data points are too few to draw conclusions for the complete FACT
light curves but they do not contradict the measured trend.

Figure 3: Top-left: Distribution of the $\sigma$-parameter; Top-right:
distribution of the $\theta$-parameter; Bottom-right:The number of Bayesian
blocks for a light curve against the $\theta$-parameters fit with a power law.

##  5 Discussion

Analyzing GeV (Fermi-LAT) and TeV (FACT) blazar light curves with Bayesian
blocks and the HOP algorithm shows that a large fraction of the flux
variations is best described with just one block. In this case the rise and
decay are not resolved enough, suggesting that the true variability (possibly
tracing back to the physical process) takes place on even smaller timescales
than days. Fermi-LAT has, for instance, observed variability within several
minutes [15] and FACT detected intranight variability for Mrk 421 [16]. Thus,
it is very likely that the variations in the daily binned GeV and TeV light
curves are superpositions of individual, smaller flux fluctuations. This is
further supported by the similarity of our results to correspondingly analyzed
Fermi-LAT light curves in monthly binning [14]. Both appear to be too large to
resolve the high-energy blazar variability. When excluding the single-block
flares, every kind of asymmetry ($-1\leq A\leq 1$) is present with the same
probability for both samples, see Fig. 2. It is unlikely that a mechanism
producing strongly positively asymmetric flares results in a light curve with
just as many negatively as positively asymmetric flux fluctuations (on the
same time scale). Possible theories for particle acceleration and emission
models need to reproduce this variety of high-energy flux fluctuations in
daily binning. $\gamma$-ray flares powered in one or several plasmoids provide
a possible explanation [17]. The difference in the $\sigma$-distributions for
the chunks and complete FACT light curves can be understood in the following
way. If there are chunks which show large amplitude variations, the extracted
$\sigma$-parameter will have a larger value. When extracting the
$\sigma$-parameter of an entire FACT light curve, the extracted $\sigma$-value
will be influenced by long-time behavior possibly containing very little
variation in flux. The anti-correlation of the numbers of Bayesian blocks and
the $\theta$-parameters, can be interpreted as follows. The smaller the
$\theta$-parameter, the larger the number of Bayesian blocks describing the
light curve (chunks). This means that the variability for these sources is
more complex and over all larger. Our findings show that the $\theta$ and
$\sigma$-parameters can be used to quantify the variability in a light curve.

We conclude that daily binning at the given sensitivities is insufficient to
infer strong constraints on the physical emission mechanisms driving
variability in LSP as well as HSP blazars. Emission models for $\gamma$-ray
radiation need to be able to reproduce the properties of the observed light
curves, particularly the fast variability, analyzed here.

## Acknowledgements

SMW is grateful for the Klaus Murmann Scholarship awarded by the Stiftung der
Deutschen Wirtschaft (sdw, German Business Foundation) as well as the warm
hospitality and great inspiration at KIPAC/SLAC and Stanford University. PRB
acknowledges DATEV Stiftung Zukunft for funding the interdisciplinary data lab
DataSphere@JMUW. JDS thanks Joe Bredekamp, the NASA Applied Information
Systems Research Program and the NASA Astrophysics Data Analysis Program
(Grant NNX16AL02G) for support. Fermi: The Fermi-LAT Collaboration
acknowledges support for LAT development, operation and data analysis from
NASA and DOE (United States), CEA/Irfu and IN2P3/CNRS (France), ASI and INFN
(Italy), MEXT, KEK, and JAXA (Japan), and the K.A. Wallenberg Foundation, the
Swedish Research Council and the National Space Board (Sweden). Science
analysis support in the operations phase from INAF (Italy) and CNES (France)
is also gratefully acknowledged. This work performed in part under DOE
Contract DE-AC02-76SF00515. FACT: The important contributions from ETH Zurich
grants ETH-10.08-2 and ETH-27.12-1 as well as the funding by the Swiss SNF and
the German BMBF (Verbundforschung Astro- und Astroteilchenphysik) and HAP
(Helmoltz Alliance for Astroparticle Physics) are gratefully acknowledged.
Part of this work is supported by Deutsche Forschungsgemeinschaft (DFG) within
the Collaborative Research Center SFB 876 "Providing Information by Resource-
Constrained Analysis", project C3. We are thankful for the very valuable
contributions from E. Lorenz, D. Renker and G. Viertel during the early phase
of the project. We thank the Instituto de Astrofísica de Canarias for allowing
us to operate the telescope at the Observatorio del Roque de los Muchachos in
La Palma, the Max-Planck-Institut für Physik for providing us with the mount
of the former HEGRA CT3 telescope, and the MAGIC collaboration for their
support.

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