T_{m}}\frac{\left|\sum_{t=m+1}^{m+k}\mathbf{x}_{t}^{\prime}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}-\mathbf{C}\right)^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(\frac{d^{2}}{m^{\min\left\\{1,\eta\right\\}}}a_{m}^{\max\left\\{1/2,\eta-1/2\right\\}}\right)=o_{P}\left(1\right),$

on account of the fact that $d=O\left(m^{1/4}\right)$. Therefore it follows
that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\mathbf{x}_{t}^{\prime}\left(\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}\left(\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\mathbf{x}_{t}^{\prime}\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}+o_{P}\left(1\right).$

We now show that

$r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}=o_{P}\left(1\right).$

Indeed

$\displaystyle r_{m}^{\eta-1/2}$ $\displaystyle\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\left[\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right]\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle\leq r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\displaystyle\left\|\sum_{t=m+1}^{m+k}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)\right\|\left\|\mathbf{C}^{-1}\right\|_{F}\displaystyle\left\|\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right\|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}.$

Using Lemmas B.2-B.4, this entails that

$\displaystyle r_{m}^{\eta-1/2}$ $\displaystyle\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\sum_{t=m+1}^{m+k}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\left[\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right]\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle=O_{P}(1)r_{m}^{\eta-1/2}d\left\|\mathbf{C}^{-1}\right\|_{F}\max_{a_{m}\leq
k\leq
T_{m}}\frac{k^{\zeta_{3}}}{\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle=O_{P}(1)r_{m}^{\eta-1/2}d\left\|\mathbf{C}^{-1}\right\|_{F}\max_{a_{m}\leq
k\leq T_{m}}\left(\frac{k}{k+m}\right)^{\zeta_{3}-\eta}\max_{a_{m}\leq k\leq
T_{m}}\frac{1}{\left(m+k\right)^{1-\zeta_{3}}}.$

By virtue of Lemma B.3, we can choose $1/2<\zeta_{3}<\min\\{1,\eta\\}$ in the
above, so that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\left[\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right]\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle=O_{P}(1)r_{m}^{\eta-1/2}dm^{\zeta_{3}-1}\left\|\mathbf{C}^{-1}\right\|_{F}\left(\frac{a_{m}}{m}\right)^{\zeta_{3}-\eta}$
$\displaystyle=O_{P}(1)\left\|\mathbf{C}^{-1}\right\|_{F}dm^{-1/2}a_{m}^{\zeta_{3}-1/2}.$

In order to bound $\left\|\mathbf{C}^{-1}\right\|_{F}$, we note that

$\left\|\mathbf{C}^{-1}\right\|_{F}\leq
d^{1/2}\left\|\mathbf{C}^{-1}\right\|_{2}=d^{1/2}\frac{1}{s_{\min}\left(\mathbf{C}\right)},$

where $\left\|\mathbf{C}^{-1}\right\|_{2}$ is the $L_{2}$-norm of a matrix.
Again by Assumption 2(iii), $s_{\min}\left(\mathbf{C}\right)>0$, which entails
that there exists a finite constant $c_{0}$ such that

(B.6) $\left\|\mathbf{C}^{-1}\right\|_{F}\leq c_{0}d^{1/2}.$

Hence it holds that

$r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\left[\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right]\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}=O_{P}\left(\frac{d^{3/2}}{m^{1/2}}a_{m}^{\zeta_{3}-1/2}\right);$

under the constraint $d=O\left(m^{1/4}\right)$, this finally entails that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\mathbf{x}_{t}^{\prime}\left[\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right]\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\frac{k}{m}\mathbf{c}_{1}^{\prime}\mathbf{C}^{-1}\displaystyle\left(\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}+o_{P}\left(1\right).$

By construction,
$\mathbf{c}_{1}^{\prime}\mathbf{C}^{-1}=\left(1,0,...,0\right)^{\prime}$, and
therefore it finally follows that

$r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\mathbf{x}_{t}^{\prime}\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)-\displaystyle\frac{k}{m}\displaystyle\sum_{t=1}^{m}\epsilon_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}=o_{P}\left(1\right).$

The desired result now follows from putting all together.

######  Lemma B.6.

We assume that Assumptions 1(i) and 4 hold. Then, on a suitably enlarged
probability space, it is possible to define two independent standard Wiener
processes $\left\\{W_{1,m}\left(k\right),k\geq 1\right\\}$ and
$\left\\{W_{2,m}\left(k\right),k\geq 1\right\\}$ such that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{1}{\sigma}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\epsilon_{t}-\displaystyle\frac{k}{m}\displaystyle\sum_{t=1}^{m}\epsilon_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|W_{1,m}\left(k\right)-\displaystyle\frac{k}{m}W_{2,m}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}+o_{P}\left(1\right).$

Henceforth, we set $\sigma=1$ for simplicity and without loss of generality.
Standard arguments entail that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq T_{m}}$
$\displaystyle\frac{\left|\left(\displaystyle\sum_{t=m+1}^{m+k}\epsilon_{t}-\displaystyle\frac{k}{m}\displaystyle\sum_{t=1}^{m}\epsilon_{t}\right)-\left(W_{1,m}(k)-\displaystyle\frac{k}{m}W_{2,m}(m)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}$
$\displaystyle=r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\left(\displaystyle\sum_{t=m+1}^{m+k}\epsilon_{t}-W_{1,m}(k)\right)-\displaystyle\frac{k}{m}\left(\displaystyle\sum_{t=1}^{m}\epsilon_{t}-W_{2,m}(m)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}$
$\displaystyle\leq r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\epsilon_{t}-W_{1,m}(k)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}$
$\displaystyle+r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\displaystyle\frac{k}{m}\left(\displaystyle\sum_{t=1}^{m}\epsilon_{t}-W_{2,m}(m)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}.$

Using (B.1) in Lemma B.1, it follows that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq T_{m}}$
$\displaystyle\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\epsilon_{t}-W_{1,m}(k)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}$
$\displaystyle=r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}k^{\zeta_{1}}\frac{k^{-\zeta_{1}}\left|\displaystyle\sum_{t=m+1}^{m+k}\epsilon_{t}-W_{1,m}(k)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}$
$\displaystyle=O_{P}\left(1\right)r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{k^{\zeta_{1}}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}$
$\displaystyle=O_{P}\left(1\right)r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}k^{\zeta_{1}-1/2}\left(\frac{m+k}{k}\right)^{\eta-1/2}$
$\displaystyle=O_{P}\left(a_{m}^{\zeta_{1}-1/2}\right)=o_{P}\left(1\right).$

Similarly, using (B.2) in Lemma B.1

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{k}{m}\frac{\left|\left(\displaystyle\sum_{t=1}^{m}\epsilon_{t}-W_{2,m}(m)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}$
$\displaystyle=r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{k}{m}\frac{m^{\zeta_{2}}}{m^{\zeta_{2}}}\frac{\left|\left(\displaystyle\sum_{t=1}^{m}\epsilon_{t}-W_{2,m}(m)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}$
$\displaystyle=O_{P}\left(1\right)r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{km^{\zeta_{2}}}{m^{3/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}.$

Hence, after some algebra, it can be shown that

$r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{k}{m}\frac{\left|\left(\displaystyle\sum_{t=1}^{m}\epsilon_{t}-W_{2,m}(m)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{m+k}\right)^{\eta}}=O_{P}\left(r_{m}^{\min\left\\{1/2,\eta-1/2\right\\}}m^{\zeta_{2}-1/2}\right)=o_{P}\left(1\right).$

The desired result now follows.

##  C. Proofs

###### Proof of Theorem 1.

Lemmas B.1-B.6, put together, entail that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{1}{\sigma}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\widehat{\epsilon}_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|W_{1,m}\left(k\right)-\displaystyle\frac{k}{m}W_{2,m}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}+o_{P}\left(1\right).$

As in the other proofs, we will set $\sigma=1$ for simplicity and without loss
of generality. We begin by noting that the distribution of
$W_{1,m}\left(\cdot\right)$ and $W_{2,m}\left(\cdot\right)$ does not depend on
$m$, so that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|W_{1,m}\left(k\right)-\displaystyle\frac{k}{m}W_{2,m}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|W_{1}\left(k\right)-\displaystyle\frac{k}{m}W_{2}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|W_{1}\displaystyle\left(\frac{k}{m}\right)-\displaystyle\frac{k}{m}W_{2}\left(1\right)\right|}{\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}.$

By standard arguments, it is easy to see that

(B.1)
$W_{1}\left(t\right)-tW_{2}\left(1\right)\overset{\mathcal{D}}{=}\left(1+t\right)W\left(\frac{t}{1+t}\right),$

for all $t$, where $W\left(\cdot\right)$ is a standard Wiener.555Indeed, this
can be verified readily by noting that both processes $W_{1}\left(s\right)$
and $W_{2}\left(s\right)$ have mean zero, are Gaussian, and have covariance
kernel given by  $\displaystyle
E\left(W_{1}\left(t\right)-tW_{2}\left(1\right)\right)\left(W_{1}\left(s\right)-sW_{2}\left(1\right)\right)$
$\displaystyle=$ $\displaystyle
E\left(W_{1}\left(t\right)W_{1}\left(s\right)\right)-stE\left(W_{2}\left(1\right)\right)^{2}$
$\displaystyle=$ $\displaystyle s\wedge t-st,$ and  $\displaystyle
E\left(\left(1+t\right)W\left(\frac{t}{1+t}\right)\left(1+s\right)W\left(\frac{s}{1+s}\right)\right)$
$\displaystyle=$
$\displaystyle\left(1+t\right)\left(1+s\right)\left(\frac{t}{1+t}\wedge\frac{s}{1+s}\right),$
respectively - it is now not hard to see that the two covariance kernels are
the same. Hence we will study

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{\left|\left(1+\displaystyle\frac{k}{m}\right)W\displaystyle\left(\frac{\frac{k}{m}}{1+\frac{k}{m}}\right)\right|}{\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}r_{m}^{\eta-1/2}\max_{a_{m}/m\leq t\leq
T_{m}/m}\frac{\left|W\displaystyle\left(\frac{t}{1+t}\right)\right|}{\displaystyle\left(\frac{t}{1+t}\right)^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}r_{m}^{\eta-1/2}\max_{a_{m}/\left(m+a_{m}\right)\leq
s\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(s\right)\right|}{s^{\eta}}.$

Define

$u=s\frac{a_{m}+m}{a_{m}};$

it follows that

$\displaystyle r_{m}^{\eta-1/2}\max_{a_{m}/\left(m+a_{m}\right)\leq s\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(s\right)\right|}{s^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}r_{m}^{\eta-1/2}\max_{1\leq u\leq
T_{m}\left(m+a_{m}\right)/\left(a_{m}\left(m+T_{m}\right)\right)}\frac{\left|W\displaystyle\left(u\frac{a_{m}}{a_{m}+m}\right)\right|}{\displaystyle\left(u\frac{a_{m}}{a_{m}+m}\right)^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}r_{m}^{\eta-1/2}\left(\frac{a_{m}}{a_{m}+m}\right)^{1/2-\eta}\max_{1\leq
u\leq
T_{m}\left(m+a_{m}\right)/\left(a_{m}\left(m+T_{m}\right)\right)}\frac{\left|W\left(u\right)\right|}{u^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}\max_{1\leq u\leq
T_{m}\left(m+a_{m}\right)/\left(a_{m}\left(m+T_{m}\right)\right)}\frac{\left|W\left(u\right)\right|}{u^{\eta}},$

having used the scale transformation of the Wiener process in the third
passage, and the definition of $r_{m}$ in the last one. Recalling (10), by
elementary arguments it holds that

$\lim_{m\rightarrow\infty}\frac{T_{m}\left(m+a_{m}\right)}{a_{m}\left(m+T_{m}\right)}\rightarrow\infty,$

whence, as $m\rightarrow\infty$, it follows that, by continuity

$\max_{1\leq u\leq
T_{m}\left(m+a_{m}\right)/\left(a_{m}\left(m+T_{m}\right)\right)}\frac{\left|W\left(u\right)\right|}{u^{\eta}}\overset{a.s.}{\rightarrow}\sup_{1\leq
u<\infty}\frac{\left|W(u)\right|}{u^{\eta}}.$

As a final remark, we reconsider the proof, showing that the limiting
distribution is determined only by the observations very close to $a_{m}$. To
show this, we divide into two intervals the domain where the maximum is taken
defining

(B.2)
$\displaystyle\Theta_{m,1}=r_{m}^{\eta-1/2}\max_{a_{m}/\left(m+a_{m}\right)\leq
s<\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}}\frac{\left|W\left(s\right)\right|}{s^{\eta}},$
(B.3)
$\displaystyle\Theta_{m,2}=r_{m}^{\eta-1/2}\max_{\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}\leq
s<T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(s\right)\right|}{s^{\eta}}.$

for some $\epsilon>0$ such that
$\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}=o\left(T_{m}/\left(m+T_{m}\right)\right)$.
We begin by studying $\Theta_{m,2}$. Repeatedly using the scale transformation
for Wiener process, we have

$\displaystyle\Theta_{m,2}\overset{\mathcal{D}}{=}r_{m}^{\eta-1/2}\max_{\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}\leq
s\leq
T_{m}/\left(m+T_{m}\right)}\frac{s^{1/2}}{s^{1/2}}\frac{\left|W\left(s\right)\right|}{s^{\eta}}$
$\displaystyle\leq
r_{m}^{\eta-1/2}\max_{\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}\leq
s\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(s\right)\right|}{s^{1/2}}\max_{\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}\leq
s\leq T_{m}/\left(m+T_{m}\right)}s^{1/2-\eta}$
$\displaystyle\leq\left(\left(\frac{a_{m}}{a_{m}+m}\right)^{1/2}\right)^{\eta-1/2}\max_{\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}\leq
s\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(s\right)\right|}{s^{1/2}}.$

We note that

$\max_{\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}\leq s\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(s\right)\right|}{s^{1/2}}\overset{\mathcal{D}}{=}\max_{1\leq
u\leq\left(T_{m}\left(m+a_{m}\right)^{1-\epsilon}\right)/\left(\left(m+T_{m}\right)a_{m}^{1-\epsilon}\right)}\frac{\left|W\left(u\right)\right|}{u^{1/2}},$

so that the Law of the Iterated Logarithm entails that

$\max_{\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}\leq s\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(s\right)\right|}{s^{1/2}}=O_{P}\left(\sqrt{\ln\ln\frac{T_{m}\left(m+a_{m}\right)^{1-\epsilon}}{\left(m+T_{m}\right)a_{m}^{1-\epsilon}}}\right)=O_{P}\left(\sqrt{\ln\ln\left(\frac{m}{a_{m}}\right)^{1-\epsilon}}\right).$

Hence, recalling (10)

$\Theta_{m,2}=O_{P}\left(\left(\left(\frac{a_{m}}{a_{m}+m}\right)^{1/2}\right)^{\eta-1/2}\sqrt{\ln\ln\left(\frac{m}{a_{m}}\right)^{1-\epsilon}}\right)=o_{P}\left(1\right).$

Using this result in conjunction with Lemmas B.1-B.6 it follows that, as
$m\rightarrow\infty$

$P\left(r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
T_{m}}\frac{1}{\sigma}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\widehat{\epsilon}_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\displaystyle\left(\frac{k}{k+m}\right)^{\eta}}=\Theta_{m,1}\right)=1.$

We now conclude our remarks by studying $\Theta_{m,1}$ in (B.2). Let

$u=s\frac{a_{m}+m}{a_{m}}.$

Then it holds that

$\displaystyle\Theta_{m,1}=$ $\displaystyle
r_{m}^{\eta-1/2}\max_{a_{m}/\left(m+a_{m}\right)\leq
s\leq\left(a_{m}/\left(m+a_{m}\right)\right)^{1-\epsilon}}\frac{\left|W\left(s\right)\right|}{s^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle
r_{m}^{\eta-1/2}\max_{1\leq
u\leq\left(\left(m+a_{m}\right)/a_{m}\right)^{\epsilon}}\frac{\left|W\displaystyle\left(u\frac{a_{m}}{a_{m}+m}\right)\right|}{\displaystyle\left(u\frac{a_{m}}{a_{m}+m}\right)^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle
r_{m}^{\eta-1/2}\left(\frac{a_{m}}{a_{m}+m}\right)^{1/2-\eta}\max_{1\leq
u\leq\left(\left(m+a_{m}\right)/a_{m}\right)^{\epsilon}}\frac{\left|W\left(u\right)\right|}{u^{\eta}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{1\leq
u\leq\left(\left(m+a_{m}\right)/a_{m}\right)^{\epsilon}}\frac{\left|W\left(u\right)\right|}{u^{\eta}},$

and

$\Theta_{m,1}\overset{\mathcal{D}}{=}\max_{1\leq
u\leq\left(\left(m+a_{m}\right)/a_{m}\right)^{\epsilon}}\frac{\left|W\left(u\right)\right|}{u^{\eta}}\overset{a.s.}{\rightarrow}\sup_{1\leq
u<\infty}\frac{\left|W(u)\right|}{u^{\eta}}.$

###### Proof of Lemma 1.

The proof of the lemma is standard, and we only report its main passages,
mainly to highlight the impact of $d$. Let
$\gamma_{j}=E\left(\epsilon_{0}\epsilon_{j}\right)$. By Assumption 1, it is
easy to see that

$\sigma^{2}=\lim_{m\rightarrow\infty}\sum_{t,s=1}^{m}E\left(\epsilon_{t}\epsilon_{s}\right)=\gamma_{0}+2\sum_{j=1}^{\infty}\gamma_{j}.$

We begin by showing that the $\gamma_{j}$s satisfy

(B.4) $\sum_{j=1}^{\infty}j\left|\gamma_{j}\right|<\infty.$

Indeed, let $\epsilon_{j}=h\left(\eta_{j},\eta_{j-1},...\right)$, and define
$\epsilon_{j,j}=h\left(\eta_{j},\eta_{j-1},...,\eta_{1},\widetilde{\eta}_{0,j,j},\widetilde{\eta}_{-1,j,j},...\right)$
where, as per Definition 1,
$\left\\{\widetilde{\eta}_{t,j,j},-\infty<t,j<\infty\right\\}$ i.i.d. copies
of $\eta_{0}$ independent of $\left\\{\eta_{t},-\infty<t<\infty\right\\}$. By
construction, $\epsilon_{0}$ and $\epsilon_{j,j}$ are independent and have
mean zero, so that $E\left(\epsilon_{0}\epsilon_{j,j}\right)=0$ and

$\displaystyle\sum_{j=1}^{\infty}j\left|E\left(\epsilon_{0}\epsilon_{j}\right)\right|=\sum_{j=1}^{\infty}j\left|E\left(\epsilon_{0}\left(\epsilon_{j}-\epsilon_{j,j}\right)\right)\right|\leq\sum_{j=1}^{\infty}j\left|\epsilon_{0}\right|_{2}\left|\epsilon_{j}-\epsilon_{j,j}\right|_{2}$
$\displaystyle\leq
c_{0}\sum_{j=1}^{\infty}j\left|\epsilon_{j}-\epsilon_{j,j}\right|_{2}\leq
c_{0}\sum_{j=1}^{\infty}j\left|\epsilon_{j}-\epsilon_{j,j}\right|_{4}\leq
c_{0}\sum_{j=1}^{\infty}j^{1-a}\leq c_{1},$

having used: the Cauchy-Schwartz inequality in the third passage; the fact
that, by Assumption 1(i) $\left|\epsilon_{0}\right|_{2}<\infty$ in the fourth
passage; the $L_{p}$-norm inequality in the fifth passage; and Assumption 1(i)
in the last one, recalling that $a>2$. We now write

$\displaystyle\widehat{\sigma}_{m}^{2}-\sigma^{2}=$
$\displaystyle\left(\widehat{\gamma}_{0}-\gamma_{0}\right)+2\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\widehat{\gamma}_{j}-\gamma_{j}\right)+2\sum_{j=1}^{H}\frac{j}{H+1}\gamma_{j}-2\sum_{j=H+1}^{\infty}\gamma_{j}$
$\displaystyle=$ $\displaystyle I+II+III+IV.$

By (B.4), it follows that

$\frac{1}{2}III\leq\frac{1}{H+1}\sum_{j=1}^{\infty}j\left|\gamma_{j}\right|\leq
c_{0}\frac{1}{H+1};$

similarly

$\frac{1}{2}IV\leq\sum_{j=H+1}^{\infty}\frac{j}{H+1}\left|\gamma_{j}\right|\leq\frac{1}{H+1}\sum_{j=1}^{\infty}j\left|\gamma_{j}\right|\leq
c_{0}\frac{1}{H+1},$

so that $III+IV=O\left(H^{-1}\right)$. We now turn to $I$ and $II$ noting that

$\displaystyle\widehat{\gamma}_{j}-\gamma_{j}=\frac{1}{m}\sum_{t=j+1}^{m}\widehat{\epsilon}_{t}\widehat{\epsilon}_{t-j}-E\left(\epsilon_{0}\epsilon_{j}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{t=j+1}^{m}\left(\epsilon_{t}\epsilon_{t-j}-E\left(\epsilon_{0}\epsilon_{j}\right)\right)+\left(\beta-\widehat{\beta}_{m}\right)^{\prime}\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}\epsilon_{t-j}$
$\displaystyle+\left(\beta-\widehat{\beta}_{m}\right)^{\prime}\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t-j}\epsilon_{t}+\left(\widehat{\beta}_{m}-\beta\right)^{\prime}\left(\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime}\right)\left(\widehat{\beta}_{m}-\beta\right),$

whence

$\displaystyle\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\widehat{\gamma}_{j}-\gamma_{j}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\frac{1}{m}\sum_{t=j+1}^{m}\left(\epsilon_{t}\epsilon_{t-j}-E\left(\epsilon_{0}\epsilon_{j}\right)\right)\right)$
$\displaystyle+\left(\beta-\widehat{\beta}_{m}\right)^{\prime}\left[\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}\epsilon_{t-j}\right)\right]$
$\displaystyle+\left(\beta-\widehat{\beta}_{m}\right)^{\prime}\left[\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t-j}\epsilon_{t}\right)\right]$
$\displaystyle+\left(\widehat{\beta}_{m}-\beta\right)^{\prime}\left[\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime}\right)\right]\left(\widehat{\beta}_{m}-\beta\right)$
$\displaystyle=$ $\displaystyle II_{a}+II_{b}+II_{c}+II_{d}.$

It holds that

$\displaystyle
E\left(\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\frac{1}{m}\sum_{t=j+1}^{m}\left(\epsilon_{t}\epsilon_{t-j}-E\left(\epsilon_{0}\epsilon_{j}\right)\right)\right)\right)^{2}$
$\displaystyle\leq$ $\displaystyle
c_{0}H\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)^{2}E\left(\frac{1}{m}\sum_{t=j+1}^{m}\left(\epsilon_{t}\epsilon_{t-j}-E\left(\epsilon_{0}\epsilon_{j}\right)\right)\right)^{2}$
$\displaystyle\leq$ $\displaystyle
c_{0}H\sum_{j=1}^{H}E\left(\frac{1}{m}\sum_{t=j+1}^{m}\left(\epsilon_{t}\epsilon_{t-j}-E\left(\epsilon_{0}\epsilon_{j}\right)\right)\right)^{2}.$

By Assumption 1(i) and the same logic as in the proof of Lemma B.2, it follows
that $\epsilon_{t}\epsilon_{t-j}$ is an $L_{2}$-decomposable Bernoulli shift
for all $j$; hence, using Proposition 4 in Berkes et al. (2011), it follows
immediately that

$H\sum_{j=1}^{H}E\left(\frac{1}{m}\sum_{t=j+1}^{m}\left(\epsilon_{t}\epsilon_{t-j}-E\left(\epsilon_{0}\epsilon_{j}\right)\right)\right)^{2}\leq
c_{0}\frac{H^{2}}{m},$

which entails that $II_{a}=O_{P}\left(Hm^{-1/2}\right)$. Standard passages,
using Assumption 2(iii), entail that

(B.5)
$\left\|\widehat{\beta}_{m}-\beta\right\|=O_{P}\left(\sqrt{\frac{d}{m}}\right);$

further

$\displaystyle
E\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}\epsilon_{t-j}\right)$
$\displaystyle\leq$ $\displaystyle
E\sum_{j=1}^{H}\left\|\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}\epsilon_{t-j}\right\|\leq
c_{0}\frac{1}{m}\sum_{j=1}^{H}\sum_{t=j+1}^{m}\left(E\left\|\mathbf{x}_{t}\right\|^{2}\right)^{1/2}\left(E\left|\epsilon_{t-j}\right|^{2}\right)^{1/2}\leq
c_{0}H;$

this bound is not the sharpest possible (one could assume
$E\left(\mathbf{x}_{t}\epsilon_{t-j}\right)=0$ and then use the Markov
inequality), but it suffices for our purposes. Using (B.5), this entails that
both $II_{b}$ and $II_{c}$ are $O_{P}\left(d^{1/2}Hm^{-1/2}\right)$. Finally

$\displaystyle\left|\left(\widehat{\beta}_{m}-\beta\right)^{\prime}\left[\sum_{j=1}^{H}\left(1-\frac{j}{H+1}\right)\left(\frac{1}{m}\sum_{t=j+1}^{m}\mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime}\right)\right]\left(\widehat{\beta}_{m}-\beta\right)\right|$
$\displaystyle\leq$
$\displaystyle\left\|\widehat{\beta}_{m}-\beta\right\|^{2}\frac{1}{m}\sum_{j=1}^{H}\sum_{t=j+1}^{m}\left|\mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime}\right|,$

and since
$\left|\mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime}\right|\leq\left\|\mathbf{x}_{t}\right\|\left\|\mathbf{x}_{t-j}\right\|$,
using the Cauchy-Schwartz inequality and the stationarity of $\mathbf{x}_{t}$

$E\frac{1}{m}\sum_{j=1}^{H}\sum_{t=j+1}^{m}\left|\mathbf{x}_{t}\mathbf{x}_{t-j}^{\prime}\right|\leq\frac{1}{m}\sum_{j=1}^{H}\sum_{t=j+1}^{m}\left(E\left\|\mathbf{x}_{t}\right\|^{2}\right)^{1/2}\left(E\left\|\mathbf{x}_{t-j}\right\|^{2}\right)^{1/2}\leq
c_{0}H.$

Thus, again using (B.5), it follows that $II_{d}=O_{P}\left(Hdm^{-1}\right)$.
The final result now follows from putting all together.

###### Proof of Theorem 2.

The proof is related to that of Theorem 2.2 in Horváth et al. (2004), and we
only report the main passages for the sake of a concise discussion. Standard
algebra entails that

$\sum_{t=m+1}^{m+\widetilde{k}}\widehat{\epsilon}_{t}=\sum_{t=m+1}^{m+\widetilde{k}}\epsilon_{t}-\left(\widehat{\beta}_{m}-\beta_{0}\right)^{\prime}\sum_{t=m+1}^{m+\widetilde{k}}\mathbf{x}_{t}+\Delta_{m}^{\prime}\sum_{t=m+1}^{m+\widetilde{k}}\mathbf{x}_{t}I\left(t>m+k^{\ast}\right).$

The proof of Theorem 1 immediately implies that

$r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+\widetilde{k}}\epsilon_{t}-\left(\widehat{\beta}_{m}-\beta_{0}\right)^{\prime}\displaystyle\sum_{t=m+1}^{m+\widetilde{k}}\mathbf{x}_{t}\right|}{g_{\eta}\left(m;\widetilde{k}\right)}=O_{P}\left(1\right).$

Also, using Lemma B.3, it is not hard to see that

$\displaystyle\Delta_{m}^{\prime}\sum_{t=m+1}^{m+\widetilde{k}}\mathbf{x}_{t}I\left(t>k^{\ast}\right)$
$\displaystyle=$
$\displaystyle\left(\widetilde{k}-k^{\ast}\right)\mathbf{c}_{1}^{\prime}\Delta_{m}+\mathbf{c}_{1}^{\prime}\Delta_{m}O_{P}\left(\left(\widetilde{k}-k^{\ast}\right)^{1/2}\right)=\mathbf{c}_{1}^{\prime}\Delta_{m}\left(a_{m}+O_{P}\left(a_{m}^{1/2}\right)\right).$

Recall that, by assumption, it holds that
$\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|>0$. Under condition (i), let
$\widetilde{k}=k^{\ast}+a_{m}$. It follows that

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\Delta_{m}^{\prime}\displaystyle\sum_{t=m+k^{\ast}+1}^{m+\widetilde{k}}\mathbf{x}_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{\widetilde{k}}{m}\right)\displaystyle\left(\frac{\widetilde{k}}{m+\widetilde{k}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
r_{m}^{\eta-1/2}\frac{a_{m}}{m^{1/2}\left(1+\displaystyle\frac{\widetilde{k}}{m}\right)\displaystyle\left(\frac{\widetilde{k}}{m+\widetilde{k}}\right)^{\eta}}\mathbf{c}_{1}^{\prime}\Delta_{m}+O_{P}\left(\mathbf{c}_{1}^{\prime}\Delta_{m}\right)$
$\displaystyle=$ $\displaystyle
a_{m}^{1/2}\mathbf{c}_{1}^{\prime}\Delta_{m}+O_{P}\left(\mathbf{c}_{1}^{\prime}\Delta_{m}\right)+o\left(a_{m}^{1/2}\mathbf{c}_{1}^{\prime}\Delta_{m}\right),$

whence the theorem follows immediately if (14) holds. Under condition (ii),
let $\widetilde{k}=\left\lfloor\left(1+c\right)k^{\ast}\right\rfloor\leq
T_{m}$ for some $c>0$; we have, using the mean value theorem

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\Delta_{m}^{\prime}\displaystyle\sum_{t=m+k^{\ast}+1}^{m+\widetilde{k}}\mathbf{x}_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{\widetilde{k}}{m}\right)\displaystyle\left(\frac{\widetilde{k}}{m+\widetilde{k}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
c_{0}r_{m}^{\eta-1/2}\frac{k^{\ast}+O_{P}\left(\left(k^{\ast}\right)^{1/2}\right)}{m^{1/2}\left(1+\displaystyle\frac{k^{\ast}}{m}\right)\displaystyle\left(\frac{k^{\ast}}{m+k^{\ast}}\right)^{\eta}}\mathbf{c}_{1}^{\prime}\Delta_{m}$
$\displaystyle=$ $\displaystyle
c_{0}r_{m}^{\eta-1/2}m^{1/2}\left(\frac{k^{\ast}}{m+k^{\ast}}\right)^{1-\eta}\mathbf{c}_{1}^{\prime}\Delta_{m},$

which again immediately yields the desired result as long as (15) holds.

###### Proof of Theorem 3.

We will study the limiting behaviour of

$r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
a_{m}}\frac{\left|Q\left(m;k\right)\right|}{g_{\eta}\left(m;k\right)}=r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\widehat{\epsilon}_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}},$

under the alternative hypothesis of (13). By standard algebra, it holds that

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\widehat{\epsilon}_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}-\displaystyle\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}.$

We begin by noting that, following the passages above, it is not hard to see
that

(B.6)
$\left\|\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}\right\|=O_{P}\left(d^{1/2}a_{m}\right).$

Using Lemmas B.2, (B.5) and (B.6), it follows that

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\left(\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}-\mathbf{C}^{-1}\right)\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle\leq$ $\displaystyle
r_{m}^{\eta-1/2}\frac{\left\|\displaystyle\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\right\|\left\|\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}-\mathbf{C}^{-1}\right\|_{F}\displaystyle\left\|\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right\|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(1\right)r_{m}^{\eta-1/2}m^{-1/2}\left(\frac{a_{m}}{m}\right)^{-\eta}d^{1/2}a_{m}\frac{d}{m^{1/2}}\left(\frac{d}{m}\right)^{1/2}=O_{P}\left(d^{2}\frac{a_{m}^{1/2}}{m}\right)=o_{P}\left(1\right),$

by having $d=O\left(m^{1/4}\right)$. Therefore

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}-\displaystyle\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}-\displaystyle\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}+o_{P}\left(1\right).$

Similarly, we note that

$E\left\|\sum_{t=m+1}^{m+a_{m}}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)\right\|^{2}=\sum_{j=2}^{d}E\left(\sum_{t=m+1}^{m+a_{m}}\left(x_{j,t}-E\left(x_{j,0}\right)\right)\right)^{2}\leq
c_{0}da_{m},$

by using the fact that $x_{j,t}-E\left(x_{j,0}\right)$ is a weakly dependent
process in the sense of Definition 1, and Proposition 4 in Berkes et al.
(2011), whence
$\left\|\sum_{t=m+1}^{m+a_{m}}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)\right\|=O_{P}\left(d^{1/2}a_{m}^{1/2}\right)$.
Therefore, it is easy to see that

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle\leq$ $\displaystyle
r_{m}^{\eta-1/2}\frac{\left\|\displaystyle\sum_{t=m+1}^{m+a_{m}}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)\right\|\left\|\mathbf{C}^{-1}\right\|_{F}\displaystyle\left\|\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right\|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(r_{m}^{\eta-1/2}m^{-1/2}\left(\frac{a_{m}}{m}\right)^{-\eta}d^{1/2}a_{m}^{1/2}d^{1/2}\left(\frac{d}{m}\right)^{1/2}\right)=O_{P}\left(\frac{d^{3/2}}{m^{1/2}}\right)=o_{P}\left(1\right),$

having used Lemmas B.2 and B.3, (B.6) and $d=O\left(m^{1/4}\right)$. This
entails that

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}-\displaystyle\sum_{t=m+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}-a_{m}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{c}_{1}^{\prime}\mathbf{C}^{-1}\mathbf{x}_{t}\epsilon_{t}\right)+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}+o_{P}\left(1\right)$
$\displaystyle=$ $\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}-\displaystyle\frac{a_{m}}{m}\left(\displaystyle\sum_{t=1}^{m}\epsilon_{t}\right)+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}+o_{P}\left(1\right),$

on account of the fact that
$\mathbf{c}_{1}^{\prime}\mathbf{C}^{-1}\mathbf{x}_{t}=1$ for all $t$. Also
note that

$r_{m}^{\eta-1/2}\frac{a_{m}}{m}\frac{\left|\displaystyle\left(\frac{1}{m^{1/2}}\sum_{t=1}^{m}\epsilon_{t}\right)\right|}{\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}=O_{P}\left(1\right)r_{m}^{\eta-1/2}\left(\frac{a_{m}}{m}\right)^{1-\eta}=O_{P}\left(\left(\frac{a_{m}}{m}\right)^{1/2}\right)=o_{P}\left(1\right),$

whence

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}-\displaystyle\frac{a_{m}}{m}\left(\displaystyle\sum_{t=1}^{m}\epsilon_{t}\right)+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}+o_{P}\left(1\right).$

Finally, using Lemma B.1, it holds that

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}-\sigma
W_{1,m}\left(a_{m}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(1\right)r_{m}^{\eta-1/2}m^{\zeta_{1}-1/2}\left(\frac{a_{m}}{m}\right)^{-\eta}=o_{P}\left(1\right),$

so that

$\displaystyle
r_{m}^{\eta-1/2}\frac{\left|\displaystyle\sum_{t=m+1}^{m+a_{m}}\epsilon_{t}+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}$
$\displaystyle=$ $\displaystyle r_{m}^{\eta-1/2}\frac{\left|\sigma
W_{1,m}\left(a_{m}\right)+\displaystyle\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{a_{m}}{m}\right)\displaystyle\left(\frac{a_{m}}{m+a_{m}}\right)^{\eta}}+o_{P}\left(1\right)$
$\displaystyle=$ $\displaystyle a_{m}^{-1/2}\left|\sigma
W_{1,m}\left(a_{m}\right)+\sum_{t=k^{\ast}+1}^{a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|+o_{P}\left(1\right),$

where the last result follows from having $a_{m}=o\left(m\right)$. Given that
the distribution of $W_{1,m}\left(\cdot\right)$ does not depend on $m$, it
follows that

$a_{m}^{-1/2}\left|\sigma
W_{1,m}\left(a_{m}\right)+\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|\overset{\mathcal{D}}{=}\left|\sigma
W\left(1\right)+a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|,$

where $W\left(\cdot\right)$ is a standard Wiener. The results above entail
that, as $m\rightarrow\infty$

$P\left(r_{m}^{\eta-1/2}\max_{a_{m}\leq k\leq
a_{m}}\frac{\left|Q\left(m;a_{m}\right)\right|}{g\left(m;a_{m}\right)}\leq
x\right)=P\left(\left|\sigma
Z+a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right|\leq
x\right)+o_{P}\left(1\right),$

for all $-\infty<x<\infty$, with $Z\sim N\left(0,1\right)$.

We are now ready to prove the main statement of the theorem. By assumption,
$\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|=c_{0}\left\|\mathbf{c}_{1}\right\|\left\|\Delta_{m}\right\|$,
where $c_{0}>0$ is the cosine of the angle between $\mathbf{c}_{1}$ and
$\Delta_{m}$. Hence, Assumption 5(ii) entails

$\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|=c_{0}\left\|\mathbf{c}_{1}\right\|\left\|\Delta_{m}\right\|=\Omega\left(d^{1/2}\Delta_{m}\right);$

henceforth, we can and will assume, without loss of generality, that
$\mathbf{c}_{1}^{\prime}\Delta_{m}>0$. We note that, using the fact that, by
Assumption 5(i), $a_{m}-k^{\ast}=\Omega\left(a_{m}\right)$, we have

(B.7) $\displaystyle
a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}$
$\displaystyle=$ $\displaystyle
a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{c}_{1}^{\prime}\Delta_{m}+a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\Delta_{m}$
$\displaystyle=$ $\displaystyle
a_{m}^{1/2}\Omega\left(d^{1/2}\Delta_{m}\right)+O_{P}\left(d^{1/2}\Delta_{m}\right),$

where the second term follows from the CLT, viz.

$\left|a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\Delta_{m}\right|\leq\left\|a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)\right\|\left\|\Delta_{m}\right\|=O_{P}\left(d^{1/2}\right)\Omega\left(\Delta_{m}\right).$

Equation (B.7), and the fact that we are assuming
$\mathbf{c}_{1}^{\prime}\Delta_{m}>0$, imply that

$\lim_{m\rightarrow\infty}P\left(\sigma
Z+a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}>0\right)=1,$

and

(B.8)
$a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\overset{\mathcal{P}}{\rightarrow}\infty,$

as $m\rightarrow\infty$. Therefore, as $m\rightarrow\infty$

$P\left(r_{m}^{\eta-1/2}\frac{\left|Q\left(m;a_{m}\right)\right|}{g_{\eta}\left(m;a_{m}\right)}\leq
x\right)=P\left(\sigma
Z+a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\leq
x\right)+o_{P}\left(1\right).$

Given that, by (B.8)

$\lim_{m\rightarrow\infty}P\left(\sigma
Z+a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\leq
c_{\alpha,\eta}\right)=\lim_{m\rightarrow\infty}P\left(Z\leq\frac{c_{\alpha,\eta}}{\sigma}-\frac{1}{\sigma}a_{m}^{-1/2}\sum_{t=m+k^{\ast}+1}^{m+a_{m}}\mathbf{x}_{t}^{\prime}\Delta_{m}\right)=0,$

putting all the above together, we finally have

$\lim_{m\rightarrow\infty}P\left(r_{m}^{\eta-1/2}\frac{\left|Q\left(m;a_{m}\right)\right|}{g_{\eta}\left(m;a_{m}\right)}\leq
c_{\alpha,\eta}\right)=0.$

We now conclude, by noting that the event $\left\\{\tau_{m}>a_{m}\right\\}$ is
equivalent to having

$\left\\{r_{m}^{\eta-1/2}\left|Q\left(m;a_{m}\right)\right|\leq
c_{\alpha,\eta}g_{\eta}\left(m;a_{m}\right)\right\\}.$

We have shown that $\lim_{m\rightarrow\infty}P\left(\tau_{m}>a_{m}\right)=0$,
whence $\lim_{m\rightarrow\infty}P\left(\tau_{m}=a_{m}\right)=1$.

###### Proof of Theorem 4.

We prove that Lemmas B.2-B.4 hold under (18). Thereafter, the proofs of
Theorems 1, 2 and 3 and Lemma 1 can be repeated verbatim. Further, for
simplicity, we report the proof for the case $p=1$, writing (18) as

(B.9) $y_{t}=\rho y_{t-1}+\mathbf{z}_{t}^{\prime}\beta^{Z}+\epsilon_{t}.$

Equation (B.9) can be expressed in recursive form

(B.10)
$y_{t}=\rho^{t}y_{0}+\sum_{j=0}^{t-1}\rho^{j}\left(\mathbf{z}_{t-j}^{\prime}\beta^{Z}+\epsilon_{t-j}\right),$

where $y_{0}$ is an initial condition. Under Assumption 6(ii), (B.9) admits a
unique stationary, non-anticipative solution given by

(B.11)
$\overline{y}_{t}=\sum_{j=0}^{\infty}\rho^{j}\left(\mathbf{z}_{t-j}^{\prime}\beta^{Z}+\epsilon_{t-j}\right)=\sum_{j=0}^{\infty}\rho^{j}u_{t-j},$

where $u_{t}=\mathbf{z}_{t}^{\prime}\beta^{Z}+\epsilon_{t}$, for any
(stochastically bounded) initial value $y_{0}$. By Assumption 1,
$x_{j,t}=h_{j}^{x}\left(\eta_{j,t}^{x},...\right)$ for $2\leq j\leq d-p$ and
$\epsilon_{t}=h^{\epsilon}\left(\eta_{t}^{\epsilon},...\right)$; hence, it
immediately follows that we can use the representation
$u_{t}=h^{u}\left(\eta_{t}^{u},...\right)$ and that $u_{t}$ satisfies
Assumption 1. Define the $\sigma$-fields
$\mathcal{F}_{t-\ell}^{t-j}=\left\\{\eta_{i}^{u}\right\\}_{i=t-\ell}^{t-j}$,
with the convention that $\mathcal{F}_{t-\ell}^{t-j}$ is empty if $j>\ell$,
and
$\mathcal{F}_{t-\min\left\\{\ell,j\right\\}}^{-\infty}=\left\\{\widetilde{\eta}_{t-\min\left\\{\ell,j\right\\}}^{u},...,\widetilde{\eta}_{-\infty}^{u}\right\\}$,
where $\widetilde{\eta}_{t}^{u}$ is an independent copy of $\eta_{t}^{u}$,
forming an i.i.d. sequence. We now define the coupling constructions

$\widetilde{u}_{t-j,\ell}=h^{u}\left(\mathcal{F}_{t-\ell}^{t-j},\mathcal{F}_{t-\min\left\\{\ell,j\right\\}}^{-\infty}\right),$

and

$\widetilde{y}_{t,\ell}=\sum_{j=0}^{\infty}\rho^{j}\widetilde{u}_{t-j,\ell}.$

Using Minkowski’s inequality, we have that

$\left|y_{t}-\widetilde{y}_{t,\ell}\right|_{4}\leq\left|y_{t}-\overline{y}_{t}\right|_{4}+\left|\overline{y}_{t}-\widetilde{y}_{t,\ell}\right|_{4}=I+II,$

with

(B.12)
$\left|y_{t}-\overline{y}_{t}\right|_{4}\leq\left|\rho^{t}y_{0}\right|_{4}+\left|\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|_{4}\leq
c_{0}\left|\rho\right|^{t}+\sum_{j=t}^{\infty}\left|\rho\right|^{j}\left|u_{0}\right|_{4}\leq
c_{1}\left|\rho\right|^{t},$

and

(B.13) $\displaystyle\left|\overline{y}_{t}-\widetilde{y}_{t,\ell}\right|_{4}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{\infty}\left|\rho\right|^{j}\left|u_{t-j}-\widetilde{u}_{t-j,\ell}\right|_{4}=\sum_{j=0}^{\ell}\left|\rho\right|^{j}\left|u_{t-j}-\widetilde{u}_{t-j,\ell}\right|_{4}+\sum_{j=\ell+1}^{\infty}\left|\rho\right|^{j}\left|u_{t-j}-\widetilde{u}_{t-j,\ell}\right|_{4}$
$\displaystyle\leq$
$\displaystyle\sum_{j=0}^{\ell}\left|\rho\right|^{j}\left|u_{t-j}-\widetilde{u}_{t-j,\ell}\right|_{4}+\left|\rho\right|^{\ell}\sum_{j=1}^{\infty}\left|\rho\right|^{j}\left(\left|u_{t-j}\right|_{4}+\left|\widetilde{u}_{t-j,\ell}\right|_{4}\right)$
$\displaystyle\leq$ $\displaystyle
c_{0}\left(\sum_{j=0}^{\ell}\left|\rho\right|^{j}\right)\ell^{-a}+c_{1}\left|\rho\right|^{\ell}\leq
c_{2}\ell^{-a},$

for some $c_{2}<\infty$. Note that equation (B.13) entails that
$\overline{y}_{t}$ also satisfies Assumption 1.

We are now ready to prove that Lemmas B.2-B.4 hold under (18). Let
$\overline{\mathbf{x}}_{t}=\left(\mathbf{z}_{t}^{\prime},\overline{\mathbf{y}}_{p,t}^{\prime}\right)^{\prime}$.
We begin with Lemma B.4, and note that

(B.14)
$\displaystyle\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}-\frac{1}{m}\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\overline{\mathbf{x}}_{t}^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{m}\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\left(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right)^{\prime}+\frac{1}{m}\sum_{t=1}^{m}\left(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right)\overline{\mathbf{x}}_{t}^{\prime}+\frac{1}{m}\sum_{t=1}^{m}\left(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right)\left(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right)^{\prime}.$

It holds that

$\displaystyle\left\|\frac{1}{m}\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\left(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right)^{\prime}\right\|_{F}^{2}=$
$\displaystyle\sum_{j,h=1}^{d}\left(\frac{1}{m}\sum_{t=1}^{m}\overline{x}_{j,t}\left(x_{h,t}-\overline{x}_{h,t}\right)\right)^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{d}\left(\frac{1}{m}\sum_{t=1}^{m}\overline{x}_{j,t}^{2}\right)\sum_{h=1}^{d}\left(\frac{1}{m}\sum_{t=1}^{m}\left(x_{h,t}-\overline{x}_{h,t}\right)^{2}\right).$

It is easy to see that

$E\sum_{j=1}^{d}\left(\frac{1}{m}\sum_{t=1}^{m}\overline{x}_{j,t}^{2}\right)\leq
c_{0}d,$

and

$E\sum_{h=1}^{d}\left(\frac{1}{m}\sum_{t=1}^{m}\left(x_{h,t}-\overline{x}_{h,t}\right)^{2}\right)\leq
c_{0}dm^{-1},$

because by (B.12), for all $h$,
$\left|x_{h,t}-\overline{x}_{h,t}\right|_{2}\leq\left|x_{h,t}-\overline{x}_{h,t}\right|_{4}\leq
c_{1}\left|\rho\right|^{t}$. Hence

$\left\|\frac{1}{m}\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\left(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right)^{\prime}\right\|_{F}=O_{P}\left(\frac{d}{m^{1/2}}\right),$

and the same can be shown for the other terms in (B.14); we note that these
bounds are not the sharpest possible (in essence, this is due to the fact that
some coordinates of $\mathbf{x}_{t}$ and $\overline{\mathbf{x}}_{t}$ may be
the same, thus reducing the dimensionality), but they suffice for our
purposes. Lemma B.4 now follows readily from repeating the original proof,
using
$\sum_{t=1}^{m}\overline{\mathbf{x}}_{t}\overline{\mathbf{x}}_{t}^{\prime}$
instead of $\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}$. We now turn
to Lemma B.2. We estimate

$\left\|\frac{1}{m}\sum_{t=1}^{m}\left(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right)\epsilon_{t}\right\|=\sum_{h=1}^{d}\left|\sum_{t=1}^{m}\left(x_{h,t}-\overline{x}_{h,t}\right)\epsilon_{t}\right|^{2}\leq\sum_{h=1}^{d}\left|\sum_{t=1}^{m}\rho^{t}y_{0}\epsilon_{t}\right|^{2}+\sum_{h=1}^{d}\left|\sum_{t=1}^{m}\left(\epsilon_{t}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right)\right|^{2},$

having used (B.10) and (B.11). Considering the first term, it holds that

$E\sum_{h=1}^{d}\left|\sum_{t=1}^{m}\rho^{t}y_{0}\epsilon_{t}\right|^{2}=\sum_{h=1}^{d}\sum_{t,s=1}^{m}\left|\rho\right|^{t+s}E\left(y_{0}^{2}\epsilon_{t}\epsilon_{s}\right)\leq\sum_{h=1}^{d}\sum_{t,s=1}^{m}\left|\rho\right|^{t+s}\left|y_{0}\right|_{4}^{2}\left|\epsilon_{t}\right|_{4}\left|\epsilon_{s}\right|_{4}\leq
c_{0}d,$

having used the Cauchy-Schwartz inequality (twice), Assumption 1, and
Assumption 6. As far as the second term is concerned, note that

$\displaystyle
E\sum_{h=1}^{d}\left|\sum_{t=1}^{m}\left(\epsilon_{t}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right)\right|^{2}$
$\displaystyle=$ $\displaystyle
E\sum_{h=1}^{d}\left|\sum_{t=1}^{m}\left(\rho^{t}\epsilon_{t}\sum_{h=0}^{\infty}\rho^{h}u_{h}\right)\right|^{2}=E\sum_{h=1}^{d}\sum_{t,s=1}^{m}\left(\rho^{t+s}\epsilon_{t}\epsilon_{s}\sum_{h,i=0}^{\infty}\rho^{h+i}u_{h}u_{i}\right)$
$\displaystyle=$
$\displaystyle\sum_{h=1}^{d}\sum_{t,s=1}^{m}\rho^{t+s}\sum_{h,i=0}^{\infty}\rho^{h+i}E\left(\epsilon_{t}\epsilon_{s}u_{h}u_{i}\right)\leq
c_{0}d,$

which follows again from Assumptions 1 and 6(ii), and Hölder’s inequality.
Hence it follows that

$\left\|\frac{1}{m}\sum_{t=1}^{m}\left(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t}\right)\epsilon_{t}\right\|=O_{P}\left(\frac{d^{1/2}}{m}\right),$

and now the desired result follows by repeating the proof of Lemma B.2 using
$\overline{\mathbf{x}}_{t}$ instead of $\mathbf{x}_{t}$. Finally, we consider
Lemma B.3, and we estimate

$\max_{1\leq k\leq
T_{m}}\frac{1}{k^{\zeta_{3}}}\left\|\sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t})\right\|\leq
c_{0}d^{1/2}\left(\max_{1\leq k\leq
T_{m}}\frac{1}{k^{\zeta_{3}}}\left|\sum_{t=m+1}^{m+k}\rho^{t}y_{0}\right|+\max_{1\leq
k\leq
T_{m}}\frac{1}{k^{\zeta_{3}}}\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|\right),$

where again we do not derive the sharpest bounds because we do not take into
account the fact that some elements of $\mathbf{x}_{t}$ and
$\overline{\mathbf{x}}_{t}$ coincide. It is easy to see that

$\max_{1\leq k\leq
T_{m}}\frac{1}{k^{\zeta_{3}}}\left|\sum_{t=m+1}^{m+k}\rho^{t}y_{0}\right|=\left|y_{0}\right|\max_{1\leq
k\leq
T_{m}}\frac{1}{k^{\zeta_{3}}}\left|\sum_{t=m+1}^{m+k}\rho^{t}\right|=O_{P}\left(1\right),$

for all $\zeta_{3}>0$. Also

$\displaystyle P\left(\max_{1\leq k\leq
T_{m}}\frac{1}{k^{\zeta_{3}}}\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|\geq
x\right)$ $\displaystyle\leq$ $\displaystyle
P\left(\max_{0\leq\ell\leq\left\lceil\ln
T_{m}\right\rceil}\max_{\exp\left(\ell\right)\leq
k\leq\exp\left(\ell+1\right)}\frac{1}{k^{\zeta_{3}}}\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|\geq
x\right)$ $\displaystyle\leq$ $\displaystyle\sum_{\ell=0}^{\left\lceil\ln
T_{m}\right\rceil}P\left(\max_{\exp\left(\ell\right)\leq
k\leq\exp\left(\ell+1\right)}\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|\geq
x\exp\left(\zeta_{3}\ell\right)\right)$ $\displaystyle\leq$ $\displaystyle
c_{0}x^{-p}\exp\left(-p\zeta_{3}\ell\right)E\max_{\exp\left(\ell\right)\leq
k\leq\exp\left(\ell+1\right)}\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|^{p}$
$\displaystyle\leq$ $\displaystyle
x^{-p}\exp\left(-p\zeta_{3}\ell\right)E\max_{1\leq
k\leq\exp\left(\ell+1\right)}\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|^{p}$

for all $p\leq 4$. Note now that, by repeated application of Minkowski
inequality and by Assumptions 1 and 6(ii)

$\displaystyle\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|_{p}$
$\displaystyle\leq$
$\displaystyle\sum_{t=m+1}^{m+k}\left|\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|_{p}\leq\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\left|\rho\right|^{j}\left|u_{t-j}\right|_{p}$
$\displaystyle\leq$
$\displaystyle\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\left|\rho\right|^{j}\left|u_{t-j}\right|_{p}\leq
c_{0}\sum_{t=m+1}^{m+k}\left|\rho\right|^{t},$

so that

$\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|_{p}^{p}\leq
c_{0}\left(\sum_{t=m+1}^{m+k}\left|\rho\right|^{t}\right)^{p};$

this is true for all $m$ and $k$, and by construction
$\left|\rho\right|^{t}\geq 0$. Hence, we can apply Theorem F in Móricz (1976),
whence

$E\max_{1\leq
k\leq\exp\left(\ell+1\right)}\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|^{p}\leq\log_{2}\left(4\exp\left(\ell+1\right)\right)\left(\sum_{t=m+1}^{m+k}\left|\rho\right|^{t}\right)^{p}\leq
c_{0}\ell,$

which entails that

$\max_{1\leq k\leq
T_{m}}\frac{1}{k^{\zeta_{3}}}\left|\sum_{t=m+1}^{m+k}\sum_{j=t}^{\infty}\rho^{j}u_{t-j}\right|=O_{P}\left(1\right),$

for all $\zeta_{3}>0$. Putting all together, for all $\zeta_{3}>0$ it holds
that

$\max_{1\leq k\leq
T_{m}}\frac{1}{k^{\zeta_{3}}}\left\|\sum_{t=m+1}^{m+k}(\mathbf{x}_{t}-\overline{\mathbf{x}}_{t})\right\|=O_{P}\left(d^{1/2}\right);$

again, Lemma B.3 now follows from repeating the original proof with
$\overline{\mathbf{x}}_{t}$ instead of $\mathbf{x}_{t}$.

###### Proof of Proposition 5.

Let $\sigma=1$ for simplicity. We begin by showing that, on a suitably
enlarged probability space, there exist two independent standard Wiener
processes $\left\\{W_{1,m}\left(k\right),k\geq 1\right\\}$ and
$\left\\{W_{2,m}\left(k\right),k\geq 1\right\\}$ such that

(B.15) $\max_{1\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\widehat{\epsilon}_{t}-\left(W_{1,m}\left(k\right)-\displaystyle\frac{k}{m}W_{2,m}\left(m\right)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}=o_{P}\left(1\right).$

Some passages are repetitive, but we report them anyway to make the proof
easier to follow. We begin by noting that

(B.16) $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\mathbf{x}_{t}^{\prime}\left(\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}-\mathbf{C}^{-1}\right)\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle\leq$ $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left\|\displaystyle\sum_{t=m+1}^{m+k}\mathbf{x}_{t}\right\|\left\|\left(\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}-\mathbf{C}^{-1}\right)\right\|_{F}\displaystyle\left\|\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right\|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq k\leq
T_{m}}\frac{d^{1/2}kdm^{-1/2}d^{1/2}m^{-1/2}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(1\right)\frac{d^{2}}{m^{1/2}}\max_{1\leq j\leq
J}\frac{1}{\widetilde{r}_{m}^{1/2-\eta_{j}}c_{\alpha,\eta_{j}}}\max_{1\leq
k\leq T_{m}}\left(\frac{k}{m+k}\right)^{1-\eta_{j}}\frac{1}{d_{a_{m},\eta}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(1\right)\frac{d^{2}}{m^{1/2}}=o_{P}\left(1\right),$

having used: Lemmas B.2 and B.4 in the third line, and the fact that
$d=o\left(m^{1/4}\right)$ in the last line. Similarly, it holds that

(B.17) $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)^{\prime}\mathbf{C}^{-1}\displaystyle\left(\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle\leq$ $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left\|\displaystyle\sum_{t=m+1}^{m+k}\left(\mathbf{x}_{t}-\mathbf{c}_{1}\right)\right\|\left\|\mathbf{C}^{-1}\right\|_{F}\displaystyle\left\|\frac{1}{m}\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right\|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq k\leq
T_{m}}\frac{k^{\zeta_{3}}d^{1/2}d^{1/2}d^{1/2}m^{-1/2}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)d^{3/2}\max_{1\leq j\leq
J}\frac{1}{\widetilde{r}_{m}^{1/2-\eta_{j}}c_{\alpha,\eta_{j}}}\max_{1\leq
k\leq
T_{m}}\left(\frac{k}{m+k}\right)^{\zeta_{3}-\eta_{j}}\left(\frac{1}{m+k}\right)^{1-\zeta_{3}}\frac{1}{d_{a_{m},\eta}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(1\right)d^{3/2}\left(\frac{1}{m}\right)^{1-\zeta_{3}}\widetilde{r}_{m}^{\zeta_{3}-1/2}=o_{P}\left(1\right),$

having used: Lemma B.3 with $\zeta_{3}=1/2+\varepsilon$ for all $\eta_{j}<1/2$
and $1/2<\zeta_{3}<\min\left\\{1,\eta_{j}\right\\}$ for all $\eta_{j}>1/2$,
Lemma B.2 and (B.6) in the third line, and the fact that
$d=o\left(m^{1/4}\right)$ in the last line. Combining (B.16) and (B.17), and
recalling that $\mathbf{c}_{1}^{\prime}\mathbf{C}^{-1}\mathbf{x}_{t}=1$, we
get

$\max_{1\leq k\leq
T_{m}}\frac{\left|\left(\displaystyle\sum_{t=m+1}^{m+k}\mathbf{x}_{t}^{\prime}\right)\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\mathbf{x}_{t}^{\prime}\right)^{-1}\left(\displaystyle\sum_{t=1}^{m}\mathbf{x}_{t}\epsilon_{t}\right)-\displaystyle\frac{k}{m}\displaystyle\sum_{t=1}^{m}\epsilon_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}=o_{P}\left(1\right),$

whence

(B.18) $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\widehat{\epsilon}_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\epsilon_{t}-\displaystyle\frac{k}{m}\displaystyle\sum_{t=1}^{m}\epsilon_{t}\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}+o_{P}\left(1\right).$

Also, by standard algebra based on Lemma B.1

$\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|\displaystyle\sum_{t=m+1}^{m+k}\epsilon_{t}-W_{1,m}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq j\leq
J}\max_{1\leq k\leq
T_{m}}\frac{k^{\zeta_{1}}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq j\leq
J}\frac{1}{\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}}\max_{1\leq k\leq
T_{m}}\frac{k^{\zeta_{1}-\eta_{j}}}{\left(m+k\right)^{1/2-\eta_{j}}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(1\right)a_{m}^{\zeta_{1}-1/2}=o_{P}\left(1\right),$

and

$\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|\displaystyle\frac{k}{m}\displaystyle\sum_{t=1}^{m}\epsilon_{t}-\displaystyle\frac{k}{m}W_{2,m}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq k\leq
T_{m}}\frac{\frac{k}{m}m^{\zeta_{2}}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle=$ $\displaystyle
m^{\zeta_{2}-1/2}O_{P}\left(1\right)\max_{1\leq j\leq
J}\frac{1}{\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}}\max_{1\leq k\leq
T_{m}}\left(\frac{k}{m+k}\right)^{1-\eta_{j}}=m^{\zeta_{2}-1/2}O_{P}\left(1\right)=o_{P}\left(1\right).$

Thence (B.15) follows. We now study

$\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|W_{1,m}\left(k\right)-\displaystyle\frac{k}{m}W_{2,m}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|W_{1}\left(k\right)-\displaystyle\frac{k}{m}W_{2}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left(1+\displaystyle\frac{k}{m}\right)\left|W\displaystyle\left(\frac{k/m}{1+k/m}\right)\right|}{\left(1+\displaystyle\frac{k}{m}\right)\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{1\leq k\leq
T_{m}}\frac{\left|W\displaystyle\left(\frac{k/m}{1+k/m}\right)\right|}{\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{k}{m+k}\right)^{\eta_{j}}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{1/m\leq t\leq
T_{m}/m}\frac{\left|W\displaystyle\left(\frac{t}{1+t}\right)\right|}{\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{t}{1+t}\right)^{\eta_{j}}},$

seeing as the distributions of $W_{1,m}\left(k\right)$ and
$W_{2,m}\left(k\right)$ do not depend on $m$ (second line), and by the same
token as in (B.1), with $W\left(\cdot\right)$ standard Wiener. Therefore,
letting $J^{\ast}$ be the set of indices $j$ for which $\eta_{j}>1/2$ and
repeatedly using the scale transform of Wiener process

$\displaystyle\max_{1/m\leq t\leq
T_{m}/m}\frac{\left|W\displaystyle\left(\frac{t}{1+t}\right)\right|}{\min_{1\leq
j\leq
J}c_{\alpha,\eta_{j}}\widetilde{r}_{m}^{1/2-\eta_{j}}d_{a_{m},\eta}\displaystyle\left(\frac{t}{1+t}\right)^{\eta_{j}}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\max\left\\{\max_{j\notin J^{\ast}}\max_{1/m\leq t\leq
T_{m}/m}\frac{\left|W\displaystyle\left(\frac{t}{1+t}\right)\right|}{c_{\alpha,\eta_{j}}\displaystyle\left(\frac{t}{1+t}\right)^{\eta_{j}}},\max_{j\in
J^{\ast}}r_{m}^{\eta_{j}-1/2}\max_{a_{m}/m\leq t\leq
T_{m}/m}\frac{\left|W\displaystyle\left(\frac{t}{1+t}\right)\right|}{c_{\alpha,\eta_{j}}\displaystyle\left(\frac{t}{1+t}\right)^{\eta_{j}}}\right\\}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\max\left\\{\max_{j\notin J^{\ast}}\max_{1/\left(m+1\right)\leq
u\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(u\right)\right|}{c_{\alpha,\eta_{j}}u^{\eta_{j}}},\max_{j\in
J^{\ast}}r_{m}^{\eta_{j}-1/2}\max_{a_{m}/\left(m+a_{m}\right)\leq u\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(u\right)\right|}{c_{\alpha,\eta_{j}}u^{\eta_{j}}}\right\\}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\max\left\\{\max_{j\notin J^{\ast}}\max_{1/\left(m+1\right)\leq
u\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(u\right)\right|}{c_{\alpha,\eta_{j}}u^{\eta_{j}}},\max_{j\in
J^{\ast}}\max_{1\leq
s\leq\left(T_{m}\left(m+a_{m}\right)\right)/\left(a_{m}\left(m+T_{m}\right)\right)}\frac{\left|W\left(s\right)\right|}{c_{\alpha,\eta_{j}}s^{\eta_{j}}}\right\\}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\max\left\\{\max_{j\notin J^{\ast}}\max_{1/\left(m+1\right)\leq
u\leq
T_{m}/\left(m+T_{m}\right)}\frac{\left|W\left(u\right)\right|}{c_{\alpha,\eta_{j}}u^{\eta_{j}}},\max_{j\in
J^{\ast}}\max_{\left(a_{m}\left(m+T_{m}\right)\right)/\left(T_{m}\left(m+a_{m}\right)\right)\leq
u\leq
1}\frac{\left|W\left(u\right)\right|}{c_{\alpha,\eta_{j}}u^{1-\eta_{j}}}\right\\}$
$\displaystyle\overset{a.s}{\rightarrow}$
$\displaystyle\max\left\\{\max_{j\notin
J^{\ast}}\sup_{0<u<1}\frac{\left|W\left(u\right)\right|}{c_{\alpha,\eta_{j}}u^{\eta_{j}}},\max_{j\in
J^{\ast}}\sup_{0<u<1}\frac{\left|W\left(u\right)\right|}{c_{\alpha,\eta_{j}}u^{1-\eta_{j}}}\right\\}\overset{\mathcal{D}}{=}\sup_{0<u<1}\frac{\left|W\left(u\right)\right|}{\min_{1\leq
j\leq J}c_{\alpha,\eta_{j}}u^{\widetilde{\eta}_{j}}},$

whence the desired result.

###### Proof of Proposition 6.

We know from the proof of Theorem 2 that the presence of power depends only on
the non-centrality induced by the break, so - following the logic in that
proof - we need to study

(B.19)
$\frac{\left(\widetilde{k}-k^{\ast}\right)\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{\widetilde{k}}{m}\right)\min_{1\leq
j\leq
J}\widetilde{r}_{m}^{1/2-\eta_{j}}\displaystyle\left(\frac{\widetilde{k}}{m+\widetilde{k}}\right)^{\eta_{j}}},$

where $\widetilde{k}$ is specified later. Under condition (i), let
$\widetilde{k}=a_{m}+k^{\ast}$; note that whenever $j\in J^{\ast}$, (B.19) is
of the same order as

$\frac{a_{m}\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|}{m^{1/2}\min_{j\in
J^{\ast}}\displaystyle\left(\frac{a_{m}}{m}\right)^{\eta_{j}}}=\frac{a_{m}^{1-\max_{j\in
J^{\ast}}\eta_{j}}\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|}{m^{1/2-\max_{j\in
J^{\ast}}\eta_{j}}},$

and seeing as

$\lim_{m\rightarrow\infty}\frac{a_{m}^{1-\max_{j\in
J^{\ast}}\eta_{j}}\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|}{m^{1/2-\max_{j\in
J^{\ast}}\eta_{j}}}\times\frac{1}{a_{m}^{1/2}\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|}=\infty,$

on account of (10), it follows that under condition (i)

$\frac{\left(\widetilde{k}-k^{\ast}\right)\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{\widetilde{k}}{m}\right)\min_{1\leq
j\leq
J}\widetilde{r}_{m}^{1/2-\eta_{j}}\displaystyle\left(\frac{\widetilde{k}}{m+\widetilde{k}}\right)^{\eta_{j}}}\geq
c_{0}a_{m}^{1/2}\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|,$

for some $c_{0}>0$, whence (14) yields the final result. Under conditions (ii)
and (iii), let
$\widetilde{k}=\left\lfloor\left(1+c_{0}\right)k^{\ast}\right\rfloor$; by
standard algebra, it is easy to see that (B.19) is proportional to

$\frac{k^{\ast}\left|\mathbf{c}_{1}^{\prime}\Delta_{m}\right|}{m^{1/2}\left(1+\displaystyle\frac{k^{\ast}}{m}\right)\min_{1\leq
j\leq
J}\widetilde{r}_{m}^{1/2-\eta_{j}}\displaystyle\left(\frac{k^{\ast}}{m+k^{\ast}}\right)^{\eta_{j}}},$

whence (21) yields the desired result.


# A data-driven approach to guide supersonic impinging jet control

Spencer L. Stahl1<EMAIL_ADDRESS>Datta V. Gaitonde1  1Department of
Mechanical and Aerospace Engineering, The Ohio State University, OH 43210, USA

###### Abstract

A data-driven framework using snapshots of an uncontrolled flow is proposed to
identify, and subsequently demonstrate, effective control strategies for
different objectives in supersonic impinging jets. The approach, based on a
dynamic mode decomposition reduced order model (DMD-ROM), determines forcing
receptivity in an economical manner by projecting flow and actuator-specific
forcing snapshots onto a reduced subspace and then evolving the results
forward in time. Since it effectively determines a linear response around the
unsteady flow in the time-domain, the method differs materially from typical
techniques that use steady basic states, such as stability or input-output
approaches that employ linearized Navier-Stokes operators in the frequency-
domain. The method naturally accounts for factors inherent to the snapshot
basis, including configuration complexity and flow parameters such as Reynolds
number. Furthermore, gain metrics calculated in the reduced subspace
facilitate rapid assessments of flow sensitivities to a wide range of forcing
parameters, from which optimal actuator inputs may be selected and results
confirmed in scale-resolved simulations or experiments. The DMD-ROM approach
is demonstrated from two different perspectives. The first concerns asymptotic
feedback resonance, where the effects of harmonic pressure forcing are
estimated and verified with nonlinear simulations using a blowing-suction
actuator. The second examines time-local behavior within critical feedback
events, where the phase of actuation becomes important. For this, a
conditional space-time mode is used to identify the optimal forcing phase that
minimizes convective instability initiation within the resonance cycle.

##  1 Introduction

Active flow control has the potential to alleviate many aerodynamic problems;
some objectives include mitigation of thermomechanical loads, separation and
unsteadiness, or flow-acoustic interactions due to supersonic jets impinging
on a ground plane, of interest in this work. Optimal control inputs are
predicated on identifying responsiveness to external forcing (receptivity to
perturbations) that depend on the underlying flow mechanisms, which are often
examined with the linearized frequency-domain Navier-Stokes equations applied
to steady basic states. Classical linear stability extracts such information
from the laminar state (Theofilis, 2011; Juniper et al., 2014), or, the time-
averaged turbulent state, whose modes are related to larger turbulent coherent
structures (Crighton & Gaster, 1976). More recently, other manners of
analyzing the mean flow have been employed with overlapping objectives to
extract singular or eigenvalue spectra with corresponding modes (Schmid, 2007;
Luchini & Bottaro, 2014; Ranjan et al., 2020). From a control standpoint,
optimal forcing-response relationships may be derived from the mean flow
through resolvent or input-output modes that maximize linear energy growth.
Example applications of control strategies derived from perturbed steady basic
states may be found in Herrmann et al. (2021, 2023).

When unsteady data is available in the form of snapshots, additional
information may be derived to discern the key mechanisms as well as to propose
control inptus. Modal decomposition techniques (Taira et al., 2020) have
become increasingly popular to represent the dynamics (model order reduction),
and their potential implications for control have also been elaborated on
(Rowley & Dawson, 2017; Brunton & Noack, 2015). In the present work, we
propose a data-driven approach (§2) to derive a forcing-response relationship
using perturbations in the time-varying flow with snapshot-based modal
decomposition serving as a linear surrogate for the dynamics. Dynamic Mode
Decomposition (DMD) provides such a representation (Schmid, 2010) and is thus
used to generate an efficient reduced order model (DMD-ROM) for the nonlinear
behavior contained in the snapshots. The method also provides a reduced basis
for projection of the input control forcing and the evolution in time of the
forced system. The procedure may be viewed as a much simplified data-driven
analog of the synchronized Large Eddy Simulation (LES) method of Adler &
Gaitonde (2018) to determine time-local perturbation growth in the turbulent
flow.

Several advantages may be delineated. Since it is data-driven, the DMD-ROM
contains the main dynamics encapsulated in the snapshots, including
configuration complexity and flow parameters such as Mach and Reynolds
numbers, large values of which can often impede more difficult linearized
Navier-Stokes driven approaches because of spurious modes. In addition, the
time-domain nature of the method eases specification of realizable actuator-
specific inputs, including the nature of perturbations, localization in space,
and phase variations in time, this last advantage also enables control
assessments of time-local events. The approach may also be applied only with
select snapshot variables; this is advantageous, for example, when data is
obtained from experiments. More crucially, even when all data is available,
say, from uncontrolled scale-resolved simulations, only a subset of variables
may be selected to construct the DMD-ROM. In the impinging jet problem of
interest, the hydrodynamic-acoustic interaction is well encapsulated in the
scale-resolved pressure field, and is therefore the only variable used.

For computational economy – an important consideration for time-domain methods
– we use a reduced subspace, constructed from projection of flow and forcing
snapshots onto a Proper Orthogonal Decomposition (POD) basis. This
significantly speeds up parametric studies of actuator locations and forcing
properties. Evaluation metrics such as energy gain for asymptotic or short-
time events are also computed in the reduced space; projection back to the
full physical space may be performed only when needed.

The capabilities of the DMD-ROM approach are highlighted in the context of a
complex supersonic impinging jet containing interacting shocks and nonlinear
instabilities. The resonant dynamics have been extensively reviewed by
Edgington-Mitchell (2019). Briefly, Kelvin-Helmholtz structures in the shear-
layer emanating from the nozzle exit impinge on the ground plane to generate
acoustic waves that travel back to the nozzle exit to complete the loop. Two
applications of the DMD-ROM are considered to address asymptotic and short-
time phenomena, respectively. The first examines the tonal behavior of the
impinging jet. For this, we employ harmonic pressure forcing at the nozzle
receptivity region to interfere with the aeroacoustic feedback resonance and
modify the most amplified tones. The forced linear response is confirmed with
post facto scale-resolved simulations featuring realizable blowing-suction
actuators with the same spectral forcing properties.

The second aspect concerns transient or time-local control to address the
problem of energy growth intrinsic to conditionally selected events occurring
over finite time. This example targets the vital acoustic feedback receptivity
events in the resonance cycle which initiate the convective shear-layer
instability. Perturbation-based insights into such transient structures may be
found in Karami et al. (2018). Here we use conditional space-time proper
orthogonal decomposition (CST-POD) method to isolate an ensemble of shear-
layer events directly from the unsteady flow (Schmidt & Schmid, 2019; Stahl et
al., 2023) for perturbation-based receptivity analysis, with the objective of
determining the optimal forcing phase to mitigate the convective instability
that perpetuates the resonance cycle.

##  2 Time-Domain DMD-ROM Forcing Methodology

The proposed method, summarized in the schematic of Fig. 1, is implemented in
four steps: (1) construction of the DMD-ROM from snapshots of the unforced
flow, (2) creation and projection on the reduced space of the forcing snapshot
time-sequence, (3) time advancement of the linearized forcing response and
gain evaluation in the reduced space and (4) expansion of the forced response
back to the full-space when desired.

Figure 1: Elements of the DMD control method. Step (1) calculation of the DMD
operator from the unforced data. Step (2) projection of forcing snapshots onto
POD modes. Step (3) overall schematic including initial condition projection
and time iteration scheme in the reduced space and (4) projection back to
physical space.

The first step (1), uses the unforced sequence of $N$ flow snapshots,
$\textbf{Q}=[q_{1},...,q_{N}]$, to model the linear operator A that marches
the system from one snapshot to the next, $q_{n+1}=\textbf{A}q_{n}$. Since Q
and A are full sized, $M\times N$ and $M\times M$ matrices, where $M$ is the
number of spatial degrees of freedom, the classical DMD approach (Schmid,
2010) is employed to obtain the reduced $N\times N$ operator,
$\tilde{\textbf{A}}$, by projection onto the POD modes, U, obtained from the
singular value decomposition of Q.

The second step (2), constructs the $M\times N$-sized forcing snapshot matrix,
$\textbf{F}=[f_{1},...,f_{N}]$, based on the chosen actuator location and
perturbation properties. For the jet examples, the most natural location for
actuators is the region around the nozzle exit. Although the forcing function
may be an arbitrary function of time, here we consider harmonic forcing
$f(t)=A\sin(\omega t+\phi)$, with $A$, $\omega$ and $\phi$ being the
amplitude, angular frequency and phase shift in time, respectively. The
reduced flow and forcing snapshots are projected onto the POD modes of the
flow, $\tilde{q}_{n}=\textbf{U}^{-1}q_{n}$ and
$\tilde{f_{n}}=\textbf{U}^{-1}f_{n}$, respectively.

The third step (3), shown in the overall scheme of Fig. 1, evolves the reduced
forced system in time,

$\tilde{q}^{\prime}_{n+1}=\tilde{\textbf{A}}\tilde{q}^{\prime}_{n}+\tilde{f}_{n},\hskip
21.68121ptn=0,1,\ldots,N$ (1)

where $\tilde{q}^{\prime}_{n+1}$ represents the linear response of the system
and $n=0$ represents the initial condition. Since interpreting the forced
results directly from the reduced coordinates can be ambiguous, in the fourth
step (4), the full space equivalent of $q^{\prime}_{n+1}$ may be reassembled
when desired by projection back to the full-space domain to obtain the forced
solution ($\textbf{Q}^{\prime}=\textbf{U}\tilde{\textbf{Q}}^{\prime}$). For
the present feed-forward study, projection back to the full-space is only
necessary after the time-iteration stage is complete. Full-space expansion
within each iteration may become necessary in feedback control to inform
changes in the forcing.

The procedure to obtain gain in the reduced space exploits features that are
consistent between it and the full-sized system. Specifically, we use the
$L2$-norm ratio of the forced and unforced solutions
$\sigma=||\tilde{\textbf{Q}}^{\prime}||_{2}/||\tilde{\textbf{Q}}||_{2}$ to
obtain a relative sense of gain. This provides an adequate assessment for the
asymptotic forcing analysis (§3.1). For the transient problem (§3.2), we use
the ratio of the $L2$-norm of individual snapshots to the initial condition
snapshot: $\sigma(t)=||\tilde{q}(t)||_{2}/||\tilde{q}_{o}||_{2}$. Since this
transient norm applies to both unforced and forced systems, the value may then
be assessed to understand the relative effects of the forcing over time.
Operations in the reduced space assure computational economy and enable
testing of a wide range of forcing parameters.

Finally, we note that the approach differs from the DMD Control (DMDC) method
of Rosenfeld & Kamalapurkar (2021), which uses forced data for system
identification to distinguish the dynamic and forcing operators. The current
DMD-ROM approach could potentially be enriched with such data when combined
with a learned control operator; however the focus here is on obtaining the
linear response of the unforced dataset to different forcing parameters,
optimal results from which may be refined with limited computationally intense
nonlinear simulations or experimental testing.

##  3 Application to Supersonic Impinging Jets

We consider the aeroacoustic resonance of a planar (spanwise homogeneous),
Mach 1.27, underexpanded jet (nozzle-width, $D=0.0254m$) impinging on a plate
surface located $4D$ away, as shown in Fig. 2(a) which also displays domain
dimensions.

Figure 2: (a) Planar impinging jet domain. (b) Instantaneous uncontrolled
pressure snapshot and DMD-ROM initial condition with acoustic probe location.
(c) nozzle forcing location (rectangle) for the DMD-ROM (pressure) and LES
(blowing-suction).

The uncontrolled (baseline) data, containing multiple resonant tones, is
acquired from an LES at a Reynolds number of $Re=5.8\times 10^{5}$ and
comprises $3{,}855$ pressure snapshots from the mid-plane of the domain
sampled at a nondimensional frequency $St=fD/a=10$, where $f$ is the frequency
(Hz) and $a=343m/s$ is the ambient speed of sound. The methodology for the
LES, including numerical scheme and mesh, follow those established in Stahl et
al. (2022).

As noted earlier, the pressure variable is suitable for the dynamics of
interest since it encompasses the hydrodynamic and acoustic phenomena that
dictate the feedback loop. A DMD-ROM based on mean-subtracted pressure
fluctuations aids in stability by ensuring all DMD eigenvalues are neutrally
stable (Towne et al., 2018). A representative baseline pressure snapshot,
shown in Fig. 2(b), illustrates the natural asymmetric (at this time instant,
predominantly blue on the right, red on the left outside the stagnation
region) side-to-side flapping of the jet and feedback acoustics. The probe
location in Fig. 2(b) monitors the acoustic feedback tones used in the rest of
the paper. The forcing is applied on rectangular regions of size $0.2D\times
0.07D$ centered at the nozzle lip exit, displayed in the inset of Fig. 2(c),
which shows both pressure forcing for the DMD-ROM and blowing-suction
actuation for the LES.

###  3.1 Example 1: Asymptotic forcing analysis

The first example illustrates the asymptotic tonal response of the forcing
model.

Figure 3:  (a) Unforced acoustic feedback tones at the probe located outside
the jet. (b) Initial deviation of the unforced LES and forced DMD-ROM (top)
with the forcing input signal (bottom). (c) Scaled DMD-ROM acoustic
spectrogram subtracted from unforced spectrogram. (d) Leading POD mode and
temporal coefficient spectra of the unforced and forced systems.

The power-spectral density (PSD) of the uncontrolled pressure fluctuations
from the probe are plotted in Fig. 3(a). Several tones are evident, the two
loudest of which are at $St=0.2$ and $St=0.4$. These correspond respectively
to asymmetric and symmetric patterns, consistent with prior studies (Stahl et
al., 2022), as elaborated further below. We consider a broad range of forcing
amplitude and frequency parameters. To illustrate the capability of the DMD-
ROM approach, we first discuss in detail the effect of sinusoidal forcing at a
frequency of $St=0.4$ and amplitude $A=0.005$ (lower part of Fig. 3(b)), which
is only approximately $10\%$ of acoustic fluctuations at the probe. The
forcing is applied to the both sides of the nozzle exit. These conditions
represent the potential manipulation of the resonance mode from the loudest
asymmetric tone at $St=0.2$ to the symmetric mode at $St=0.4$. A short-time
comparison of the unforced and forced DMD-ROM probe signals, top part of Fig.
3(c), displays a gradual deviation of the pressure over time due to the
forcing.

The long term, asymptotic influence of the forcing on the resonance tones is
displayed in Fig. 3(c) with a spectrogram of the probe signals. To highlight
tonal differences, the DMD-ROM results are scaled by the gain ($\sigma=1.44$)
and subtracted from the unforced spectrogram; absolute gain behavior is
addressed later. The spectrogram demonstrates that the $St=0.4$ tone increases
with time, while the $St=0.2$ tone diminishes. Changes to the overall flow-
field are shown in Fig. 3(d) by the leading POD modes of the unforced and
forced systems, along with their corresponding unscaled temporal coefficient
spectra. The results demonstrate that the dominant mode without control is
asymmetric at $St=0.2$, while that in the forced case is symmetric at
$St=0.4$. Clearly the forcing resonates with the symmetric impinging mode,
which overtakes the nominal flapping mode as the leading dynamic.

To confirm the results from the model, an LES with control is employed by
applying perturbations at $St=0.4$ along the spanwise length of the nozzle
using a realizable blowing-suction actuator. As depicted in Fig. 2(c),
actuation is applied on the inner side of the nozzle lip by specifying choked
flow based on the main impinging jet nozzle condition.

Figure 4:  Validation of the LES blowing-suction at $St=0.4$ showing the (a)
controlled LES spectrogram and (b) leading POD mode and temporal coefficient
spectra.

The acoustic probe spectrogram from the LES results is presented in Fig. 4(a).
As in the DMD-ROM, the initially louder $St=0.2$ tone diminishes with time and
is replaced by the $St=0.4$ tone as the forcing dominates the solution. While
the frequencies and overall dynamics are predicted favorably, the magnitude of
the tones are overpredicted by the DMD-ROM approach. This is a consequence of
its linear nature, which precludes non-linear saturation mechanisms and thus
motivated the gain scaling in Fig. 3(c) to isolate the observed trends. The
leading POD mode and its PSD obtained from the LES are shown in Fig. 4(b). A
comparison with POD modes of Fig. 3(d), demonstrate a similarity between the
controlled LES and DMD-ROM forced results, reflecting the fact that the
leading mode is now symmetric, as opposed to the asymmetric unforced case.

The gain behavior is now examined for a range of forcing frequencies and
amplitudes. Figure 5(a) shows the gain from $St=0.03$ to $1.1$ and amplitudes
$A=0.001$ to $0.015$.

Figure 5: (a) DMD-ROM gain as a function of forcing amplitude and frequency.
(b) Forcing amplitude at resonance tones.

As noted earlier, for this problem, the gain (§2) measures the ratio of the
L2-norm for all forced snapshots to unforced snapshots and therefore captures
the asymptotic trends in total energy growth over the time period of the
unforced data. Several streaks are observed, corresponding to frequencies
amplified by the flow, among which the resonance tones are particularly
susceptible and thus prominent. Figure 5(b) selects some of these amplified
tones to isolate gain versus forcing amplitude; the previous detailed results
at $St=0.4$ forcing are marked with a red line. For some frequencies, e.g.,
$St=0.2$ and $0.6$, growth is observed at all amplitudes. For others,
including $St=0.4$, a larger minimum amplitude is required to observe the
initiation of gain, after which the behavior is similar and rises linearly on
the log-log scale. This behavior is attributable to amplitudes that are too
low to grow appreciably against the fluctuating snapshots over the finite time
period of the ROM. Therefore, larger amplitudes are preferred for sensitivity
studies to better educe these relative growth rates. On the other hand, some
frequencies in Fig. 5(a) display no realizable gain ($\sigma=1$) for all
amplitudes, indicating these forcing conditions have little influence on the
flow. Of course, forcing characteristics that inhibit pressure fluctuations
($\sigma<1$) are of great interest from a practical perspective, but the
present approach, like other linear methods, highlights only growing modes.
However, the upcoming transient analysis discusses a method to inhibit growth
by interfering with time-local events.

###  3.2 Example 2: Transient receptivity analysis

To demonstrate the versatility of the DMD-ROM approach, the statistically
stationary fluctuations of the prior example are replaced by a conditional
space-time mode that isolates the sequence of acoustic wave arrival followed
by growth of the convective shear-layer instability. For this, the DMD-ROM is
cast as a transient receptivity analysis subject to external forcing, where
the gain is measured over a short time-horizon. The CST-POD calculation is
derived as described in Stahl et al. (2023). Briefly, the unforced pressure
signal at the probe (shown earlier in Fig. 2(a)) is analyzed to conditionally
identify finite-time events and corresponding snapshots. A few peak amplitudes
in the pressure signal are shown in Fig. 6(a) for reference, representing
acoustic waves passing over the probe. Each CST-POD event time-window spans
$\Delta T=10$ ($100$ snapshots) and is centered between two feedback cycles,
capturing $74$ total events across the entire data set to generate the
ensemble. A sample event is shown in Fig. 6(b) with the probe pressure (left
axis) plotted with the black solid curve.

Figure 6: (a) Acoustic feedback events from pressure probe. (b) Isolated event
from the pressure signal (black) in comparison to the CST-POD transient gain
(shaded) through two feedback cycles. (c) CST-POD mode progression in time.

The CST-POD mode is obtained from a local spatial domain near the nozzle where
the events are sampled; a few representative snapshots are depicted in Fig.
6(c) to highlight the upstream moving acoustic wave and the ensuing shear-
layer instability. The transient gain of the CST-POD mode is normalized by the
first CST-POD snapshot and is shown by the shaded curve in Fig. 6(b),
illustrating the timing between the acoustic event ($t=0$) and peak shear-
layer growth ($t=+2$) in the feedback cycle. This transient gain represents
the baseline for evaluating the flow response to different external forcing
parameters.

The DMD-ROM forcing for this example case is similar to the previous case, but
is only applied on the nozzle side where the CST-POD mode was derived. Since
the CST-POD data is not statistically stationary, both U and
$\tilde{\textbf{A}}$ are calculated from the full variable without subtracting
the mean. In addition to forcing frequency and amplitude, the phase now
becomes an important factor because of the short time horizon. Results using
the previous $St=0.4$ forcing with an optimal amplitude and phase are first
presented before examining the entire parameter space. The choice of the
amplitude is arbitrary since it is referenced to the CST-POD mode; in this
case $A=0.5$ captures the desired trends in the neighborhood of the local
minimum. The optimal phase is $\phi=150^{\circ}$ based on the parameter sweep
presented below.

Figure 7: (a) Transient gain of the unforced and forced feedback cycles
($St=0.4$, $A=0.5$, and $\phi=150^{\circ}$). (b) Transient gain as a function
of forcing phase. (c) Overall gain as a function of phase and amplitude
($St=0.4$). (d) Overall gain as a function of frequency and phase ($A=0.5$).

Figure 7(a) displays the forcing (lower part of the figure) and gains
associated with unforced and forced responses. The results show that the
transient gain is significantly reduced at the peak of the shear-layer
instability. Figures 7(b-d) illustrate the shear-layer receptivity as a
function of forcing phase. In (b), the transient gain shows phase effects for
a constant $St=0.4$ forcing. At $\phi=150^{\circ}$, the forcing influences the
natural instability processes and dampens the peak at $t=+2$. Figure 7(c)
plots the overall gain ($\sigma$) as a function of amplitude and phase for
$St=0.4$. As with the asymptotic case of Example 1, the gain increases
linearly with forcing amplitude; this holds for all other frequencies as well.
However, in this transient case, conditions where gain is reduced ($\sigma<1$)
become more apparent and are of interest. Figure 7(d) explores the larger
parameter space by plotting the overall gain as frequency and phase are
varied. As expected, the lower frequencies display the largest modulation in
feedback instability growth; these are on the order of the resonant
frequencies and have fewer cycles within the CST-POD time-window. In contrast,
higher-frequencies have little influence on receptivity and are less affected
by phase.

##  4 Discussion

A data-driven framework is presented to discern the linear response of a
turbulent flow with a view towards control analyses. The nonlinear system is
modeled with a dynamic mode decomposition reduced order model (DMD-ROM).
Prescribed forcing conditions are projected onto the same reduced space as the
unsteady flow, and its effects obtained in the time-domain. Computational
efficiency is ensured by calculating the gain due to forcing in the reduced
subspace, which enables rapid scanning of large ranges of forcing parameters,
from which candidates may be expanded to the physical space for further study
and verified with scale-resolved methods.

The key advantages of the method are that, being data-driven, it may be
applied to any flowfield for which time-resolved snapshots are available,
regardless of geometry and flow parameters, without need to solve the
linearized Navier-Stokes equations. Furthermore, the time-domain nature of the
technique facilitates examination of the forced evolution in the unsteady flow
itself, eases actuator perturbation inputs and examination of transient events
where phase becomes important.

Application of the model to the complex physics associated with a resonating
supersonic impinging jet forced at the nozzle receptivity region, successfully
predicts the switch from flapping to symmetric resonance modes, as validated
with a full three-dimensional, nonlinear simulation. For transient events, the
effects of forcing at different phases within the resonant feedback cycle
provides insight into means of diminshing or amplifying convective
instabilities with respect to the critical events within the cycle. In both
examples, thousands of forcing parameters were tested across amplitude,
frequency, and phase variables, highlighting the performance and utility of
the method.

Acknowledgments: The authors acknowledge support from the Collaborative Center
for Aeronautical Sciences and the Office of Naval Research.

Declaration of interest: The authors declare no conflict of interest.

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