in $\|\cdot\|_1$-norm and  uniformly in $\omega$. Furthermore, we have by Assumption <ref> $\|\Sigma_n(\omega)-f(\omega)\|_{\max}=O(1/n)$ and $\|\Sigma_n^{-1}(\omega)-f^{-1}(\omega)\|_{1}=O(\spar/n)$. Additionally, note that we have by (<ref>) $\sup_\omega\|1/(n h) \sum_{k=1}^n \kappa_M(\omega-\omega_k) [Z(\omega_k) Z^H(\omega_k)-\Sigma_n(\omega)]\|_{\max}=\sup_\omega\|\fsp(\omega)-E \fsp(\omega)\|_{\max}=\bp (\gps{x})$ and $\sup_\omega \|\Sigma_n^{-1}(\omega)\|_1=O(\spar)$. From this we have

\begin{align*}
&\frac{\sum_{k=1}^n \kappa_M(\omega-\omega_k) e_v^\top \Sigma_n^{-1}(\omega) [Z(\omega_k) Z^H(\omega_k)-\Sigma_n(\omega)]
I_{p;-v} [\hat \gamma_{-v,u}(\omega)- \gamma_{-v,u}(\omega)]}{\sqrt{Mn} DN \Sigma_{n,v,v}^{-1}(\omega)}  \\
&\leq \sqrt{n/M}\|\Sigma_n^{-1}(\omega)\| \|\fsp(\omega)-E \fsp(\omega)\|_{\max} \|[\hat \gamma_{-v,u}(\omega)- \gamma_{-v,u}(\omega)]\|_1 |DN \Sigma_{n,v,v}^{-1}(\omega)|\\
&\leq (n/M)^{1/4}\spar \|\fsp(\omega)-E \fsp(\omega)\|_{\max} (n/M)^{1/4}\|[\hat \gamma_{-v,u}(\omega)- \gamma_{-v,u}(\omega)]\|_1 |DN \Sigma_{n,v,v}^{-1}(\omega)|\\
&= \bp(\gps {((x-\gpbias)/(\spar (n/M)^{1/4}))^{1/2}}+ \gps 1),
\end{align*}

where  the above equality  holds uniformly in $\omega$. The other  terms  of $\delta(\omega)$ are of the same order and, therefore,

$$\sup_\omega |\delta(\omega)|=\bp(\gps {((x-\gpbias)/(\spar (n/M)^{1/4}))^{1/2}}+ \gps 1).$$

§.§ Error bounds for Gaussian approximations

In all Lemmas of  this subsection,  Assumption <ref> to <ref> hold true.

For $l=1,\dots,M/N-1$ and if $\Theta_{0,\tau}<\infty$ and for some sequence $g_n$ $d_{m,\tau}=\sum_{t=0}^\infty \min (\delta_{t,\tau},\Psi_{m+1,\tau})=o(g_n)$ , we have for some $q \leq \tau $

\max_l \frac{1}{\sqrt{Mn}} |g_{n}^{[u,v]}(\omega_l^\prime)-g_{n,m}^{[u,v]}(\omega_l^\prime)|=\bp( (\rho^m/x)^{q/2})

$E(\frac{1}{\sqrt{Mn}} |g_{n}^{[u,v]}(\omega_l^\prime)-g_{n,m}^{[u,v]}(\omega_l^\prime)|)^{q/2}=O(d_{m,q}^{q/2})$.

Following the proof of Lemma A.1 in [29], we have by Lemma A.2 in [47] and some $q \leq \tau $

$$E(\frac{1}{\sqrt{Mn}} |g_{n}^{[u,v]}(\omega_l^\prime)-g_{n,m}^{[u,v]}(\omega_l^\prime)|)^{q/2}=O(d_{m,q}^{q/2}).$$

Under Assumption <ref> we have $d_{m,q}=\sum_{t=0}^\infty \min(\delta_{t,q}^{[\max]},(\sum_{j=m}^\infty (\delta_{j,q}^{[max]})^2)^{1/2})\leq C \rho^m$. Hence,  by Markov's inequality

\begin{align*}
P(\max_l \frac{1}{\sqrt{Mn}} |g_{n}^{[u,v]}(\omega_l^\prime)-g_{n,m}^{[u,v]}(\omega_l^\prime)| \geq x)\leq C (\rho^m/x)^{q/2}.
\end{align*}

For $q\leq \tau/2$ it holds true that

\begin{align*}
E( \frac{1}{\sqrt{Mn}}\max_l |\bar g_{n,m}(l)^{[u,v]} & -g_{n,m}(l)^{[u,v]}|)^q  \leq C_\tau m^{2q} \big[(Mn)^{-1+q/\tau}\\
&  +M^{-q/2}(Mn)^{-1+(2q-1)/\tau+(q^2-q)/\tau^2}\big]
\end{align*}

\frac{1}{\sqrt{Mn}}\max_l |\bar g_{n,m}(l)^{[u,v]}-g_{n,m}(l)^{[u,v]}|=\bp(C_\tau m^{3/2\tau-2} ((x \sqrt{Mn})^{-\tau/2} n+m^{\tau/2}x^{-\tau} (Mn)^{-\tau/2+1})).

First note that since the kernel is bounded, $\bar g_{n,m}(l)$ can be written as
$\bar g_{n,m}(l)=\sum_{t=2}^n \bar U_{t,l,m} \sum_{s=1}^{t-1} a_{n,t-s} \bar V_{s,l,m}$ for some bounded coefficients $a_{n,i} \in \C$. A similar expression for  $g_{n,m}(l)$ holds true. We then  obtain
for some constant $C>0$, that,

\begin{align*}
\max_l | \bar g_{n,m}(l)^{[u,v]}-g_{n,m}(l)^{[u,v]}|\leq& C \max_l \sum_{t=2}^n |\bar U_{t,l,m}-U_{t,l,m}| \sum_{s=\max(1,t-M)}^{t-1} |  V_{s,l,m} | \\
C \max_l \sum_{t=2}^n |U_{t,l,m}| \sum_{s=\max(1,t-M)}^{t-1} |V_{s,l,m}-\bar V_{s,l,m}|\\
&+C \max_l \sum_{t=2}^n |\bar V_{t,l,m}-V_{t,l,m}| \sum_{s=\max(1,t-M)}^{t-1} |  U_{s,l,m} | \\
C \max_l \sum_{t=2}^n |V_{t,l,m}| \sum_{s=\max(1,t-M)}^{t-1} |U_{s,l,m}-\bar U_{s,l,m}|.
\end{align*}

The four terms  of the above inequality, can be treated by the same arguments. We focus here on the first only. Note that for all $t$ it holds true that $E |\max_l U_{t,l,m}|^\tau<\infty $, $E |\max_l V_{t,l,m}|^\tau<\infty$. Hence, by Hölder's inequality and Markov's inequality

\begin{align*}
E (\max_l |\bar U_{t,l,m}-U_{t,l,m}|)^q &= E \max_l |U_{t,l,m}|^q \ind(|U_{t,l,m}|> (Mn)^\iota)\\
&\leq  E |U_{t,l,m}|^\tau P(|U_{t,l,m}|\geq (Mn)^\iota)^{1-q/\tau}=O((Mn)^{-1+q/\tau}).
\end{align*}

Using  $m$-dependency and Hölder's inequality, we obtain

\begin{align*}
&(1/(Mn))^{q/2}E \left(\max_l \sum_{t=2}^n |\bar U_{t,l,m}-U_{t,l,m}| \sum_{s=\max(1,t-M)}^{t-1} |  V_{s,l,m} |)\right)^q \\
\leq& C_\tau (1/(Mn))^{q/2} E \left(\max_l \sum_{t=m+1}^n |\bar U_{t,l,m}-U_{t,l,m}| \sum_{s=\max(1,t-M)}^{t-m} |  V_{s,l,m} |\right)^q \\
&+C_\tau E \left( \max_l \sum_{t=2}^n |\bar U_{t,l,m}-U_{t,l,m}| \sum_{s=\max(1,t-m+1)}^{t-1} |  V_{s,l,m}|\right)^q \\
\leq& C_\tau (1/(Mn))^{q/2} \Bigg[(Mn)^{q/2} m^{2(q-1)} (Mn)^{-1+q/\tau}+\\
&+n^{q/2}m^{2q-1}(Mn)^{-1+(\tau/q-1)^{-1}+q/\tau-(q/\tau)/(\tau/q-1)}\Bigg] \\
&\leq C_\tau m^{2q} [(Mn)^{-1+q/\tau}+M^{-q/2}(Mn)^{-1+(\tau/q-1)^{-1}+q/\tau-(q/\tau)/(\tau/q-1)}].
\end{align*}

This proves  the first assertion of the lemma. For the second assertion, we  use the same splitting arguments as above and get

\begin{align*}
\max_l& \sum_{t=2}^n |\bar U_{t,l,m}-U_{t,l,m}| \sum_{s=\max(1,t-M)}^{t-1} |  V_{s,l,m} |\\
=& \sum_{t=m+1}^n \max_l |\bar U_{t,l,m}-U_{t,l,m}| \sum_{s=\max(1,t-M)}^{t-m} |  V_{s,l,m} |
\\
&+ \sum_{t=2}^n \max_l |\bar U_{t,l,m}-U_{t,l,m}| \sum_{s=\max(1,t-m+1)}^{t-1} |  V_{s,l,m}|\\
=&: \sum_{t=m+1}^n W_t+\sum_{t=2}^n \tilde W_t,
\end{align*}

with an obvious notation for $W_t$ and $\widetilde W_t$. We obtain by Lemma 2 in [29] and  for $Q=\tau/2$ and the arguments used above, that

\begin{align*}
P(\sum_{t=m+1}^n W_t \geq x) &\leq C_Q (m/x^2 E (\sum_{t=m+1}^n W_t)^2)^Q+C_Q m^{\tau-1}/x^\tau \sum_{t=m+1}^n E W_t^\tau \\
& \leq C_Q (m/x^2 m^2 (Mn)^{2/\tau})^Q+C_Q m^{\tau-1}x^{-\tau} n M m^{\tau-1}\\
& \leq C_\tau m^{2\tau-2}x^{-\tau} n M.
\end{align*}

Furthermore, we have for $\widetilde W_t$

\begin{align*}
P(\sum_{t=m+1}^n \widetilde W_t \geq x) &\leq C_Q (m/x^2 E (\sum_{t=m+1}^n \widetilde W_t)^2)^Q+C_Q m^{\tau/2-1}/x^{\tau/2} \sum_{t=m+1}^n E \widetilde W_t^{\tau/2} \\
& \leq C_Q (m/x^2 m^2 M^{-1} (Mn)^{(2\tau-1)^{-1}})^Q+C_Q m^{\tau/2-1}x^{-\tau/2} n m^{\tau-1}\\
& \leq C_\tau m^{3/2\tau-2}x^{-\tau/2} n.
\end{align*}

Putting this together with $x=x\sqrt{Mn}$ gives the second assertion.

We have

E (\max_l \sum_{j=1}^{k_n+1} v_j(l)^{[u,v]})^2\leq C n(M)^{1-\beta} m^{4}

E (\max_l \sum_{j=1}^{k_n} u_j(l)^{[u,v]})^2\leq C Mn m^{4}.

First note that by independence
E (\sum_{j=1}^{k_n+1} \max_l v_j(l)^{[u,v]})^2=\sum_{j=1}^{k_n+1} E (\max_l v_j(l)^{[u,v]})^2.

For $j=1,\dots,k_n$ and similarly to the proof of Lemma <ref>,   the properties of the  kernel $K$ and the   $m$-dependence of $\bar U_{t,l,m},V_{t,l,m}$, we have

\begin{align*}
E (\max_l v_j(l)^{[u,v]})^2&\leq C E(\sum_{t\in I_j} \max_l \bar U_{t,l,m} )^2 E (\sum_{s=\max(1,t-M)}^{t-4m} K((t-s)/M) \exp(-i(t-s)\omega_l^\prime)\bar V_{s,l,m})^2\\
&+C_\tau  E( \sum_{t\in I_j} \max_l \bar U_{t,l,m} \sum_{s=t-4m+1}^{t-1} K((t-s)/M) \exp(-i(t-s)\omega_l^\prime) \bar V_{s,l,m})^{2} \\
&\leq C |I_j|M m^{3}+|I_j|m^{2q-1} \leq C_q |I_j| m^{3}(M+m).
\end{align*}

Similar arguments apply to $u_j(l)$ with $I_j$ replaced by $H_j$. We have  $|I_j|=CM+m, j=1,\dots,k_n$, $|I_{k_n+1}|< C(M)^{1+\beta}+M+m, \beta>0$, $|H_j|=C(M)^{1+\beta}$, and $k_n=\floor{n/(|I_1|+|H_1|)}$.

\begin{align*}
\max_l E (\sum_{j=1}^{k_n+1} v_j(l)^{[u,v]})^2&\leq C (n/(1+|H_1|/|I_1|) m^{3}(M+m))
\leq C_q n(M)^{1-\beta} m^{4}
\end{align*}

\begin{align*}
\max_l E (\sum_{j=1}^{k_n+1} u_j(l)^{[u,v]})^2&\leq C Mn m^{4}.
\end{align*}

It holds true  that

\begin{align*}
\sqrt{1/(Mn)} \max_l & |\Gamma^{[u,v]}(l)|\\
& =\bp(C \log(n)^{C_\tau} (x n^{1/2-(a+2)/\tau-2(1+a)/\tau^2})^{-\tau}+x^{-C\tau} (Mn)^{-\tau}),
\end{align*}

where  $\Gamma^{[u,v]}(l)=\sum_{j=1}^{k_n+1} v_j^{[u,v]}(l).$

Note first that

\begin{align*}
P(\max_l \sqrt{1/(Mn)} |\Gamma^{[u,v]}(l)|>x) & \leq \sum_{l} P( |\Gamma^{[u,v]}(l)|> \sqrt{Mn}x)\\
& \leq M P( |\Gamma^{[u,v]}(l)|> \sqrt{Mn}x).
\end{align*}

We focus on the term on the RHS of the last inequality. For this, we use  Lemma 2 in [29] and obtain for any $Q$ and some positive constants $C_Q$

P(|\Gamma(l)^{[u,v]}|\geq \sqrt{Mn} x) \leq C (\frac{\sum_{j=1}^{k_n+1} E |v_j^{[u,v]}(l)|^2}{Mn x})^Q+C \sum_{j=1}^{k_n+1} P(|v_j^{[u,v]}(l)|\geq C_q \sqrt{Mn} x).

For the first term on the RHS we have by Lemma <ref>

$$\frac{\sum_{j=1}^{k_n+1} E |v_j^{[u,v]}(l)|^2}{Mn x }\leq C (n^{-\beta a}m^2 x).$$

Hence, for some $Q=C \tau$ we have that this term is of order $O( x^{-C\tau} (Mn)^{-\tau}).$

Furthermore, by Lemma A.5 in [47], see also Proposition 3 in [29] (note that we do not use the replacement of $m$ by $n$ for the second and third term on the RHS of Proposition 3), we have with  $M=(Mn)^{\iota}, k=M+m$ and  $ x=\sqrt{Mn} x, y=4 \log((nxM)^{\tau})$, that

\begin{align*}
P(|v_j^{[u,v]}(l)|\geq& C_q \sqrt{Mn}x)\leq 2 (Mnx)^{-\tau}\\
&+C nm^2 (Mn)^{2\iota} \Big( (Mn)^{-1} x^{-2} \log((nxM)^{\tau})^{-2} m^3 [(Mn)^{2\iota}+M+m]M\Big)^Q\\
&+C n^2m^2 (Mn)^{2\iota} \max_{Z \in \{U,V\}}P\left(|\bar Z_{0,l,m}|\geq \frac{C  \log((nxM)^{\tau})^{-1} x\sqrt{ Mn }}{m^2 ((Mn)^{\iota}+(M+m))^{1/2}}\right)\\
\end{align*}

Consider  the second term on the RHS and recall that $M=n^{a}$.

\begin{align*}
\frac{m^3 [(Mn)^{2\iota}+M+m]M}{Mn x^{2} \log((Mn)^{\tau})^{2}}\leq x^{-2} \log(n)^{C_\tau} (n^{-1+a}+n^{(1-a)(-1+2/\tau)+a(4/\tau-1)}).
\end{align*}

Hence, for $Q=C \tau$ we obtain $II=O( x^{-C\tau} (Mn)^{-\tau}) $.

For the third term we use Markov's inequality and obtain for $\tau\geq 8$

\begin{align*}
III&\leq C n^2m^2 (Mn)^{2\iota} \max_{Z \in \{U,V\}} E |\bar Z_{0,l,m}|^\tau \left(\frac{m^2 ((Mn)^{\iota}+(M+m))^{1/2}}{ \log((Mn/x)^{\tau}) x\sqrt{Mn}}\right)^\tau \\
& \leq C \log(n)^{C_\tau}  n^{-\tau/2+2+(1+a)2/\tau}x^{-\tau}=C \log(n)^{C_\tau} (x n^{1/2-2/\tau-2(1+a)/\tau^2})^{-\tau}.
% &\leq C \log(n)^{3(\zeta+d_\xi)(1+\tau)} [n^{2+2/\tau +1-\tau/2+a(2/\tau+1-\tau/2+\tau/4-3/2}
% +n^{2+2/\tau-\tau/2+a(2/\tau+\tau/2-\tau/2+\tau/4-3/2)}]\\
% &=\leq C \log(n)^{3(\zeta+d_\xi)(1+\tau)} [n^{-(\tau/2-3-2/\tau)-a(\tau/4+1/2-2/\tau)}
% +n^{-(\tau/2-2-2/\tau)(1-a)-a(\tau/4+5/2-4/\tau)}]\\
% &=o(1).
\end{align*}

Let $\xi$ be defined as in  Remark <ref> of the main paper  and let
$$\hat u_j(l)^{[u,v]}=u_j(l)^{[u,v]} \ind(|u_j(l)^{[u,v]}|\leq \sqrt{Mn} \xi)- E (u_j(l)^{[u,v]} \ind(|u_j(l)^{[u,v]}|\leq \sqrt{Mn} \xi)$$
be a truncated version of $u_j(l)^{[u,v]}$, $j=1,\dots,k_n, l=1,\dots,M/N-1$.
We have

\begin{align*}
\frac{1}{\sqrt{Mn}}\max_l |&  \sum_{j=1}^{k_n} \hat u_j(l)^{[u,v]}- u_j(l)^{[u,v]} |\\
& = \bp \Big( \frac{\log(n)^{C_\tau} }{(x n^{1/2(1-1/4a)-1/\tau(2-a)-2(1+a)/\tau^2})^{\tau}}+x^{-C\tau} \frac{1}{(Mn)^{\tau}}\Big)
\end{align*}

\begin{align*}
\frac{1}{Mn} \max_l  &  \sum_{j=1}^{k_n} E|\hat u_j(l)^{[u,v]}- u_j(l)^{[u,v]}|^2\\
& =O\Big(\frac{\log(n)^{C_\tau}}{ (\xi n^{1/2(1-1/4a)-1/\tau(2-a)-2(1+a)/\tau^2})^{\tau}}+\xi^{-C\tau} \frac{1}{(Mn)^{\tau}}\Big)
\end{align*}

Let $\tilde Z_j(l)=u_j(l)^{[u,v]} \ind(|u_j(l)^{[u,v]}|> \sqrt{Mn} \xi)$. We  follow the lines of proof of Lemma <ref>. The main difference is that $u_j$ consist of $p_n=(M)^{1+\beta}$ elements whereas in Lemma <ref>,  $v_j$ consist of only $M$ elements. We obtain

P(|\sum_{j=1}^{k_n} \tilde Z_j(l) |\geq \sqrt{Mn} x) \leq C \Big(\frac{\sum_{j=1}^{k_n+1} E |\tilde Z_j(l)|^2}{Mn x }\Big)^Q+C \sum_{j=1}^{k_n+1} P(|\tilde Z_j(l)|\geq C_q \sqrt{Mn} x). $$

We have by Cauchy-Schwarz's  inequality, that $E |\tilde Z_j(l)|^2 \leq (E(u_j(l)^{[u,v]})^4 P(|u_j(l)^{[u,v]}|\geq C_q \sqrt{Mn} \xi))^{1/2}$.
Furthermore, we have with similar arguments as in Lemma <ref> $E(u_j(l)^{[u,v]})^4\leq C (|H_j|M)^2 m^8$. This  means that
(\frac{\sum_{j=1}^{k_n+1} E |\tilde Z_j(l)|^2}{Mn x })^Q \leq (P(|u_j(l)^{[u,v]}|\geq C_q \sqrt{Mn} \xi))^{Q/2}.
In the above derivation eight  moments are required. Note that since $Q$ can be chosen large, it is also possible to use Hölder's inequality so that only $4+\delta$ for some $\delta>0$ moments are required.

We  first investigate  $P(|\tilde Z_j(l)|\geq C_q \sqrt{Mn} x)$. Similar to the arguments used for $P(|v_j^{[u,v]}(l)|\geq C_q \sqrt{Mn} x)$ in the proof of Lemma <ref>,  we have by Lemma A.5 in [47], see also Proposition 3 in [29],

with $M=(Mn)^{\iota}, k=(M)^{1+\beta}, x=\sqrt{Mn} x, y=4 \log((nxM)^{\tau})$

\begin{align*}
P(|u_j^{[u,v]}(l)|\geq& C_q \sqrt{Mn}x)\leq 2 (nx/(h))^{-\tau}\\
&+C nm^2 (Mn)^{2\iota} \Big( (Mn)^{-1} x^{-2} \log((nxM)^{\tau})^{-2} m^3 [(Mn)^{2\iota}+(M)^{1+\beta}]M\Big)^Q\\
&+C n^2m^2 (Mn)^{2\iota} \max_{Z \in \{U,V\}}P\left(|\bar Z_{0,l,m}|\geq \frac{C  \log((nxM)^{\tau})^{-1} x\sqrt{ Mn }}{m^2 ((Mn)^{\iota}+(M)^{1+\beta})^{1/2}}\right)\\
\end{align*}

Consider  the second term on the RHS above. Recall  that $M=n^{a}$ and $a\leq 2/3$.

\begin{align*}
(Mn)^{-1}& x^{-2} \log((Mn)^{\tau})^{-2} m^3 [(Mn)^{2\iota}+(M)^{1+\beta}]M\\
&\leq x^{-2} \log(n)^{C_\tau} (n^{-1+a(1+\beta)}+n^{(1-a)(-1+2/\tau)+a(4/\tau-1)}).
\end{align*}

Note that $n^{-1+a(1+\beta)}=O(k_n)=n^{8/(5\tau)}$. Hence, for $Q=C_\tau$ we obtain $II=O( x^{-C_\tau} (Mn)^{-\tau})$.

For the third term we use Markov's inequality and obtain for $\tau\geq 8$

\begin{align*}
III&\leq C n^2m^2 (Mn)^{2\iota} \max_{Z \in \{U,V\}} E |\bar Z_{0,l,m}|^\tau \left(\frac{m^2 ((Mn)^{\iota}+(M)^{1+\beta})^{1/2}}{ \log((n/(hx))^{\tau}) x\sqrt{Mn}}\right)^\tau \\
& \leq C \log(n)^{C_\tau}  n^{-\tau/2+\beta a/2 \tau+2+(1+a)2/\tau}x^{-\tau}=C \log(n)^{C_\tau} (x n^{1/2(1-1/4a)-1/\tau(2-a)-2(1+a)/\tau^2})^{-\tau}
% &\leq C \log(n)^{3(\zeta+d_\xi)(1+\tau)} [n^{2+2/\tau +1-\tau/2+a(2/\tau+1-\tau/2+\tau/4-3/2}
% +n^{2+2/\tau-\tau/2+a(2/\tau+\tau/2-\tau/2+\tau/4-3/2)}]\\
% &=\leq C \log(n)^{3(\zeta+d_\xi)(1+\tau)} [n^{-(\tau/2-3-2/\tau)-a(\tau/4+1/2-2/\tau)}
% +n^{-(\tau/2-2-2/\tau)(1-a)-a(\tau/4+5/2-4/\tau)}]\\
% &=o(1).
\end{align*}

Since $1/\xi=\log(n)^{C_\tau}$,  $Q$ can chosen as $Q= C_\tau$ such that $C (\frac{\sum_{j=1}^{k_n+1} E |\tilde Z_j(l)|^2}{Mn x })^Q=O( x^{-C\tau} (Mn)^{-\tau})$.

§.§ Covariance structure of lag-window estimators

Let $\fspc(\omega)=\frac{M}{n}  \sum_{k=1}^{n} \kappa_M(\omega-\omega_k)  (Z(\omega_k)Z^{H}(\omega_k)-\Sigma_n(\omega_k))$ and $u,v \in \{1,\dots,p\}$. Under Assumption <ref>,<ref>,<ref>, we have

\sup_\omega |n/M\var(e_{v}^\top \fspc(\omega) e_{u})-C_{K_2}(f_{v,v}(\omega)f_{u,u}(\omega)+\ind(\omega \in \pi \Z)f_{v,u}(\omega)^2)|=O(1/M).

We express the occurring fourth order moments in terms of covariances and cumulants,
see among others Section 5.1 in [35]. To elaborate,  we have

\begin{align*}
n/M\var(e_{v}^\top \fsp(\omega) e_{u})=I_1+I_2+I_3,
\end{align*}

\begin{eqnarray*}
I_{1}&=&  \frac{M}{n} \sum_{k_1,k_2=1}^n \kappa_M(\omega-\omega_{k_1}) \kappa_M(\omega-\omega_{k_2})
\cov(e_v^\top Z_n(\omega_{k_1}), e_v^\top Z_n(\omega_{k_2})) \cov(Z_n(\omega_{k_1})^H e_u, Z_n(\omega_{k_2})^H e_u)\\
I_{2}  &=&\frac{M}{n} \sum_{k_1,k_2=1}^n \kappa_M(\omega-\omega_{k_1}) \kappa_M(\omega-\omega_{k_2})
\cov(e_v^\top Z_n(\omega_{k_1}),Z_n(\omega_{k_2})^H e_u)   \cov(Z_n(\omega_{k_1})^H e_u, e_v^\top Z_n(\omega_{k_2}) ) \\
I_{3} &=& \frac{M}{n} \sum_{k_1,k_2=1}^n \kappa_M(\omega-\omega_{k_1}) \kappa_M(\omega-\omega_{k_2}) \cum(
e_{u}^\top Z_n(\omega_{k_1}),
e_{v}^\top Z_n(\omega_{k_1}),e_{u}^\top Z_n(\omega_{k_2}),e_{v}^\top Z_n(\omega_{k_2})).
\end{eqnarray*}

By Theorem 4.1 in [39] and the summability of fourth order cumulants, i.e.,
\mathbb{Z}}|\cum( e_{a_1}^\top X_{0},e_{a_2}^\top X_{\tau_1},e_{a_3}^\top X_{\tau_2}, e_{a_4}^\top X_{\tau_3})|<\infty.$$
This implies uniformly for all $\omega_{1},\dots,\omega_{4} \in [0,2\pi]$

$$\max_{a_1,\dots,a_4}\left|\cum[e_{a_1}^\top Z_n(\omega_1),
e_{a_2}^\top Z_n(\omega_2),e_{a_3}^\top Z_n(\omega_3),e_{a_4}^\top Z_n(\omega_4)] \right|  = O(n^{-1}),$$

due to  Assumption <ref>,
we have that $I_3=O(1/M)$. Following the arguments of [35],  Section 5.1, see also Section 5.4 in [6], we obtain uniformly for all $\omega \in [0,2\pi]$

\begin{align*}
I_{1}=&\frac{M}{n} \sum_{k_1,k_2=1}^n \kappa_M(\omega-\omega_{k_1}) \kappa_M(\omega-\omega_{k_2}) \frac{1}{(2 \pi n)^2} \sum_{t_1,t_2=1}^n \Gamma_{v,v}(t_1-t_2) \exp( i(t_1 \omega_{k_1} -t_2 \omega_{k_2})\\
&\times \sum_{\tau_1,\tau_2=1}^n \Gamma_{u,u}(\tau_1-\tau_2) \exp( -i(\tau_1 \omega_{k_1} -\tau_2 \omega_{k_2}))\\
=&\frac{M}{n} \sum_{k_1}^n \kappa_M(\omega-\omega_{k_1})^2  (f_{u,u}(\omega_{k_1})+O(1/n))(f_{u,u}(\omega_{k_1})+O(1/n))\\
% =&\frac{M}{n} \sum_{k_1,k_2=1}^n \kappa_M(\omega-\omega_{k_1}) \kappa_M(\omega-\omega_{k_2})
% \frac{1}{2\pi} \sum_{h=-n+1}^{n-1} (1-|h|/n) \Gamma_{v,v}(h) \exp(i \omega_1 h)\ind(\omega_1-\omega_2) \\
% &\times \frac{1}{2\pi} \sum_{h=-n+1}^{n-1} (1-|h|/n) \Gamma_{u,u}(h) \exp(-i \omega_1 h)\ind(\omega_1-\omega_2) \\
% =&\frac{M}{n} \sum_{k_1=1}^n \kappa_M(\omega-\omega_{k_1})^2
% (f_{v,v}(-\omega_{k_1})+O(1/n))(f_{u,u}(\omega_{k_1})+O(1/n)) \\
%=& \frac{M}{n} \sum_{k_1,k_2=1}^n \kappa_M(\omega-\omega_{k_1}) \kappa_M(\omega-\omega_{k_2}) (f_{u,u}(\omega_{k_1}) f_{v,v}(\omega_{k_1})+O(1/n))\\
=& C_{K_2}f_{u,u}(\omega)f_{v,v}(\omega)+O(1/M).
\end{align*}

\begin{align*}
I_{2}=&\frac{M}{n} \sum_{k_1,k_2=1}^n \kappa_M(\omega-\omega_{k_1}) \kappa_M(\omega-\omega_{k_2}) \frac{1}{(2 \pi n)^2} \sum_{t_1,t_2=1}^n \Gamma_{v,u}(t_1-t_2) \exp( i(t_1 \omega_{k_1} +t_2 \omega_{k_2})\\
&\times \sum_{\tau_1,\tau_2=1}^n \Gamma_{u,v}(\tau_1-\tau_2) \exp( i(\tau_1 \omega_{k_1} +\tau_2 \omega_{k_2}))\\
%=& \frac{M}{n} \sum_{k_1,k_2=1}^n \kappa_M(\omega-\omega_{k_1}) \kappa_M(\omega-\omega_{k_2}) (\ind(\omega \in \pi \Z)f_{v,u}(\omega_{k_1}) f_{v,u}(\omega_{k_1})+O(1/n))\\
=& \ind(\omega \in \pi \Z)C_{K_2}f_{v,u}(\omega)f_{v,u}(\omega)+O(1/M).
\end{align*}

Under Assumption  <ref> to <ref> the following assertions hold true. If $K$ is the uniform kernel, then

\frac{1}{Mn} \max_{l_1,l_2=1,\dots,M,l_1\not=l_2} | \cov(\sum_{j=1}^{k_n} \hat u_j(l_1)^{[u,v]},
\sum_{j=1}^{k_n} \hat u_j(l_2)^{[u,v]})| =O(1/M+1/k_n).

For other kernels, we have

\frac{1}{Mn} | \cov(\sum_{j=1}^{k_n} \hat u_j(l_1)^{[u,v]},
\sum_{j=1}^{k_n} \hat u_j(l_2)^{[u,v]})| =O(1/M+1/k_n+(M(\omega_{l_1}^\prime+\omega_{l_2}^\prime))^{-r}+(M(\omega_{l_1}^\prime-\omega_{l_2}^\prime))^{-r}).

First note that  by independence
\cov(\sum_{j=1}^{k_n} \hat u_j(l_1)^{[u,v]},
\sum_{j=1}^{k_n} \hat u_j(l_2)^{[u,v]})= \sum_{j=1}^{k_n}\cov( \hat u_j(l_1)^{[u,v]}, \hat u_j(l_2)^{[u,v]}).

\begin{align*}
\sum_{j=1}^{k_n}&\cov( \hat u_j(l_1)^{[u,v]}, \hat u_j(l_2)^{[u,v]})=\\
&\sum_{j=1}^{k_n} [\cov( u_j(l_1)^{[u,v]}, u_j(l_2)^{[u,v]})
-\cov(u_j(l_1)^{[u,v]},\hat u_j(l_2)^{[u,v]}- u_j(l_2)^{[u,v]}) \\
&-\cov(\hat u_j(l_1)^{[u,v]}- u_j(l_1)^{[u,v]},u_j(l_2)^{[u,v]})
+\cov(\hat u_j(l_1)^{[u,v]}- u_j(l_1)^{[u,v]},\hat u_j(l_2)^{[u,v]}- u_j(l_2)^{[u,v]})\\
=& I+II+III+IV
\end{align*}

For $II,III$ and $IV$ note that by Lemma <ref> and $\tau \geq 8$,
Mn^{-1} \max_l \sum_{j=1}^{k_n} \var(\hat u_j(l)^{[u,v]}- u_j(l)^{[u,v]})=O(M^{-2}).
Hence, we obtain by Cauchy-Schwarz's and Hölder's inequality, that

\begin{align*}
II&=\frac{1}{Mn} \sum_{j=1}^{k_n} \cov(\hat u_j(l_1)^{[u,v]}- u_j(l_1)^{[u,v]},u_j(l_2)^{[u,v]}) \\
&\leq (\sum_{j=1}^{k_n} \frac{1}{Mn} \var(\hat u_j(l_1)^{[u,v]}- u_j(l_1)^{[u,v]}))^{1/2}
(\sum_{j=1}^{k_n} \frac{1}{Mn} \var (u_j(l_2)^{[u,v]}))^{1/2}=O(M^{-1}).
\end{align*}

Similar arguments apply to $III$ and $IV$.

For $I$ note that

\begin{align*}
u_j(l)^{[u,v]}=& \sum_{t \in H_j} (\sum_{s=1}^{t-1} K((t-s)/M)\exp(-i (t-s) \omega_l^\prime) [\bar U_{t,l,m}\bar V_{s,l,m}- E U_{t,l,m}\bar V_{s,l,m}]\\
&+\sum_{s=1}^{t-1} K((t-s)/M)\exp(i (t-s) \omega_l^\prime) [\bar V_{t,l,m}\bar U_{s,l,m}-E\bar V_{t,l,m}\bar U_{s,l,m}])
\end{align*}

Evaluating  $\cov( u_j(l_1)^{[u,v]}, u_j(l_2)^{[u,v]})$ using the the above decomposition leads to  four terms, i.e.,

\begin{align*}
&\cov( u_j(l_1)^{[u,v]}, u_j(l_2)^{[u,v]})=\\
&\sum_{t\in \Z} \sum_{s_1,s_2=-M}^M K(\frac{s_1}{M}) K(\frac{s_2}{M}) \cov(\bar U_{0,l_1,m}\bar V_{-s_1,l_1,m}, \bar U_{t,l_2,m}\bar V_{t-s_2,l_2,m})\exp(-i s_1 \omega_{l_1}^\prime+s_2 \omega_{l_2}^\prime) +O(\frac{1}{k_n})\\
&=\sum_{t\in \Z} \sum_{s_1,s_2=-M}^M K(\frac{s_1}{M}) K(\frac{s_2}{M}) \exp(-i s_1 \omega_{l_1}^\prime+s_2 \omega_{l_2}^\prime)\big[\cov(\bar U_{0,l_1,m},\bar U_{t,l_2,m}) \cov(\bar V_{-s_1,l_1,m},\bar V_{t-s_2,l_2,m})\\
&+\cov(\bar U_{0,l_1,m},\bar V_{t-s_2,l_2,m})\cov(\bar V_{-s_1,l_1,m}, \bar U_{t,l_2,m})
+\cum(\bar U_{0,l_1,m},\bar V_{-s_1,l_1,m}, \bar U_{t,l_2,m},\bar V_{t-s_2,l_2,m})\big] +O(\frac{1}{k_n}).
\end{align*}

By the  summability of the  fourth order cumulants, see also Theorem 4.1 in [39], we have that the cumulant term is of order $O(1/M)$.
Furthermore, we have with the decay conditions of the autocovariance, smoothness of the kernel and the decay condition of the Fourier coefficients of the kernel

\begin{align*}
I.1=& f_{\bar U_{l_1,m},\bar U_{l_2,m}}(\omega_{l_1}^\prime)f_{\bar V_{l_1,m},\bar V_{l_2,m}}(\omega_{l_1}^\prime) \frac{1}{M} \sum_{s=-M}^M K^2(s/M) \exp(-is (\omega_{l_1}^\prime-\omega_{l_2}^\prime)+O(1/M)\\
& \leq C (M(\omega_{l_1}^\prime-\omega_{l_2}^\prime))^{-r}+O(1/M)
\end{align*}

\begin{align*}
I.2=& f_{\bar U_{l_1,m},\bar V_{l_2,m}}(\omega_{l_1}^\prime)f_{\bar V_{l_1,m},\bar U_{l_2,m}}(\omega_{l_1}^\prime) \frac{1}{M} \sum_{s=-M}^M K^2(s/M) \exp(-is (\omega_{l_1}^\prime+\omega_{l_2}^\prime)+O(1/M)\\
& \leq C (M(\omega_{l_1}^\prime+\omega_{l_2}^\prime))^{-r}+O(1/M)
\end{align*}

For $K$ being the uniform kernel and $\omega_l^\prime=\pi l/M$, we have
$\frac{1}{M} \sum_{s=-M}^M K^2(s/M) \exp(-is (\omega_{l_1}^\prime-\omega_{l_2}^\prime)=\ind(\omega_{l_1}^\prime=\omega_{l_2}^\prime)+O(1/M)$ and
$\frac{1}{M} \sum_{s=-M}^M K^2(s/M) \exp(-is (\omega_{l_1}^\prime+\omega_{l_2}^\prime)=\ind(\omega_{l_1}^\prime+\omega_{l_2}^\prime\in 2M \Z)+O(1/M)$.

§.§ Additional  Lemmas

Let $Z_l, l=1,\dots,d,d \geq 1$ be independent Gaussian random vectors and $Z_l \sim \mathcal{N}(0,I_2)$.
Furthermore, let $\gamma_n,g_n,h_n^{-1}>0$, $\gamma_n \leq 2 \log(d)$ and $W_n$ be a random variable with $P(|W_n|>\gamma_n)=g_n$.
Set $G_d(t)=(1-(1-\exp(-\max(0,t)/2))^d$. Then,

\begin{align}
\sup_{0\leq t \leq h_n} \left| \frac{P(\max_l \|Z_l\|^2+W_n \geq t)}{G_d(t)}-1\right|\leq(g_n \max(2,\exp(h_n/2)/d)+|\gamma_n| \max(2,\exp(|\gamma_n|/2)).
\label{eq.bound.gaussian.gt}
\end{align}

We compute derivatives under the condition $d \geq 2$. The same arguments with simpler derivatives can also be applied in the case $d=1$.
We have $P(\max_l \|Z_l\|^2+W_n\geq t)\leq P(\max_l \|Z_l\|^2 \geq t -\gamma_n)+P(|W_n|>\gamma_n)$ and by Lemma <ref>,  $P(\max_l \|Z_l\|^2+W_n\geq t)\geq P(\max_l \|Z_l\|^2 \geq t +\gamma_n)-P(|W_n|>\gamma_n)$. Thus,

\begin{align*}
\left| \frac{P(\max_l \|Z_l\|^2+W_n \geq t)}{G_d(t)}-1\right|\leq g_n/G_d(t)+G_d(t)^{-1} | G_d(t \pm \gamma_n)-G_d(t)|.
\end{align*}

Note that $(G_d(t))^{-1}$ is monoton increasing in $t$, where $G_d(0)^{-1}=1$. Additionally, we have with

\begin{align}
(1+x)^d\leq 1/(1-dx) \text{ for }x\in [-1,0]    \label{eq.inequality1}
\end{align}

\begin{align}
G_d(t)^{-1}\leq \exp(t/2)/d+1.   \label{eq.bound.Gt}
\end{align}

We consider two cases $t\leq 2 \log(d)$ and $ t > 2\log(d)$. In the first case, we further have
$G_d(t)^{-1} \leq 2$ and in the second, $G_d(t)^{-1} \leq 2 \exp(t/2)/d$. With this and using the monotonicity of $(G_d(t))^{-1}$, we get  the first summand of the  bound given in (<ref>), that is,  $g_n \max(2,\exp(h_n/2)/d)$. Regarding  the second summand of the same bound, note first that we have a positive and a negative case. Only in the case $t \leq \gamma_n$ the two cases need to be treated separately. We begin with this case and consider $t \leq \gamma_n$. Then, we  have $G_d(t)^{-1} | G_d(t - \gamma_n)-G_d(t)|\leq 2(G_d(0)-G_d(\gamma_n))$. Further, we have by the mean value theorem for some $\eps \in [0,\gamma_n]$ and the inequality (<ref>)

\begin{align*}
2(G_d(0)-G_d(\gamma_n)) &\leq |\gamma_n|  d (1-\exp(-\eps/2))^{d-1} \exp(-\eps/2)    \\
&\leq |\gamma_n| \frac{1}{\exp(\eps/2)/d+(d-1)/d} \leq \gamma_n d/(d-1) \leq 2 \gamma_n.
\end{align*}

With similar arguments, we obtain for the case $\gamma_n\leq t \leq 2\log(d)$ that for  $\eps \in [0,\gamma_n]$ (not necessarily the same $\eps$ for the positive and negative  case)

\begin{align*}
G_d(t)^{-1} | G_d(t \pm \gamma_n)-G_d(t)| &\leq  2 |\gamma_n| 2 d (1-\exp(-(t \pm \eps)/2))^{d-1} \exp(-(t \pm \eps)/2) \\
& \leq |\gamma_n| \frac{1}{\exp( (t \pm \eps)/2)/d +(d-1)/d} \leq |\gamma_n| d/(d-1) \leq 2 \gamma_n.
\end{align*}

The same arguments applied to the case $t \leq \gamma_n$ and we obtain  $G_d(t)^{-1} | G_d(t + \gamma_n)-G_d(t)$.
Now, let $t > 2 \log(d)$. Then,  by the mean value theorem we have for some $\eps \in [0,\gamma_n]$ (not necessarily the same $\eps$ for the positive  and negative  case)

\begin{align*}
G_d(t)^{-1} | G_d(t \pm \gamma_n)-G_d(t)| & \leq  |\gamma_n| \exp(t/2) (1-\exp(-(t \pm \eps)/2))^{d-1} \exp(-(t \pm \eps)/2) \\
&\leq |\gamma_n| \exp(|\gamma_n|/2)
\end{align*}

G_d(t)^{-1} | G_d(t \pm \gamma_n)-G_d(t)| \leq |\gamma_n| \max(2,\exp(|\gamma_n|/2)),

where  this bound is independent of $t$ and holds for all $t\geq 0$.

Let $A\in \R^{p\times p}$ be a  positive definite matrix and $v \in \{1,\dots,p\}$. Then, we have for the inverse of the sub-matrix $A_{-v,-v}=I_{p;-v}^\top A I_{p;-v}$ the following

\begin{align} \label{eq.lem.inv}
(I_{p;-v}^\top A I_{p;-v})^{-1}=I_{p;-v}^\top A^{-1} I_{p;-v}-\frac{I_{p;-v}^\top A^{-1} e_v e_v^\top A^{-1} I_{p;-v}}{e_v^\top A^{-1} e_v}
\end{align}

Note that $I_p=I_{p;-v}^\top I_{p;-v}+e_v e_v^\top$ and $I_{p;-v} e_v=0$. (<ref>) is equivalent to

\begin{align*}
I_{p;-v}^\top I_{p;-v} (e_v^\top A^{-1} e_v)=& I_{p;-v}^\top A I_{p;-v} I_{p;-v}^\top A^{-1} I_{p;-v} e_v^\top A^{-1} e_v -I_{p;-v}^\top A I_{p;-v} I_{p;-v}^\top A^{-1} e_v e_v^\top A^{-1} I_{p;-v} \\ & \pm I_{p;-v}^\top A e_v e_v^\top A^{-1} I_{p;-v} e_v^\top
A^{-1} e_v\\
=&I_{p;-v}^\top I_{p;-v} e_v^\top A^{-1} e_v+ I_{p;-v}^\top A e_v e_v^\top A^{-1} e_v e_v^\top A^{-1} I_{p;-v}-I_{p;-v}^\top A e_V e_v^\top A^{-1} I_{p;-v} e_v^\top A^{-1} e_v \\
=&I_{p;-v}^\top I_{p;-v} e_v^\top A^{-1} e_v.
\end{align*}

For random variables $X,Y$ and $t \in \R, \eps>0$, we have

P(X+Y \geq t) \geq P(X \geq t +\eps)-P(|Y| > \eps)

\begin{align*}
P(X \geq t +\eps) &= P(X \geq t +\eps,|Y| >\eps) +P(X \geq t +\eps,|Y| \leq \eps) \\
\leq P(|Y| >\eps)+P(X \geq t - Y)
\end{align*}

§ AUXILIARY LEMMAS

In this section, we present some auxiliary lemmas which are used  to prove  the   main results of this paper. Lemma <ref> is the key lemma used to express the de-biased partial coherence as a
quadratic form and  uses results   given in Subsection <ref> of the supplementary file. The expression as  quadratic form is the starting point to obtain a Gaussian approximation, where the steps  involved are described in more detail in Remark <ref> bellow. The corresponding lemmas
can be found  in Subsection <ref> of the supplementary file. Lemma <ref>  deals with  the uniform convergence of the maximum of the de-biased partial coherence over a growing grid of frequencies. To establish this  result,   convergences rates of  the covariance structure of the lag-window estimator and of the quadratic forms involved are required. The  results used in this context are stated as auxiliary lemmas in Subsection <ref> of the supplementary file. Lemma <ref> is the key ingredient to prove Theorem <ref>, Theorem <ref> and Theorem <ref>.

To proceed, we fix some additional notation  used in the proofs.  $\{Z_n\}$ denotes  a sequence of random variables and $\{f_n(\cdot)\}$ a sequence of functions. If $P(|Z_n|\geq x) \leq C f_n(x)$, for some constant $C>0$ and for all $n$, then the notation  $Z_n=\bp(f_n(x))$ is used. Note that for $A_n=\bp(f_n(x)), B_n=\bp(g_n(x))$, we have $A_n+B_n=\bp((f_n(x/2)+g_n(x/2)))$ since $P(|A_n+B_n|\geq x) \leq P(|A_n|\geq x/2)+P(|B_n|\geq x/2)$. Similarly, $A_n\cdot B_n=\bp((f_n(\sqrt{x})+ g_n(\sqrt{x})))$ and  for $Z_n=\bp(f_n(x))$ and  $x_n$ a sequence, we have

Let $\omega_l^\prime=\pi l N/M, l=1,\dots,M/N-1$. Under Assumption <ref>, <ref>, <ref>, and <ref>, we have

\begin{align*}
\sqrt{n/M}\sup_l&\left|(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime))-
\Big[ \frac{1}{n}T_{n,u,v}(l)-\frac{1}{2n}\rho_{u,v}(\omega_l^\prime)\big[ T_{n,v,v}(l)+ T_{n,u,u}(l)\big]\Big]\right|\\
&=\bp(\gpbetade x),
\end{align*}

$$\gpbetade x=(\gps {((x-\gpbias)/(\spar (n/M)^{1/4}))^{1/2}}+ \gps 1),$$

\begin{equation}
\label{eq.Tn-app}
T_{n,u,v}(l)=\frac{1}{2\pi} \sum_{u=-n+1}^{n-1} K(u/M) \sum_{t=\max(1,1-u)}^{\min(n,n-u)} [U_{t+u,l} V_{t,l}-E U_{t+u,l} V_{t,l}]\exp(-i u \omega_l^\prime),
\end{equation}

and, for $l=1,\dots,M/N-1$,

$$U_{t,l}=e_u^\top f^{-1}(\omega_l^\prime) X_t(f_{u,u}^{-1}(\omega_l^\prime))^{-1/2}
\ \ \mbox{and}
\ \ V_{t,l}=e_v^\top f^{-1}(\omega_l^\prime) X_t(f_{v,v}^{-1}(\omega_l^\prime))^{-1/2}.$$

Under Assumption <ref>,
the processes $\{U_{t,l}\}$ and $ \{V_{t,l}\}$ possesses  the same functional dependence as the process  $\{X_t\}$.

Let $g(x_1,x_2,x_3,x_4,x_5,x_6)=1/2((x_1+i x_2)\sqrt{x_5/x_6}+(x_3-i x_4)\sqrt{x_6/x_5})$. Then,

$$\hat \rho^{(de)}_{u,v}(\omega)=g(Re(\hat \beta_{u,\tilde v}^{(de)}(\omega)),Im(\hat \beta_{u,\tilde v}^{(de)}(\omega)),Re(\hat \beta_{v,\tilde u}^{(de)}(\omega)),Im(\hat \beta_{v,\tilde u}^{(de)}(\omega)),\hat f_{v,v}^{-1}(\omega),\hat f_{u,u}^{-1}(\omega)).$$

Using a Taylor expansion of $\hat \rho^{(de)}_{u,v}(\omega)$ around $\rho_{u,v}(\omega)$, we get  that the terms given by the derivatives of $x_5$ and $x_6$ are of order $O(1/n)$ due to $\beta_{v,\tilde u}(\omega)\sqrt{\Sigma_{n,v,v}^{-1}(\omega) \Sigma_{n,u,u}^{-1}(\omega)}=
(\beta_{u,\tilde v}(\omega))^{(C)}\sqrt{\Sigma_{n,u,u}^{-1}(\omega)/\Sigma_{n,v,v}^{-1}(\omega)}\Sigma_{n,u,u}^{-1}(\omega)$ and $\sup_\omega \|\Sigma_n(\omega)-f(\omega)\|_{\max}=O(1/n)$.  Therefore,

\begin{align}
\hat \rho^{(de)}_{u,v}(\omega)-\rho_{u,v}(\omega)=&\frac{1}{2}\Big[
(\hat \beta_{u,\tilde v}^{(de)}(\omega)-\beta_{u,\tilde v}(\omega))\sqrt{\frac{f_{u,u}^{-1}(\omega)}{f_{v,v}^{-1}(\omega)}}+(\hat \beta_{v,\tilde u}^{(de)}(\omega)-\beta_{v,\tilde u}(\omega))^{(C)}\sqrt{\frac{f_{v,v}^{-1}(\omega)}{f_{u,u}^{-1}(\omega)}}\Big]
\nonumber
\\&+
Error_1(\omega), \label{eq.proof.rho}
\end{align}

where $\sup_\omega Error_1(\omega)=O(\sup_\omega \|\fsp(\omega)- E\fsp(\omega)\|_{\max}^2)=\bp(\gps{\sqrt{x}})$. Furthermore, we have by Lemma <ref> for $u,v=1,\dots,p$,

\begin{align*}
\sqrt{n/M}&(\hat \beta_{v,\tilde u}^{(de)}(\omega)-\beta_{v,\tilde v}(\omega))=\delta(\omega)+\\
&\sum_{k=1}^n \frac{\kappa_M(\omega-\omega_k)}{\sqrt{nM }(f_{v,v}^{-1}(\omega))^2} e_v^\top f^{-1}(\omega) [Z(\omega_k) Z^H(\omega_k)-f(\omega)]f^{-1}(\omega)[e_u f_{v,v}^{-1}(\omega)-e_v f_{v,u}^{-1}(\omega)]\\
\frac{\sqrt{n/M}}{(f_{v,v}^{-1}(\omega))^2} e_v^\top f^{-1}(\omega) [\fsp(\omega)-f(\omega)]f^{-1}(\omega)[e_u f_{v,v}^{-1}(\omega)-e_v f_{v,u}^{-1}(\omega)],
\end{align*}

where $\sup_w |\delta(\omega)|=\bp(\gpbetade x)$. Define $\hat h_{u,v}(\omega):=e_v^\top f^{-1}(\omega) [\flw(\omega)-f(\omega)]f^{-1}(\omega)e_u.$ Inserting this into (<ref>) we  obtain

\begin{align*}
\sqrt{n/M}(\hat \rho^{(de)}_{u,v}(\omega)-\rho_{u,v}(\omega))=&
\sqrt{n/M}\Big[ \frac{\hat h_{u,v}(\omega)}{\sqrt{f_{v,v}^{-1}(\omega)f_{u,u}^{-1}(\omega)}}+\frac{\rho_{u,v}(\omega)}{2}
[\frac{\hat h_{u,u}(\omega)}{f_{u,u}^{-1}(\omega)}+\frac{\hat h_{v,v}(\omega)}{f_{v,v}^{-1}(\omega)}\big]\Big]
\\&+
\sqrt{n/M} Error_1(\omega)\\
&+\frac{\delta(\omega)}{2}\Big(\sqrt{\frac{f_{v,v}^{-1}(\omega)}{f_{u,u}^{-1}(\omega)}}-\sqrt{\frac{f_{u,u}^{-1}(\omega)}{f_{v,v}^{-1}(\omega)}} \Big).
\end{align*}

We have
$$\sup_\omega \sqrt{n/M} Error_1(\omega)=\bp(\gps{\sqrt{x/(n/M)^{1/4}}})=\bp(\gpbetade x).$$

Now, we take a closer look at $\hat h_{u,v}(\omega_l^\prime)$. For $l=1,\dots,M/N-1$ we define  $U_{t,l}=e_u^\top f^{-1}(\omega_l^\prime) X_t/(f_{u,u}^{-1}(\omega_l^\prime) )^{1/2}$ and $V_{t,l}=e_v^\top f^{-1}(\omega_l^\prime) X_t/(f_{v,v}^{-1}(\omega_l^\prime) )^{1/2}$.
Since by Assumption <ref> $\sup_\omega \|f^{-1}\|_2$ is bounded and $\max_{\|v\|_2=1} E | v^\top X|^\tau<\infty$, we have that $E |U_{t,l}|^\tau<\infty$ and $E|V_{t,l}|^\tau<\infty$ and that $\{U_{t,l}\}$ and $ \{V_{t,l}\}$ posses the same functional dependence as $\{X_t\}$. Furthermore, recall the definition of $ T_{n,u,v}(l)$ in (<ref>).

Then,  by (<ref>) we have
$\max_l \sqrt{n/M}|\hat h_{u,v}(\omega_l^\prime)-1/n T_{n,u,v}(l)|=o(\gpbias)$. Thus

\begin{align*}
\sqrt{n/M}\sup_\omega\Big|&(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime))-
\Big[1/n T_{n,u,v}(l)-1/2 \rho_{u,v}(\omega_l^\prime)[1/n T_{n,v,v}(l)+1/n T_{n,u,u}(l)]\Big]\Big|\\
&=\bp(\gpbetade x).
\end{align*}

The strategy  used to  proof the results presented in Lemma <ref>,
is to first approximate the random variables $U_{t,l} $ and $ V_{t,l} $ defined in Lemma <ref>,  by $m$-dependent and bounded random variables. The corresponding sums are then split into chunks of big and small blocks so that the big blocks are independent from each other. For this, we follow ideas used in  [29].
We  can  approximate $ \{X_t\}$ by the  $m$-dependent process $\{X_{t,m}=E (X_t | \eps_{t-m}, \dots, \eps_t)\}$
and denote by  $d_{m,\tau}$ the approximation error. By  the assumptions  made, we have  $d_{m,\tau}=\sum_{t=0}^\infty \min (\delta_{t,\tau},\Psi_{m+1,\tau})=O(\rho^m)$ for some $0<\rho<1$;  see Lemma <ref>. Consequently, for  $\eps>0$ small and $d_\xi>0$,  we set  $m=\big[\log((d_\xi+\eps)\log(n))/(-\log(\rho))\big]$, which implies  that $d_{m,\tau}=o(\log(n)^{-d_\xi})$.

Replacing  $X_t$ by  $X_{t,m}$ in the construction of $U_{t,l} $ and $V_{t,l} $ used in   Lemma <ref>,
leads to the random sequences  $\{U_{t,l,m}\}$,   $\{V_{t,l,m}\}$ and $T_{n,u,v,m}(l)$. We further define

$$g_n^{[u,v]}(\omega_l^\prime)=T_{n,u,v}(l)-E(T_{n,u,v}(l))- \sum_{t=1}^n \big(U_{t,l} V_{t,l}-E U_{t,l}V_{t,l}\big)$$
$$g_{n,m}^{[u,v]}(\omega_l^\prime)=T_{n,u,v,m}(l)-E(T_{n,u,v,m}(l))- \sum_{t=1}^n\big( U_{t,l,m} V_{t,l,m}-E U_{t,l,m}V_{t,l,m})\big.$$

The next step in our proof is to truncate $ U_{t,l,m}$ and $ V_{t,l,m}$ in order to obtain bounded random variables. For this  we consider

$$U_{t,l,m}^\prime=U_{t,l,m} \ind(|U_{t,l,m}| \leq (Mn)^\iota) \text{ and } V_{t,l,m}^\prime=V_{t,l,m} \ind(|V_{t,l,m}| \leq (Mn)^\iota),$$

where $\iota=1/\tau$. We then focus on  the centered random variables
$$\bar{U}_{t,l,m}=U_{t,l,m}^\prime-E U_{t,l,m}^\prime \text{ and }\bar{V}_{t,l,m}=V_{t,l,m}^\prime-E V_{t,l,m}^\prime.$$

For the construction of the big and small blocks,  let $\rho_n=\floor{(M)^{1+\beta}}, q_n=M+m,$ and $k_n=\floor{n/(p_n+q_n)}$. Note that $k_n=O(n^{1-a(1+\beta)})$ and recall that $M=n^a$. Since $k_n=n^{\eps}$ for some $\eps>0$ is needed, we set $\beta=1/4-2\iota$ and $a\leq 4/7$.

To proceed we  split the interval $[1,n]$ into alternating big and small blocks  $H_j$ and $I_j$ of lengths $p_n$ and $q_n$,  respectively, where  for $ 1 \leq j\leq k_n$,

$$H_j=[(j-1)(p_n+q_n)+1,jp_n+(j-1)q_n], \ \ I_j=[jp_n+(j-1)q_n+1, j(p_n+q_n)], $$
and  $I_{k_n+1}=[k_n(p_n+q_n)+1,n]$ is the remaining block. We set

\begin{align} \label{eq.Ybar}
\bar Y_{t,m}^{[u,v]}(l)=&\frac{1}{2\pi}\bar U_{t,l,m}\sum_{s=1}^{t-1} K((t-s)/M)\exp(-i (t-s) \omega_l^\prime) \overline V_{s,l,m} \nonumber \\
&+\frac{1}{2\pi}\bar V_{t,l,m}\sum_{s=1}^{t-1} K((t-s)/M)\exp(i (t-s) \omega_l^\prime) \overline U_{s,l,m}.
\end{align}

The above  lag-window estimator uses the sample autocovariances between $\overline{U}_{t,l,m}$ and $\overline{V}_{t,l,m}$ for lags $u=-n+1,\dots,n+1$. The first term on the right hand side
of the above equation  corresponds to such sample autocovariances for lags $u\geq 1$,  while the  second term on the right hand side of (<ref>), to lags  $u\leq -1$. Observe that
the sample autocovariance at lag $u=0$ is removed; also  see the construction of $g_n^{[u,v]}(\omega_l^\prime)$.
Using $     \bar Y_{t,m}^{[u,v]}(l)$ we further introduce the random variables

$$u_j(l)^{[u,v]}=\sum_{t \in H_j} (\bar Y_{t,m}^{[u,v]}-E\bar Y_{t,m}^{[u,v]})$$

for $1 \leq j \leq k_n$, and

$$v_j(l)^{[u,v]}=\sum_{t \in I_j} (\bar Y_{t,m}^{[u,v]}-E\bar Y_{t,m}^{[u,v]}),$$

for $1 \leq j \leq k_n+1 $.
We then have

\begin{align*}
\frac{1}{2\pi}& \sum_{u=-n+1}^{n-1} K(u/M) \sum_{t=\max(1,1-u)}^{\min(n,n-u)} [\bar U_{t+u,l,m} \bar V_{t,l,m}-E \bar U_{t+u,l,m} \bar V_{t,l,m}]\exp(-i u \omega_l^\prime)\\
%&- E \frac{1}{2\pi} \sum_{u=-n+1}^{n-1} K(u/M) \sum_{t=\max(1,1-u)}^{\min(n,n-u)} [\bar U_{t+u,l,m} \bar V_{t,l,m}-E \bar U_{t+u,l,m} \bar V_{t,l,m}]\exp(-i u \omega_l^\prime)\\
&-\sum_{t=1}^n (\bar U_{t,l,m} \bar V_{t,l,m}-E \bar U_{t,l,m} \bar V_{t,l,m})\\
&=\sum_{j=1}^{k_n} u_j(l)^{[u,v]}+\sum_{j=1}^{k_n+1} v_j(l)^{[u,v]}=:\bar g_{n,m}(l)^{[u,v]}.
\end{align*}

Note that since $K((t-s)/M)=0$ for $|t-s|>M$ and the sequences $\{U_{t,l,m}\}$,  and $ \{V_{t,l,m}\}$ are $m$-dependent, the random variables $u_{j_1}(l)^{[u,v]}$ and $u_{j_2}(l)^{[u,v]}$ are mutually independent for $j_1 \not = j_2$.

In the lemmas of Subsection <ref>, we prove
that the  introduced approximations by $m$-dependent and bounded random variables are valid. We then focus on the behavior of the sum
$ \sum_{j=1}^{k_n} u_j(l)^{[u,v]}$ involving  the big blocks  $u_j(l)^{[u,v]}$.
We show for $\omega_l^\prime=\pi l M/N, l=1,\dots,M/N-1$, that,

\begin{align*}
\sqrt{n/M}(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime))& =
\sqrt{1/(Mn)} \sum_{j=1}^{k_n} \left(u_j(l)^{[u,v]} -    \frac{\rho_{u,v}(\omega_l^\prime)}{2}[u_j(l)^{[u,u]}+u_j(l)^{[v,v]}]\right)\\
& \ \ \ +Remainder,
\end{align*}

where,  the Remainder is asymptotically negligible.
We then  approximate

$$ \left(u_j(l)^{[u,v]} -    \frac{\rho_{u,v}(\omega_l^\prime)}{2}[u_j(l)^{[u,u]}+u_j(l)^{[v,v]}]\right)$$

by a Gaussian vector. Toward this goal, we  need to introduce another truncation. To elaborate, let

$$w_j(l)=u_j(l)^{[u,v]} -    1/2 \rho_{u,v}(\omega_l^\prime)[u_j(l)^{[u,u]}+u_j(l)^{[v,v]}], $$

$\xi=(\log(n^{d_\xi}))^{-\zeta_\xi}$, $ \zeta_\xi \in [2,3/2+\Nkernel)$ and  $ d_\xi= 2(b+1)$. We then consider  the following truncated version of $w_j(l)^{[u,v]}$, $j=1,\dots,k_n, l=1,\dots,M/N-1$,

$$\hat w_j(l)^{[u,v]}=w_j(l)^{[u,v]} \ind(|w_j(l)^{[u,v]}|\leq \sqrt{Mn}\xi)- E (w_j(l)^{[u,v]} \ind(|w_j(l)^{[u,v]}|\leq \sqrt{Mn}\xi).$$

Let $\omega_l^\prime=\pi l N/M, l \in \mathcal{L}$ and
$T_{n,u,v}(l)$ as defined in Remark <ref>, and $Z_l$ i.i.d. with $Z_l\sim \mathcal{N}(0,I_2)$. We have under Assumption <ref> to <ref> and  for $h_n>0$, that,

\begin{align*}
\sup_{0\leq t \leq h_n}& \left|\frac{P(\max_l 1/(Mn)\chi(l)^{[u,v]}\geq t)}{P(\max_l \|Z_l\|_2 \geq t)}-1\right| \leq C((\log(n^{d_\xi}))^{-\zeta_\xi+3/2}+\max(2,\exp(h_n/2)) \\
&\times \log(n)^{C_\tau}\Big[\gpbetade{\log(n)} + (n^{1/2(1-1/4a)-1/\tau(2-a)-2(1+a)/\tau^2})^{-\tau}\Big],
\end{align*}

where $\chi(l)^{[u,v]}:=\chi_{(u,v)}(\omega_l^\prime)$.

Without loss of generality, we set $\mathcal{L}=\{1,\dots,M/N-1\}$. We first establish  the following equality

P(\max_l \chi(l)^{[u,v]}\geq t)=P(\max_l \|\tilde Z_l\|_2^2+W_n \geq t),

where $\tilde Z_l$ are $2$-dimensional standard Gaussians random variables and $W_n$ denotes an  error term  satisfying for some $\delta>0$,  $P(|W_n|>\delta)\leq \Delta(\delta)$. To elaborate, let
$$R_l:=\sqrt{n/M}((\Re(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)),
\Im(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)))^\top$$

such that
$\chi(l)=: R_l^\top \hat V^{-1}(\omega_l^\prime) R_l.$
Also let $\bar Z_l=(V^{-1}(\omega_l^\prime))^{1/2} \tilde Z_l$. We can then  write

\begin{align*}
\chi(l)^{[u,v]}=& \bar Z_l^\top V^{-1}(\omega_l^\prime) \bar Z_l + (\bar Z_l-R_l)^\top V^{-1}(\omega_l^\prime)(\bar Z_l-R_l)
-\bar Z_l V^{-1}(\omega_l^\prime)(\bar Z_l-R_l)\\
&-(\bar Z_l-R_l)^\top V^{-1}(\omega_l^\prime)\bar Z_l+
R_l^\top (\hat V^{-1}(\omega_l^\prime)-V^{-1}(\omega_l^\prime)) R_l\\
&=: \|\tilde Z_l\|_2^2+ W_n.
\end{align*}

That is, we need to determine the order of $W_n$,  i.e. of $\Delta(\cdot)$. Note that by Lemma <ref> we have the following

$$\max_l \|R_l\|_2=\bp(\gpbetade x+ \gps{x/\sqrt{n/M}}),$$

\begin{align*}
\|(\hat V(\omega_l^\prime)-V(\omega_l^\prime))\|_2 & \leq C \|\sqrt{M/n} R_l\|_2=\bp(\gpbetade {x\sqrt{n/M}}+ \gps{x\sqrt{n/M}}),
\end{align*}

$$\|\hat V^{-1}(\omega_l^\prime)-V^{-1}(\omega_l^\prime)\|_2\leq C \|\hat V(\omega_l^\prime)-V(\omega_l^\prime)\|_2/(1- C \|\hat V(\omega_l^\prime)-V(\omega_l^\prime)\|_2).$$

Furthermore, since $(V^{-1}(\omega_l^\prime))^{1/2} Z_l \sim \mathcal{N}(0,I_2)$ and $d\leq M$, we have
$$P(\max_l \|(V^{-1}(\omega_l^\prime))^{1/2} Z_l\|_2 \geq x)\leq M P(\|(V^{-1}(\omega_l^\prime))^{1/2} Z_1\|_2 \geq x)=M \exp(-\sqrt{x}/2).$$

It remains to establish the order of $\max_l \|(V^{-1}(\omega_l^\prime))^{1/2}(Z_l-R_l)\|_2$. We will show that  $(\bar Z_l-R_l)^\top V^{-1}(\omega_l^\prime)(\bar Z_l-R_l)$ is the dominating term.
To determine the order of $\max_l \|(V^{-1}(\omega_l^\prime))^{1/2}(Z_l-R_l)\|_2$ we split this term  into three  terms  and derive for each of them its  order which is denoted $\Delta_i(\cdot),i=1,2,3$.

For this note first that as outlined in Remark <ref>, we can approximate our random variables by bounded, $m$-dependent variables. That is we approximate $R_l$ by $\sum_{j=1}^{k_n} (\Re(\hat w_j(l)^{[u,v]}), \Im(\hat w_j(l)^{[u,v]}))^\top$.

Putting the results of Lemma <ref>
to Lemma <ref> together gives,

\begin{align*}
\|&\max_l (V^{-1}(\omega_l^\prime))^{1/2} [R_l-(\sum_{j=1}^{k_n} (\Re(\hat w_j(l)^{[u,v]}), \Im(\hat w_j(l)^{[u,v]}))^\top]\|_2=\bp(\gpbetade x+\\
&C \log(n)^{C_\tau} (x n^{1/2(1-1/4a)-1/\tau(2-a)-2(1+a)/\tau^2})^{-\tau}+x^{-C\tau} (Mn)^{-\tau}+\frac{n}{x^{\tau/2}(Mn)^{\tau/4}})=:\bp(\Delta_1(x))
\end{align*}

For  the second term on the right hand side of the previous equation, we use a Gaussian approximation. We rewrite the statistic by using bounded, $m$-dependent variables as follows

\begin{align*}
\max_l& (V^{-1}(\omega_l^\prime))^{1/2} \sum_{j=1}^{k_n} (\Re(\hat w_j(l)^{[u,v]}), \Im(\hat w_j(l)^{[u,v]}))^\top= \max_l ( \sum_{j=1}^{k_n} \tilde e_l^\top \tilde A_j),
\end{align*}

where $\tilde e_j = (e_j \otimes (1,1)^\top)\in \R^{2d}$, and $\tilde A_j$ is a $2d$-dimensional vector given by

\tilde A_j=\begin{pmatrix}
[(\sigma_1(1) \Re(\hat w_j(1)^{[u,v]}) + \sigma_3(1) \Im(\hat w_j(1)^{[u,v]})] \\
[(\sigma_3(1) \Re(\hat w_j(1)^{[u,v]}) + \sigma_2(1) \Im(\hat w_j(1)^{[u,v]})] \\ \vdots \\ [(\sigma_1(d) \Re(\hat w_j(d)^{[u,v]}) + \sigma_3(d) \Im(\hat w_j(d)^{[u,v]})] \\
[(\sigma_3(d) \Re(\hat w_j(d)^{[u,v]}) + \sigma_2(d) \Im(\hat w_j(d)^{[u,v]})]
\end{pmatrix},

\sigma_1(l) & \sigma_3(l) \\
\sigma_3(l) & \sigma_2(l)
\end{pmatrix}.

Note first that for all $l$ we have $\|\tilde e_l^\top \tilde A_j\|_2\leq \|(V^{-1}(\omega_l^\prime))\|_2 |\hat w_j^{[u,v]}(l)|\leq 2 C\sqrt{Mn} \xi$. This means that $\sum_{j=1}^{k_n} \tilde e_l^\top \tilde A_j$ is a sum of bounded independent random vectors of dimension $2$. We can apply Fact 2.2 in [16] and  define a $2d$-dimensional Gaussian vector $Z=(Z_l)_{l=1,\dots,d}$, where each $Z_l$ is $2$-dimensional, with mean zero and the variance of the entire $2d$-dimensional Gaussian vector equals  $\var(Z)=1/(Mn)\var(\sum_{j=1}^{k_n} \tilde A_j)$. Furthermore, due to the boundedness condition $|\hat w_j^{[u,v]}(l)|\leq C\sqrt{Mn} \xi$,  the Gaussian vector is a close approximation in the sense that we have for the 2-dimensional marginal distributions

P( \| \sqrt{1/(Mn)} \sum_{j=1}^{k_n} \tilde e_l^\top \tilde A_j-Z_l\|_2 \geq x) \leq c \exp( - x/( c \xi)),

where $c$ is a  positive constant.
With this, we have further for $x>0$,

\begin{align*}
P&(\max_l \| \sqrt{1/(Mn)}\sum_{j=1}^{k_n} \tilde e_l^\top \tilde A_j-Z_l\|_2 \geq x) \leq
\sum_{l=1}^d P( \| \sqrt{1/(Mn)}\sum_{j=1}^{k_n} \tilde e_l^\top \tilde A_j-Z_l\|_2
\geq x)
\\
&\leq C \exp( - x /( C \xi)+\log(d))=:\Delta_2(x).
\end{align*}

Regarding  the third term, we  have $\var(Z)=1/(Mn) \var(\sum_{j=1}^{k_n} \tilde A_j)$.

Let $H=1/(Mn)\var(\sum_{j=1}^{k_n} \tilde A_j)$, i.e., $Z \sim \mathcal{N}(0,H)$ and  $\tilde Z=H^{-1/2} Z$, that is,   $\tilde Z \sim \mathcal{N}(0,I_{2d})$. Then, we have  for $x>0$

\begin{align*}
P&(\max_l (\| \tilde e_l^\top (H^{1/2} - I_{2d}) \tilde Z \|_2 \geq x) \leq
\sum_{l=1}^d P( \| (e_{2(l-1)+1}^\top+e_{2(l-1)+2}^\top) (H^{1/2} - I_{2d}) \tilde Z \|_2 \geq x)  \\
&\leq  \sum_{l=1}^d (P( |e_{2(l-1)+1}^\top (H^{1/2} - I_{2d}) \tilde Z| \geq x/2)+
P( |e_{2(l-1)+2}^\top (H^{1/2} - I_{2d}) \tilde Z| \geq x/2)).
% &= \sum_{l=1}^d [(1-\Phi(x/(2(H_{(2l-1,2l-1)}-1)^{1/2})))+(1-\Phi(x/(2(H_{(2l,2l)}-1)^{1/2})))],
\end{align*}

The two probabilities can be bounded by the same arguments. We focus here on the first. Note that $e_{2(l-1)+1}^\top (H^{1/2} - I_{2d}) \tilde Z  \sim N(0,\|e_{2(l-1)+1}^\top (H^{1/2} - I_{2d})\|_2^2)$. Additionally, we have

\|e_{2(l-1)+1}^\top (H^{1/2} - I_{2d})\|_2 \leq \|e_{2(l-1)+1}^\top (H - I_{2d})\|_2\|(H^{1/2}+  I_{2d})^{-1}\|_2.

Since $H$ is a variance matrix and by definition positive semi-definite, we have $\|(H^{1/2}+  I_{2d})^{-1}\|_2\leq 1$. Furthermore,

$$\|e_{2(l-1)+1}^\top (H - I_{2d})\|_2^2=\sum_{r=1,r\not=2(l-1)+1}^{2d}
(e_{2(l-1)+1}^\top H e_r)^2+(e_{2(l-1)+1}^\top H e_{2(l-1)+1}^\top - 1)^2

$e_{2(l-1)+1}^\top H e_{2(l-1)+1}^\top=e_1^\top \var(\tilde e_l^\top \sum_{j=1}^{k_n} \tilde A_j) e_1 $. The term $\sum_{r=1,r\not=2(l-1)+1}^{2d}
(e_{2(l-1)+1}^\top H e_r)^2$ contains the covariance between different frequencies and we use Lemma <ref> to determine its order. For  other kernels than
the uniform one, the additional terms $1/[(r \pm (2l-1)+1)]N^{-1}]$ appear. Note that the terms  with $r$ are $\ell_2$-summable and that no periodicity in terms of the Fourier coefficients occur.

(e_{2(l-1)+1}^\top H e_r)^2=O(M(M^{-2}+k_n^{-2})+N^{-2r})=O(n^{-a}+n^{(3+2\beta)a-2}+N^{-2r}).$$
For $a <4/7$ and $ \beta <1/4$ we have $O(n^{-a}+n^{(3+2\beta)a-2})=o(n^{-\tilde \upsilon})$ for some $\tilde \upsilon>0$. For the variance term, we have

$$\tilde e_l^\top \sum_{j=1}^{k_n} \tilde A_j=(\Sigma^{-1}(\omega_l))^{1/2}
\sum_{j=1}^{k_n}
\begin{pmatrix}
\Re\Big(\hat u_j(l)^{[u,v]}- \rho_{u,v}(\omega_l^\prime)/2[\hat u_j(l)^{[u,u]}+\hat u_j(l)^{[v,v]}]\Big)\\
\Im\Big(\hat u_j(l)^{[u,v]}- \rho_{u,v}(\omega_l^\prime)/2[\hat u_j(l)^{[u,u]}+\hat u_j(l)^{[v,v]}]\Big)
\end{pmatrix}.

Additionally, we have by Lemma <ref> to

Lemma <ref>, that,

\begin{align*}
\max_{1\leq l \leq d}1/(Mn)\var\Big(&\sum_{j=1}^{k_n}\Big[\hat u_j(l)^{[u,v]}- \rho_{u,v}(\omega_l^\prime)/2[\hat u_j(l)^{[u,u]}+\hat u_j(l)^{[v,v]}]\Big]\\
&-\Big(1/n T_{n,u,v}(l)-1/2 \rho_{u,v}(\omega_l^\prime)[1/n T_{n,v,v}(l)+1/n T_{n,u,u}(l)\Big)\Big)=o(n^{-\upsilon^\prime}),
\end{align*}

where $0<\upsilon^\prime<\beta a$. The same rate also is obtained for the real and imaginary part. Additionally, we have by Lemma <ref>

\max_l \|\var
\begin{pmatrix}
\Re\Big(1/n T_{n,u,v}(l)-1/2 \rho_{u,v}(\omega_l^\prime)[1/n T_{n,v,v}(l)+1/n T_{n,u,u}(l)\Big)\\
\Im\Big(1/n T_{n,u,v}(l)-1/2 \rho_{u,v}(\omega_l^\prime)[1/n T_{n,v,v}(l)+1/n T_{n,u,u}(l)\Big)
\end{pmatrix}
\|_2=O(M^{-1}).

Thus, we get
$\max_l \|1/(Mn) \var(\tilde e_l^\top \sum_{j=1}^{k_n} \tilde A_j) -I_2\|_2=O(1/M)+o(n^{-\upsilon^\prime})=o(n^{-\upsilon^\prime})
$. Consequently, we have with $N=\log^{\Nkernel/r}(M)$ $\|e_{2(l-1)+1}^\top (H - I_{2d})\|_2=O(\log^{-\Nkernel}(M))$.

With this  bound and additional tail bounds for Gaussian random variables, see among others Appendix A in [10], we have

\begin{align*}
P&(\max_l (\| \tilde e_l^\top (H^{1/2} - I_{2d}) \tilde Z \|_2 \geq x))\leq C(
\exp(- (x \log^\Nkernel(M))^2+\log(d)) \frac{1}{x \log^\Nkernel(M)})=:\Delta_3(x).
\end{align*}

Now we set $x=(\log(n^{d_\xi}))^{-\zeta_\xi+3/2}$ with $d_\xi, \zeta_\xi$ as in Remark <ref>, especially $\zeta_\xi\geq 2$ and $\zeta_\xi < 3/2+\Nkernel$. Then,
$\Delta_3=O(\exp(-C \log(n)^{1+\eps})$,  where $ \eps>0$ and especially, we get that $\Delta_1$ obeys  the slowest rate among the three terms  $\Delta_1,\Delta_2$ and $\Delta_3$. Hence,
P(\|(V^{-1}(\omega_l^\prime))^{1/2}(Z_l-R_l)\|_2\geq x)\leq C\Delta_1(x)
and we get

P(\max_l \chi(l)^{[u,v]}\geq t)=P(\max_l \|\tilde Z_l\|_2^2+W_n \geq t),

where $W_n$ denotes the error and

$$P(|W_n|>x)\leq C P(\|(V^{-1}(\omega_l^\prime))^{1/2}(Z_l-R_l)\|_2^2>x) \leq C \Delta_1(\sqrt{x}).$$

Now we apply Lemma <ref> and  we obtain with $G_d(t)=(1-(1-\exp(-t/2))^d$,

\begin{align*}
\sup_{0\leq t   \leq h_n} \big| &  \frac{P(\max_l\tilde \chi(l)^{[u,v]}\geq t)}{G_d(t)} -1 \big|  \leq C \Delta_1\big(\log(n^{d_\xi})^{-\zeta_\xi/2+3/4}\big)\\
& \ \  \times \max(2,\exp(h_n/2))+ 2 \log(n^{d_\xi})^{-\zeta_\xi+3/2} \\
&\leq C \max(2,\exp(h_n/2)) \log(n)^{C_\tau}\Big(\gpbetade{\log(n)} \\
& \ \ +(n^{1/2(1-1/4a)-1/\tau(2-a)-2(1+a)/\tau^2})^{-\tau}+n^{-\tau/4(1+a)+1}\Big)
+ 2 \log(n^{d_\xi})^{-\zeta_\xi+3/2}.
\end{align*}

Let $\fspc(\omega)=\frac{M}{n}  \sum_{k=1}^{n} \kappa_M(\omega-\omega_k)  (Z(\omega_k)Z^{H}(\omega_k)-\Sigma_n(\omega_k))$ and $u_1,u_2,v_1,v_2\in\{1,\dots,p\}$. Under Assumption <ref>,<ref> and <ref>, we have

\begin{align*}
\sup_\omega|&n/M\cov(e_{v_1}^\top \fspc(\omega) e_{v_2}, e_{u_1}^\top \fspc(\omega) e_{u_2}) \\
&-C_{K_2} \Big(f_{v_1,u_1}(\omega)f_{u_2,v_2}(\omega)+ \ind(\omega \in \pi \Z)f_{v_1,u_2}(\omega)f_{v_2,u_1}(\omega) \Big)|=O(1/M)
\end{align*}

\begin{align*}
\sup_\omega |&n/M\cov(e_{v_1}^\top f^{-1}(\omega) \fspc(\omega) f^{-1}(\omega) e_{v_2}, e_{u_1}^\top f^{-1}(\omega) \fspc(\omega) f^{-1}(\omega) e_{u_2})\\
&- C_{K_2} \Big(f_{v_1,u_1}^{-1}(\omega)f_{u_2,v_2}^{-1}(\omega)+ \ind(\omega \in \pi \Z)f_{v_1,u_2}^{-1}(\omega)f_{v_2,u_1}^{-1}(\omega) \Big)|=O(1/M).
\end{align*}

Let $\tilde f_{ax_1+by_1,cx_2+dy_2}:=(a e_{v_1}^\top+be_{u_1}^\top) \fspc(\omega) (c^{(C)} e_{v_2}+d^{(C)} e_{u_2})$.
Note that $E \fspc(\omega)=0$ and $\cov(\tilde f_{x_1,x_2}, \tilde f_{y_1,y_2})=E e_{v_1}^\top \fspc(\omega) e_{v_2} e_{u_2}^\top \fspc(\omega) e_{u_1}.$ Furthermore, we have

\begin{align*}
4&(\var \tilde f_{x_1,x_2}+\var \tilde f_{y_1,y_2}+\var \tilde f_{x_1,y_2}+\var \tilde f_{y_1,x_2})-\\&
\var \tilde f_{x_1-y_1,x_2+y_2}-\var \tilde f_{x_1+y_1,x_2-y_2}-\var \tilde f_{ix_1-y_1,ix_2+y_2}-\var \tilde f_{ix_1+y_1,ix_2-y_2}-\\&i(\var \tilde f_{x_1-y_1,x_2-iy_2}+
\var \tilde f_{x_1+y_1,x_2+iy_2}-\var \tilde f_{x_1-iy_1,x_2-y_2}-\var \tilde f_{x_1+iy_1,x_2+iy_2})
\\=&8Ee_{v}^\top \fspc(\omega) e_{v} e_{u}^\top \fspc(\omega) e_{u}=8\cov(\tilde f_{x_1,x_2}, \tilde f_{y_1,y_2}).
\end{align*}

The sesquilinearity of the spectral density implies $\var(\tilde f_{ax_1+by_1,cx_2+dy_2})=C_{K_2}[(|a|^2 \tilde f_{x_1,x_1}+|b|^2\tilde f_{y_1,y_1}+ab^{(C)}\tilde f_{x_1,y_1}+ba^{(C)}\tilde f_{y_1,x_1})(|c|^2 \tilde f_{x_2,x_2}+|d|^2\tilde f_{y_2,y_2}+cd^{(C)}\tilde f_{x_2,y_2}+dc^{(C)}\tilde f_{y_2,x_2})+\ind(\omega \in \pi \Z)(ac^{(C)} \tilde f_{x_1,x_2}+bd^{(C)}\tilde f_{y_1,y_2}+bc^{(C)}\tilde f_{x_1,y_2}+ad^{(C)}\tilde f_{y_1,x_2})^2]$. Then, by  Lemma <ref> and   some algebra, we get
$\cov(\tilde f_{x_1,x_2}, \tilde f_{y_1,y_2})=\tilde f_{x_1,y_1} \tilde f_{y_2,x_2}+\ind(\omega \in \pi \Z) \tilde f_{x_1,y_2} \tilde f_{x_2,y_1}$ from which the first assertion of the lemma  follows.
For the second assertion, note that

\begin{align*}
\cov(e_{v_1}^\top f^{-1}(\omega) \fsp(\omega) f^{-1}(\omega) e_{v_2}, e_{u_1} f^{-1}(\omega) \fsp(\omega) f^{-1}(\omega) e_{u_2})
\sum_{s_1,s_2=1}^p \sum_{t_1,t_2=1}^p f_{u_1,s_1}^{-1}f_{s_2,u_2}^{-1}f_{t_1,v_1}^{-1}f_{v_2,t_2}^{-1}
\cov( e_{s_1}^\top \fsp(\omega) e_{s_2} , e_{t_1}^\top \fsp(\omega) e_{t_2}). \end{align*}

Inserting in the above expression the first assertion of the lemma  leads to its second assertion.

Under the conditions of Lemma <ref> we have

\max_l \|\frac{n}{M}\var
\begin{pmatrix}
\Re(1/n T_{n,u,v}(l)-1/2 \rho_{u,v}(\omega_l^\prime)[1/n T_{n,v,v}(l)+1/n T_{n,u,u}(l)) \\
\Im(1/n T_{n,u,v}(l)-1/2 \rho_{u,v}(\omega_l^\prime)[1/n T_{n,v,v}(l)+1/n T_{n,u,u}(l))
\end{pmatrix}

\begin{align*}
1-\Re(\rho_{u,v}(\omega))^2 & -\Re(\rho_{u,v}(\omega))\Im(\rho_{u,v}(\omega)) \\
-\Re(\rho_{u,v}(\omega))\Im(\rho_{u,v}(\omega)) & 1-\Im(\rho_{u,v}(\omega))^2 \\
\end{pmatrix}.
\end{align*}

By Lemma <ref> and the definition of $T_{n,u,v}(l)$ for $l,l_2=1,\dots,M/N-1$, we have that

|\frac{1}{Mn}\cov(T_{n,u_1,v_1}(l), T_{n,u_2,v_2}(l_2))-  \frac{\ind(l=l_2) C_{K_2}f^{-1}_{v_2,v_1}(\omega_l^\prime)f^{-1}_{u_1,u_2}(\omega_l^\prime)}{(f^{-1}_{u_1,u_1}(\omega_l^\prime)f^{-1}_{u_2,u_2}(\omega_l^\prime)f^{-1}_{v_1,v_1}(\omega_l^\prime)f^{-1}_{v_2,v_2}(\omega_l^\prime))^{1/2}}|=O(\frac{1}{M}).

Both assertions of the lemma  then  follow by sesquilinarity of the covariance.

§ PROOFS OF MAIN RESULTS

The assertion follows by Lemma <ref>, Lemma <ref> and following the  arguments of Lemma <ref> for $d=1$.

The assertion follows by Lemma <ref> with $h_n=2(a+2b)\log(n)$ and Assumption <ref>.

The assertions follows by Lemma <ref> and Lemma <ref>.

Under $H_0$, we have
\begin{align}
P(T_n^{[u,v]}\geq x) \leq P\Big(\max_l
\frac{n}{M} \begin{pmatrix}
\Re(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)) \\
\Im(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)) \\
\end{pmatrix}
\hat V^{-1}(\omega_l^\prime)
\begin{pmatrix}
\Re(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)) \\
\Im(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)) \\
\end{pmatrix}
\geq x\Big). \label{eq.H0.Tn}
\end{align}

To see this, let $\tilde \omega \in \{\omega_l^\prime,l\in \mathcal{L}\}$ such that $|\hat \rho^{(de)}_{u,v}(\tilde \omega)|>\delta$. If  such a $\tilde \omega$ does not exist, then $T_n^{[u,v]}=0$ and the above statement holds true. Since $\hat V^{-1}(\omega_l^\prime)$ is positive (semi)-definite for all $l$, it  suffices to show that $|\hat \rho^{(de)}_{u,v}(\tilde \omega)-\delta \exp(i\arg(\hat \rho^{(de)}_{u,v}(\tilde \omega)))|\leq |\hat \rho^{(de)}_{u,v}(\tilde \omega)-\rho_{u,v}(\tilde \omega)|$. Let $\hat \rho^{(de)}_{u,v}(\tilde \omega)=\delta^\prime \exp(i\lambda^\prime)$ and $\rho_{u_v}(\tilde \omega)=\tilde \delta\exp(i \tilde \lambda)$, where $\delta^\prime,\tilde \delta \in [0,1]$. Under $H_0$ and since $|\hat \rho^{(de)}_{u,v}(\tilde \omega)|>\delta$, we have $\delta^\prime>\delta$ and $\tilde \delta\leq \delta$. Then,
$|\hat \rho^{(de)}_{u,v}(\tilde \omega)-\delta \exp(i\arg(\hat \rho^{(de)}_{u,v}(\tilde \omega)))|=|\delta^\prime \exp(i \lambda^\prime)-\delta \exp(i\lambda^\prime)|=\delta^\prime-\delta$. Furthermore,
$|\hat \rho^{(de)}_{u,v}(\tilde \omega)-\rho_{u,v}(\tilde \omega)|\geq |\hat \rho^{(de)}_{u,v}(\tilde \omega)|-|\rho_{u,v}(\tilde \omega)|=\delta^\prime-\tilde \delta \geq \delta^\prime-\delta$. This implies (<ref>). The assertion follows then by Lemma <ref>.

Let $H_0=|\mathcal{H}_0|\leq q$ and set $b_n=2 \log(qd)$. Note first that by the inequalities  $(1+x)^d\leq 1/(1-dx)$ for $x\in [-1,0]$ and $(1+x)^d \geq 1+dx$ for $x\geq -1$, we have  the following  bounds: $1\leq q G_d(b_n)=q (1-(1-1/(qd))^d)\geq q/(q+1)\geq 1/2$. Recall  that

\begin{align*}
\widehat t = \operatorname{inf} \Big\{ 0 \leq t \leq 2 \log(dq) : \frac{G_d(t) q}{\max(1,\sum_{ (u,v) \in \mathcal{Q}} \ind(T_n^{(u,v)} \geq t))} \leq \alpha \Big\}.
\end{align*}

Our goal is to show that

$$E \left( \frac{\sum_{(u,v) \in \mathcal{H}_0} \ind(T_n^{[u,v]} \geq \hat t)}{\max(\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{[u,v]} \geq \hat t),1)}\right)=E\left( \frac{\sum_{(u,v) \in \mathcal{H}_0} \ind(T_n^{[u,v]} \geq \hat t) G_d(\hat t) q}{\max(\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{[u,v]} \geq \hat t),1) G_d(\hat t) q}\right)\leq \alpha.$$

Consider the expression  $\max(\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{[u,v]} \geq \hat t),1)$.
Let $\mathcal{H} (2 \sqrt{M/n b_n})=\mathcal{H}_a$. We have by assumption that $|\mathcal{H}_a|\geq \log(\log(n))$. Furthermore,

$$\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{(u,v)} \geq b_n)\geq \sum_{(u,v) \in \mathcal{H}_a} \ind(T_n^{(u,v)} \geq b_n).

We then have, for $(u,v) \in \mathcal{H}_a$, that

\begin{align*}
P&(T_n^{(u,v)} \geq b_n) = P((T_n^{(u,v)})^{1/2} \geq \sqrt{b_n}) \\
&\geq P(\sqrt{n/M}\max_l \inf_\lambda (\lambda_{\min} (\hat V^{-1}(\omega_l^\prime)))^{1/2} |\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\delta \exp(i \lambda)| \geq \sqrt{b_n})\\
&\geq P(\sqrt{n/M}\max_l (\lambda_{\min} (\hat V^{-1}(\omega_l^\prime)))^{1/2} (|\rho_{u,v}(\omega_l^\prime)|-\delta)\\
&\qquad \qquad
-\sqrt{n/M}\max_l (\lambda_{\min} (\hat V^{-1}(\omega_l^\prime)))^{1/2}|\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)|
\geq \sqrt{b_n})\\
&\geq P(\sqrt{n/M}\max_l (\lambda_{\min} (\hat V^{-1}(\omega_l^\prime)))^{1/2} (|\rho_{u,v}(\omega_l^\prime)|-\delta)-
(n/M \max_l  \chi(l)^{[u,v]})^{1/2}
\geq \sqrt{b_n})\\
&\geq 1-P(\max_l \chi_l^{[u,v]} \geq b_n) \\
&\geq 1- G_d(b_n)+(P(\max_l \chi_l^{[u,v]} \geq b_n)-G_d(b_n))\geq 1-(1+o(1))/q,
\end{align*}

where the second to last inequality follows by Lemma <ref>. Note that $V(\omega_l^\prime)$ has the eigenvalues $
and $C_{K_2}(1-|\rho_{u,v}(\omega_l^\prime)|^2)^2/2$, which imply that  $\lambda_{\min}(\hat V^{-1}(\omega_l^\prime)) = 2(C_{K_2}(1-|\hat \rho_{u,v}(\omega_l^\prime)|^2))^{-1}\geq 2$.

This means that for $n$ large enough,  we have for some generic constant $C>0$

\begin{align*}
P\Big(\sum_{(u,v) \in {\mathcal H}_a} \ind(T_n^{[u,v]} \geq b_n) \geq \log(\log(n))\Big)\geq 1 - |H_a| C /q=1-C(1-|\mathcal{H}_0|/q).
\end{align*}

By assumption $1-|\mathcal{H}_0|/q=o(1)$ and therefore,  it yields
with high probability that $\sum_{(u,v) \in \mathcal{H}_a} \ind(T_n^{(u,v)} \geq b_n)\geq \log(\log(n))$. Together with $q G_d(b_n)\leq 1$ we get for $n$ large enough  with high probability, that,

$$\frac{G_d(b_n) q}{\max(1,\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{(u,v)} \geq b_n))}\leq C \log(\log(n))^{-1} < \alpha.$$

Consequently, $P(\hat t \in [0,b_n))\to 1$ for $n\to \infty$.

If $\hat t$ is chosen so that $\hat t< b_n$, then $(G_d(\hat t) q)/(\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{[u,v]} \geq \hat t)) \leq \alpha$. Hence, we have by using the definition of $\hat t$

\begin{align*}
E& \left( \frac{\sum_{(u,v) \in \mathcal{H}_0} \ind(T_n^{[u,v]} \geq \hat t)}{\max(\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{[u,v]} \geq \hat t),1)}\right)\\
=&E \left(\ind(\hat t < b_n)  \frac{\sum_{(u,v) \in \mathcal{H}_0} \ind(T_n^{[u,v]} \geq \hat t)}{ G_d(\hat  t) q}
\frac{G_d(\hat  t) q}{\max(\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{[u,v]} \geq \hat  t),1)}
\right)\\
&+ E \left( \ind(\hat t \geq b_n) \frac{\sum_{(u,v) \in \mathcal{H}_0} \ind(T_n^{[u,v]} \geq \hat t)}{\max(\sum_{(u,v) \in \mathcal{Q}} \ind(T_n^{[u,v]} \geq \hat t),1)}\right)\\
&\leq \alpha/q\sum_{(u,v) \in \mathcal{H}_0} \sup_{0\leq t < b_n}  \frac{P(T_n^{[u,v]} \geq t)  }{G_d(t) }+ P(\hat t = b_n) \\
&\leq \alpha/q\sum_{(u,v) \in \mathcal{H}_0} \sup_{0\leq t < b_n}  \frac{P(\max_l 1/(Mn)\chi(l)^{[u,v]}\geq t) }{G_d(t) }+ P(\hat t = b_n) \\
&= \alpha/q|\mathcal{H}_0| \sup_{0\leq t < b_n}  \frac{P(\max_l \|Z_l\|_2+W\geq t)}{G_d(t) }+ o(1) = \alpha/q|\mathcal{H}_0|+o(1),
\end{align*}

where for the second to last inequality note that we have, by the arguments of the proof of Theorem <ref>, that,  for $1\leq u < v \leq p$

\begin{align*}
P_{H_0}(T_n^{[u,v]} \geq t) &\leq P\Big(\max_l
\frac{n}{M} \begin{pmatrix}
\Re(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)) \\
\Im(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)) \\
\end{pmatrix}
\hat V^{-1}(\omega_l^\prime)
\begin{pmatrix}
\Re(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)) \\
\Im(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\rho_{u,v}(\omega_l^\prime)) \\
\end{pmatrix}
\geq {t}\Big) \\
&=P(\max_l \frac{1}{Mn}\chi(l)^{[u,v]}\geq t).
\end{align*}

Furthermore,  for the last equality note that  by Lemma <ref>

\begin{align*}
\sup_{0\leq t \leq b_n}& \left|\frac{P(\max_l 1/(Mn)\chi(l)^{[u,v]}\geq t)}{G_d(t)}-1\right| \leq C((\log(n^{d_\xi}))^{-\zeta_\xi+3/2}+qd\log(n)^{C_\tau} \\
&\times \Big[\gpbetade{\log(n)} + (n^{1/2(1-1/4a)-1/\tau(2-a)-2(1+a)/\tau^2})^{-\tau}\Big])
%C((\log(n^{d_\xi}))^{-\zeta_\xi+3/2}+qM\Big[\gpbetade{\log(n)^{-\zeta_\xi+3/2}} \\
%&+\log(n)^{C_\tau} ( (n^{1/2(1-1/4a)-1/\tau(2-a)-2(1+a)/\tau^2})^{-\tau}+(Mn)^{-\tau})\Big])=o(1),
\end{align*}

because of Assumption <ref>. The assertion of the theorem follows then because  $q/|\mathcal{H}_0|=1+o(1)$.

§ AN ESTIMATOR OF $F^{-1}$

In this section we  elaborate on  an example of an estimator of $ f^{-1}$ which satisfies  the  requirements of Assumption <ref>, we  consider  a  CLIME type  estimator as in [17], see also [7] for details about CLIME type estimators. Notice that similar results can be obtained  using estimators based on  graphical lasso or other types of regularization.
Let $\operatorname{CLIME}_\lambda(A)$ denote the CLIME estimator applied  to an  input matrix $A$ with  tuning parameter $\lambda$ and obtained as   the solution of the optimization problem $\min \|B\|_1$ such that  $\|AB-I\|_{\max}\leq \lambda$,  plus a possible correction for  symmetry. Then, an estimator of  the inverse  spectral density matrix  at frequency $\omega$  based on the lag-window estimator $\fsp$ and using the tuning parameter $\lambda$,  is   given by

\begin{align} \label{eq.fhat.inverse}
\hat f^{-1}(\omega):=(1,i)\left(
\operatorname{CLIME}_\lambda\left(
\begin{pmatrix}
Re(\fsp(\omega)) & Im(\fsp(\omega)) \\
-Im(\fsp(\omega)) & Re(\fsp(\omega))
\end{pmatrix}\right)\right)\begin{pmatrix}
\end{pmatrix}/2;
\end{align}

with   $\fsp(\omega)$ as  given in  (<ref>).
[50],  Theorem 5.1 and Remark 4, showed that the above   estimator

\begin{align}
P(\sup_\omega \|\hat f^{-1}(\omega)-f^{-1}(\omega)\|_{\max} \geq x ) \leq \gps {x/\sup_\omega \|f^{-1}(\omega)\|_1^2}. \label{eq.rates.max.spec.inv}
\end{align}

For the class of  so-called weak sparse inverse spectral density matrices, the above result also can be extended to other matrix norms. To elaborate,   let  $\mathcal{G}_r(w(p))$ be  the set of  inverse spectral density matrices which are weakly sparse within a small $\ell_r$ ball, that is,

\begin{align}
\mathcal{G}_r(w(p)):=\Bigg\{& f^{-1} : [0,2\pi] \to \C^{p\times p}\, \Bigg| \, \sup_\omega \max_j \sum_{i=1}^p | f^{-1}_{i,j}(\omega)|^r \leq w(p)  \Bigg\}. \label{eq.finv.sparse}
\end{align}

Then, $\sup_\omega \|f^{-1}(\omega)\|_1 \leq C w(p)$ and by the arguments of [7],
it can be shown  that  for all $f^{-1}\in \mathcal{G}_r(w(p))$ and  any $l \in [1,\infty]$,

\begin{align}
P(\sup_\omega \|\hat f^{-1}(\omega)-f^{-1}(\omega)\|_l
\geq x ) & \leq \gps {\frac{x}{w(p)\sup_\omega \|f^{-1}(\omega)\|_1^2}} \nonumber \\
& \leq \gps {x/w(p)^3} . \label{eq.rates.spec.inv.freq}
\end{align}

§ ADDITIONAL SIMULATION RESULTS

VARMA$(1,1)$ process

VMA$(5)$ process

Detection of  non-zero partial coherences for the VARMA$(1,1)$ process with parameters as  in (a) and the VMA$(5)$ process with parameters as  in (d). The upper triangular part of every plot presents results for  $n=512$ and the lower triangular part for  $n=4096$. A bluish dot represents a not  discovery of a non-zero partial coherence, where the darker the blue color is the lower is the correct detection rate. A reddish dot represents a false discovery of a zero partial coherence, where  the darker the red color is the higher is the false detection rate. The left column of plots present  results for the testing procedure while

the right column of plots present results of the  regularizing procedure.

We visualize in Figure <ref> the results for the VARMA$(1,1)$ process with parameters as specified  in (a) and the VMA$(5)$ process with parameters as in (d)  and  for  $n=512$ and $n=4096$ observations. In this figure, a blank plot  describes the best possible situation while colored dots indicate some form of errors. Highly blue-saturated dots indicate that the underlying positive partial coherence is not correctly detected in most of the cases. Highly red-saturated dots indicate that a non existent partial coherence is identified  as positive.
Consider first the results for the VARMA$(1,1)$ process.
From Figure <ref>  we clearly   see the benefits of an increasing  sample size. The higher power of  is recognized  by the far fewer blue dots appearing in the left part of Figure <ref> compared to those in the right part  of the same figure. Since red dots do not appear in the plots corresponding to , it seems that this method does not produce any  systematic error in identifying non existent partial coherences. To elaborate, the three highest false detection rates for an individual partial coherence in both sample sizes together are about $3.9\%$, $ 1.6\%$ and  $1.4\%$. In contrast to this,   seems to have a systematic error in identifying  zero partial coherences  as non  zero. For the sample size $n=512$, the three largest false detection rates are $99.9\%$, $98.7\%$, and $98.7\%$. Even for the sample size of $n=4096$ observations, some zero partial coherences are wrongly detected by this method in $100\%$ of the cases. For the VMA$(5)$ processes, both approaches give reasonable results and no approach seems to have  a systematic error. However,  outperforms  in terms of power as it can be seen in this figure by the overall smaller number of blue  dots of    compared to those of .

Table <ref> reports size and power results
obtained using the simulation set up described in Section <ref> for the case $\delta=0.2$.

4c|     2c|

2c$\alpha=0.05$     2c|$\alpha=0.1$     2c|

$p$    $n$      FDR     Power     FDR     Power     FDR     Power

8|c|DGP: VARMA $(1,1)$
41cm50   190      0.00(0.02)     0.37(0.02)     0.00(0.02)     0.40(0.02)     0.00(0.02)     0.32(0.02)

512      0.00(0.02)     0.57(0.03)     0.00(0.02)     0.59(0.03)     0.05(0.03)     0.55(0.03)

2048      0.00(0.00)     0.81(0.02)     0.00(0.00)     0.81(0.02)     0.19(0.03)     0.72(0.02)

4096      0.00(0.00)     0.87(0.02)     0.00(0.00)     0.87(0.02)     0.16(0.03)     0.80(0.02)

41cm100   190        0.00(0.00)     0.11(0.02)     0.00(0.00)     0.11(0.02)     0.00(0.01)     0.06(0.01)

512        0.00(0.00)     0.30(0.02)     0.00(0.00)     0.31(0.02)     0.04(0.02)     0.36(0.03)

2048       0.00(0.00)     0.53(0.02)     0.00(0.00)     0.54(0.02)     0.07(0.02)     0.62(0.02)

4096       0.00(0.00)     0.63(0.02)     0.00(0.00)     0.64(0.02)     0.09(0.01)     0.71(0.01)

41cm200   190      0.07(0.14)     0.41(0.02)     0.07(0.15)     0.42(0.02)     0.00(0.00)     0.21(0.02)

512        0.00(0.03)     0.62(0.01)     0.00(0.03)     0.63(0.01)     0.06(0.01)     0.63(0.01)

2048       0.00(0.00)     0.80(0.01)     0.00(0.00)     0.81(0.01)     0.08(0.01)     0.70(0.01)

4096       0.00(0.00)     0.84(0.01)     0.00(0.00)     0.84(0.01)     0.11(0.01)     0.76(0.01)

8|c|DGP: VMA $(5)$
41cm50    190       0.00(0.00)     0.39(0.04)     0.00(0.00)     0.39(0.04)     0.00(0.01)     0.26(0.05)

512       0.00(0.00)     0.49(0.03)     0.00(0.00)     0.49(0.04)     0.00(0.00)     0.49(0.04)

2048      0.00(0.00)     0.72(0.03)     0.00(0.00)     0.72(0.03)     0.00(0.00)     0.70(0.05)

4096      0.00(0.00)     0.82(0.03)     0.00(0.00)     0.83(0.03)     0.01(0.02)     0.85(0.04)

41cm100    190       0.01(0.05)     0.11(0.01)     0.01(0.06)     0.11(0.01)     0.00(0.00)     0.03(0.01)

512       0.00(0.00)     0.18(0.01)     0.00(0.00)     0.19(0.01)     0.00(0.00)     0.16(0.02)

2048      0.00(0.00)     0.39(0.01)     0.00(0.00)     0.40(0.01)     0.00(0.00)     0.39(0.02)

4096      0.00(0.00)     0.54(0.01)     0.00(0.00)     0.55(0.01)     0.00(0.00)     0.53(0.02)

41cm200    190        0.00(0.00)     0.04(0.00)     0.00(0.00)     0.04(0.00)     0.00(0.00)     0.02(0.00)

512        0.00(0.00)     0.08(0.00)     0.00(0.00)     0.09(0.01)     0.00(0.00)     0.09(0.01)

2048       0.00(0.00)     0.18(0.01)     0.00(0.00)     0.19(0.01)     0.00(0.00)     0.20(0.01)

4096       0.00(0.00)     0.28(0.01)     0.00(0.00)     0.28(0.01)     0.00(0.00)     0.31(0.01)

Empirical false discovery rate and power for the case $\delta=0.2$ and for different models, different levels, and different sample sizes.

As it is seen from the above table,  for the case $\delta=0.2$,  has in all situations an empirical FDR of $0\%$. The  reason for this  lies in  the construction of the test statistic and the behavior of $ G_d(t)q$ used in the determination of the threshold value $\widehat{t}$. To elaborate, if   $\delta>0$ and   due to the indicator function appearing in the definition of  $ T^{(u,v)}_n$, the test statistic accumulates under the null hypothesis,
more  point mass around  zero. In fact, the larger $\delta$ is the less  test statistics  $ T^{(u,v)}_n$ will exceed $\delta$, i.e., the more $T_n^{(u,v)}$ values will be equal to zero.  The asymptotic distribution used  to determine critical values of this test, however, only forms
an upper bound; also  see the Proof of Theorem <ref>.  Hence  for $\delta>0$, $G_d(t)q$ becomes  a crude estimator of the expected number of falsely rejected null hypotheses and as consequence,  the FDR is forced towards zero. Nevertheless, even under such a  conservative behavior of the empirical FDR,    the empirical power of   outperforms in all cases the corresponding  power of  .

§ MULTIPLE BANDWIDTHS

In this section, we summarize the main  steps involved in the practical implementation of our inference procedure if different bandwidth are used for each time series.
A possible adaptive bandwidth rule is given in

Step 1:  Select a filter $\Phi(z)$ to prewhiten  the vector time series at hand.

Step 2:  Select a kernel $K$ and an individual bandwidth (truncation lag)  $M_{u,v}$ for each (cross)-spectral density $f_{u,v}, u,v=1,\dots,p$.

Step 3:  Set $N_{1,u,v}=\lfloor M_{u,v}/N_{u,v}\rfloor$ and use the individual grid of frequencies $\mathcal{L}_{u,v}=\{\omega_l^\prime=l \pi/N_{1,u,v}   \in \mathcal{W} :l \in {1,\dots,N_{1,u,v}-1} \}$ to cover the frequency band $ {\mathcal W}$ of interest, where  $N_{u,v}=\log^{2/r}(M_{u,v})$ and  $r$ is determined by the decay behavior of the Fourier coefficients of the kernel $K$ used; see Assumption 2 in Section 6. Set $\mathcal{L}=\bigcup \mathcal{L}_{u,v}$

Step 4:   Estimate the inverse spectral density matrix $f^{-1}$ at the  frequencies $\omega_l^\prime$ for every $ l \in \mathcal{L}$.

Step 5   Compute for all $(u,v) \in \mathcal{Q}$ and for all $l \in \mathcal{L}_{u,v}$, the de-biased estimator
$\hat \rho_{u,v}^{(de)}(\omega_l^\prime)$ and  the test statistic $T_n^{(u,v)}$, where

\begin{align*}
T_n^{(u,v)}=&\ind\big(\max_{l \in \mathcal{L}_{u,v}}  |\hat \rho^{(de)}_{u,v}(\omega_l^\prime)|> \delta\big) \times  \label{eq.test.statistic}\\
& \max_{l \in \mathcal{L}_{u,v}}\Big\{ \frac{n}{M_{u,v}} \begin{pmatrix}
\Re(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\delta \exp(i \tilde \omega_l)) \\
\Im(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\delta \exp(i \tilde \omega_l)) \\
\end{pmatrix}
\hat V^{-1}_{(u,v)}(\omega_l^\prime)
\begin{pmatrix}
\Re(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\delta \exp(i \tilde \omega_l)) \\
\Im(\hat \rho^{(de)}_{u,v}(\omega_l^\prime)-\delta \exp(i \tilde \omega_l)) \\
\end{pmatrix}\Big\}, \nonumber
\end{align*}

$\tilde \omega_l=\arg(\hat \rho^{(de)}_{u,v}(\omega_l^\prime))$,

\begin{align*}
\hat V^{-1}_{(u,v)}(\omega)=\frac{2}{C_{K_2}(1-|\hat\rho_{u,v}(\omega)|^2)^2}\begin{pmatrix}
1-\Im(\hat\rho_{u,v}(\omega))^2 & \Re(\hat\rho_{u,v}(\omega)\Im(\hat\rho_{u,v}(\omega) \\
\Re(\hat \rho_{u,v}(\omega)\Im(\hat\rho_{u,v}(\omega) & 1-\Re(\hat\rho_{u,v}(\omega))^2 \\
\end{pmatrix}.
\end{align*}

Step 6:   Set $d_{u,v}=|\mathcal{L}_{u,v}|$,  $G_{d_{u,v}}(t)=1-(1-\exp(-t/2))^{d_{u,v}}$ and calculate  the threshold

$$\widehat t = \operatorname{inf} \{ 0 \leq t \leq 2 \log(dq) : \frac{\sum_{(u,v) \in \mathcal{Q}} G_{d_{u,v}}(t)}{\max(1,\sum_{ (u,v) \in \mathcal{Q}} \ind(T_n^{(u,v)} \geq t))} \leq \alpha \}.$$

Step 7:   For each  $(u,v) \in \mathcal{Q}$ reject $H_0^{(u,v)}$ if $T_n^{(u,v)} \geq \widehat t$.

[1]
Bach, F. R. and Jordan, M. I. (2004).
Learning graphical models for stationary time series.
IEEE Transactions on Signal Processing, 52(8):2189–2199.

[2]
Barber, R. F. and Candès, E. J. (2015).
Controlling the false discovery rate via knockoffs.
The Annals of Statistics, 43(5):2055–2085.

[3]
Barigozzi, M. and Farnè, M. (2022).
An algebraic estimator for large spectral density matrices.
Journal of the American Statistical Association.

[4]
Benjamini, Y. and Hochberg, Y. (1995).
Controlling the false discovery rate: a practical and powerful
approach to multiple testing.
Journal of the Royal Statistical Society: Series B
(Methodological), 57(1):289–300.

[5]
Benjamini, Y. and Yekutieli, D. (2001).
The control of the false discovery rate in multiple testing under
The Annals of Statistics, pages 1165–1188.

[6]
Brillinger, D. (2001).
Time Series: Data Analysis and Theory.
Classics in Applied Mathematics. Society for Industrial and Applied

[7]
Cai, T., Liu, W., and Luo, X. (2011).
A constrained l1 minimization approach to sparse precision matrix
Journal of the American Statistical Association,

[8]
Cai, T. T. and Liu, W. (2016).
Large-scale multiple testing of correlations.
Journal of the American Statistical Association,

[9]
Cai, T. T., Ren, Z., and Zhou, H. H. (2016).
Estimating structured high-dimensional covariance and precision
matrices: Optimal rates and adaptive estimation.
Electronic Journal of Statistics, 10(1):1–59.

[10]
Chatterjee, S. (2014).
Superconcentration and related topics, volume 15.

[11]
Chen, X., Xu, M., and Wu, W. B. (2013).
Covariance and precision matrix estimation for high-dimensional time
The Annals of Statistics, 41(6):2994–3021.

[12]
Dahlhaus, R. (2000).
Graphical interaction models for multivariate time series 1.
Metrika, 51(2):157–172.

[13]
Dette, H. and Paparoditis, E. (2009).
Bootstrapping frequency domain tests in multivariate time series with
an application to comparing spectral densities.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 71(4):831–857.

[14]
Eichler, M. (2008).
Testing nonparametric and semiparametric hypotheses in vector
stationary processes.
Journal of Multivariate Analysis, 99(5):968–1009.

[15]
Eichler, M. (2012).
Graphical modelling of multivariate time series.
Probability Theory and Related Fields, 153(1):233–268.

[16]
Einmahl, U. and Mason, D. M. (1997).
Gaussian approximation of local empirical processes indexed by
Probability Theory and Related Fields, 107(3):283–311.

[17]
Fiecas, M., Leng, C., Liu, W., and Yu, Y. (2019).
Spectral analysis of high-dimensional time series.
Electronic Journal of Statistics, 13(2):4079–4101.

[18]
Fiecas, M. and Ombao, H. (2016).
Modeling the evolution of dynamic brain processes during an
associative learning experiment.
Journal of the American Statistical Association,

[19]
Gray, D. (2014).
Central european foreign exchange markets: a cross-spectral analysis
of the 2007 financial crisis.
The European Journal of Finance, 20(6):550–567.

[20]
Hannan, E. J. (1970).
Multiple time series.
John Wiley & Sons.

[21]
Javanmard, A. and Montanari, A. (2014).
Confidence intervals and hypothesis testing for high-dimensional
The Journal of Machine Learning Research, 15(1):2869–2909.

[22]
Jung, A. (2015).
Learning the conditional independence structure of stationary time
series: A multitask learning approach.
IEEE Transactions on Signal Processing, 63(21):5677–5690.

[23]
Koopmans, L. H. (1995).
The spectral analysis of time series.

[24]
Krampe, J. and Margaritella, L. (2021).
Dynamic factor models with sparse var idiosyncratic components.
arXiv preprint arXiv:2112.07149.

[25]
Krampe, J. and Paparoditis, E. (2021).
Sparsity concepts and estimation procedures for high-dimensional
vector autoregressive models.
Journal of Time Series Analysis, 42(5-6):554–579.

[26]
Krampe, J. and Subba Rao, S. (2022).
Inverse covariance operators of multivariate nonstationary time
arXiv preprint arXiv:2202.00933.

[27]
Li, S., Cai, T. T., and Li, H. (2023).
Transfer learning in large-scale gaussian graphical models with false
discovery rate control.
Journal of the American Statistical Association,

[28]
Liu, W. (2013).
Gaussian graphical model estimation with false discovery rate
The Annals of Statistics, pages 2948–2978.

[29]
Liu, W. and Wu, W. B. (2010).
Asymptotics of spectral density estimates.
Econometric Theory, pages 1218–1245.

[30]
Medkour, T., Walden, A. T., and Burgess, A. (2009).
Graphical modelling for brain connectivity via partial coherence.
Journal of Neuroscience Methods, 180(2):374–383.

[31]
Politis, D. N. (2003).
Adaptive bandwidth choice.
Journal of Nonparametric Statistics, 15(4-5):517–533.

[32]
Politis, D. N. (2011).
Higher-order accurate, positive semidefinite estimation of
large-sample covariance and spectral density matrices.
Econometric Theory, 27(4):703–744.

[33]
Priestley, M. (1988).
The spectral analysis of time series.

[34]
R Core Team (2021).
R: A Language and Environment for Statistical Computing.
R Foundation for Statistical Computing, Vienna, Austria.

[35]
Rosenblatt, M. (1985).
Stationary sequences and random fields.
Springer Science & Business Media.

[36]
Rosuel, A., Loubaton, P., and Vallet, P. (2022).
On the asymptotic distribution of the maximum sample spectral
coherence of gaussian time series in the high dimensional regime.
Journal of Multivariate Analysis, page 105124.

[37]
Schneider-Luftman, D. (2016).
p-value combiners for graphical modelling of eeg data in the
frequency domain.
Journal of Neuroscience Methods, 271:92–106.

[38]
Schneider-Luftman, D. and Walden, A. T. (2016).
Partial coherence estimation via spectral matrix shrinkage under
quadratic loss.
IEEE Transactions on Signal Processing, 64(22):5767–5777.

[39]
Shao, X. and Wu, W. B. (2007).
Local whittle estimation of fractional integration for nonlinear
Econometric Theory, 23(5):899–929.

[40]
Sun, Y., Li, Y., Kuceyeski, A., and Basu, S. (2018).
Large spectral density matrix estimation by thresholding.
arXiv preprint arXiv:1812.00532.

[41]
Trujillo, L. T., Stanfield, C. T., and Vela, R. D. (2017).
The effect of electroencephalogram (eeg) reference choice on
information-theoretic measures of the complexity and integration of eeg
Frontiers in Neuroscience, 11:425.

[42]
Tugnait, J. K. (2022).
On sparse high-dimensional graphical model learning for dependent
time series.
Signal Processing, 197:108539.

[43]
van de Geer, S., Bühlmann, P., Ritov, Y., and Dezeure, R. (2014).
On asymptotically optimal confidence regions and tests for
high-dimensional models.
The Annals of Statistics, 42(3):1166–1202.

[44]
Wang, H., Li, B., and Leng, C. (2009).
Shrinkage tuning parameter selection with a diverging number of
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 71(3):671–683.

[45]
Wu, W. B. (2005).
Nonlinear system theory: Another look at dependence.
Proceedings of the National Academy of Sciences,

[46]
Wu, W.-B. and Wu, Y. N. (2016).
Performance bounds for parameter estimates of high-dimensional linear
models with correlated errors.
Electronic Journal of Statistics, 10(1):352–379.

[47]
Wu, W. B. and Zaffaroni, P. (2018).
Asymptotic theory for spectral density estimates of general
multivariate time series.
Econometric Theory, 34(1):1–22.

[48]
Xu, H., Wang, D., Zhao, Z., and Yu, Y. (2022).
Change point inference in high-dimensional regression models under
temporal dependence.

[49]
Zhang, C.-H. and Zhang, S. S. (2014).
Confidence intervals for low dimensional parameters in high
dimensional linear models.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 76(1):217–242.

[50]
Zhang, D. and Wu, W. B. (2021).
Convergence of covariance and spectral density estimates for
high-dimensional locally stationary processes.
The Annals of Statistics, 49(1):233–254.

[1]
Bach, F. R. and Jordan, M. I. (2004).
Learning graphical models for stationary time series.
IEEE Transactions on Signal Processing, 52(8):2189–2199.

[2]
Barber, R. F. and Candès, E. J. (2015).
Controlling the false discovery rate via knockoffs.
The Annals of Statistics, 43(5):2055–2085.

[3]
Barigozzi, M. and Farnè, M. (2022).
An algebraic estimator for large spectral density matrices.
Journal of the American Statistical Association.

[4]
Benjamini, Y. and Hochberg, Y. (1995).
Controlling the false discovery rate: a practical and powerful
approach to multiple testing.
Journal of the Royal Statistical Society: Series B
(Methodological), 57(1):289–300.

[5]
Benjamini, Y. and Yekutieli, D. (2001).
The control of the false discovery rate in multiple testing under
The Annals of Statistics, pages 1165–1188.

[6]
Brillinger, D. (2001).
Time Series: Data Analysis and Theory.
Classics in Applied Mathematics. Society for Industrial and Applied

[7]
Cai, T., Liu, W., and Luo, X. (2011).
A constrained l1 minimization approach to sparse precision matrix
Journal of the American Statistical Association,

[8]
Cai, T. T. and Liu, W. (2016).
Large-scale multiple testing of correlations.
Journal of the American Statistical Association,

[9]
Cai, T. T., Ren, Z., and Zhou, H. H. (2016).
Estimating structured high-dimensional covariance and precision
matrices: Optimal rates and adaptive estimation.
Electronic Journal of Statistics, 10(1):1–59.

[10]
Chatterjee, S. (2014).
Superconcentration and related topics, volume 15.

[11]
Chen, X., Xu, M., and Wu, W. B. (2013).
Covariance and precision matrix estimation for high-dimensional time
The Annals of Statistics, 41(6):2994–3021.

[12]
Dahlhaus, R. (2000).
Graphical interaction models for multivariate time series 1.
Metrika, 51(2):157–172.

[13]
Dette, H. and Paparoditis, E. (2009).
Bootstrapping frequency domain tests in multivariate time series with
an application to comparing spectral densities.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 71(4):831–857.

[14]
Eichler, M. (2008).
Testing nonparametric and semiparametric hypotheses in vector
stationary processes.
Journal of Multivariate Analysis, 99(5):968–1009.

[15]
Eichler, M. (2012).
Graphical modelling of multivariate time series.
Probability Theory and Related Fields, 153(1):233–268.

[16]
Einmahl, U. and Mason, D. M. (1997).
Gaussian approximation of local empirical processes indexed by
Probability Theory and Related Fields, 107(3):283–311.

[17]
Fiecas, M., Leng, C., Liu, W., and Yu, Y. (2019).
Spectral analysis of high-dimensional time series.
Electronic Journal of Statistics, 13(2):4079–4101.

[18]
Fiecas, M. and Ombao, H. (2016).
Modeling the evolution of dynamic brain processes during an
associative learning experiment.
Journal of the American Statistical Association,

[19]
Gray, D. (2014).
Central european foreign exchange markets: a cross-spectral analysis
of the 2007 financial crisis.
The European Journal of Finance, 20(6):550–567.

[20]
Hannan, E. J. (1970).
Multiple time series.
John Wiley & Sons.

[21]
Javanmard, A. and Montanari, A. (2014).
Confidence intervals and hypothesis testing for high-dimensional
The Journal of Machine Learning Research, 15(1):2869–2909.

[22]
Jung, A. (2015).
Learning the conditional independence structure of stationary time
series: A multitask learning approach.
IEEE Transactions on Signal Processing, 63(21):5677–5690.

[23]
Koopmans, L. H. (1995).
The spectral analysis of time series.

[24]
Krampe, J. and Margaritella, L. (2021).
Dynamic factor models with sparse var idiosyncratic components.
arXiv preprint arXiv:2112.07149.

[25]
Krampe, J. and Paparoditis, E. (2021).
Sparsity concepts and estimation procedures for high-dimensional
vector autoregressive models.
Journal of Time Series Analysis, 42(5-6):554–579.

[26]
Krampe, J. and Subba Rao, S. (2022).
Inverse covariance operators of multivariate nonstationary time
arXiv preprint arXiv:2202.00933.

[27]
Li, S., Cai, T. T., and Li, H. (2023).
Transfer learning in large-scale gaussian graphical models with false
discovery rate control.
Journal of the American Statistical Association,

[28]
Liu, W. (2013).
Gaussian graphical model estimation with false discovery rate
The Annals of Statistics, pages 2948–2978.

[29]
Liu, W. and Wu, W. B. (2010).
Asymptotics of spectral density estimates.
Econometric Theory, pages 1218–1245.

[30]
Medkour, T., Walden, A. T., and Burgess, A. (2009).
Graphical modelling for brain connectivity via partial coherence.
Journal of Neuroscience Methods, 180(2):374–383.

[31]
Politis, D. N. (2003).
Adaptive bandwidth choice.
Journal of Nonparametric Statistics, 15(4-5):517–533.

[32]
Politis, D. N. (2011).
Higher-order accurate, positive semidefinite estimation of
large-sample covariance and spectral density matrices.
Econometric Theory, 27(4):703–744.

[33]
Priestley, M. (1988).
The spectral analysis of time series.

[34]
R Core Team (2021).
R: A Language and Environment for Statistical Computing.
R Foundation for Statistical Computing, Vienna, Austria.

[35]
Rosenblatt, M. (1985).
Stationary sequences and random fields.
Springer Science & Business Media.

[36]
Rosuel, A., Loubaton, P., and Vallet, P. (2022).
On the asymptotic distribution of the maximum sample spectral
coherence of gaussian time series in the high dimensional regime.
Journal of Multivariate Analysis, page 105124.

[37]
Schneider-Luftman, D. (2016).
p-value combiners for graphical modelling of eeg data in the
frequency domain.
Journal of Neuroscience Methods, 271:92–106.

[38]
Schneider-Luftman, D. and Walden, A. T. (2016).
Partial coherence estimation via spectral matrix shrinkage under
quadratic loss.
IEEE Transactions on Signal Processing, 64(22):5767–5777.

[39]
Shao, X. and Wu, W. B. (2007).
Local whittle estimation of fractional integration for nonlinear
Econometric Theory, 23(5):899–929.

[40]
Sun, Y., Li, Y., Kuceyeski, A., and Basu, S. (2018).
Large spectral density matrix estimation by thresholding.
arXiv preprint arXiv:1812.00532.

[41]
Trujillo, L. T., Stanfield, C. T., and Vela, R. D. (2017).
The effect of electroencephalogram (eeg) reference choice on
information-theoretic measures of the complexity and integration of eeg
Frontiers in Neuroscience, 11:425.

[42]
Tugnait, J. K. (2022).
On sparse high-dimensional graphical model learning for dependent
time series.
Signal Processing, 197:108539.

[43]
van de Geer, S., Bühlmann, P., Ritov, Y., and Dezeure, R. (2014).
On asymptotically optimal confidence regions and tests for
high-dimensional models.
The Annals of Statistics, 42(3):1166–1202.

[44]
Wang, H., Li, B., and Leng, C. (2009).
Shrinkage tuning parameter selection with a diverging number of
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 71(3):671–683.

[45]
Wu, W. B. (2005).
Nonlinear system theory: Another look at dependence.
Proceedings of the National Academy of Sciences,

[46]
Wu, W.-B. and Wu, Y. N. (2016).
Performance bounds for parameter estimates of high-dimensional linear
models with correlated errors.
Electronic Journal of Statistics, 10(1):352–379.

[47]
Wu, W. B. and Zaffaroni, P. (2018).
Asymptotic theory for spectral density estimates of general
multivariate time series.
Econometric Theory, 34(1):1–22.

[48]
Xu, H., Wang, D., Zhao, Z., and Yu, Y. (2022).
Change point inference in high-dimensional regression models under
temporal dependence.

[49]
Zhang, C.-H. and Zhang, S. S. (2014).
Confidence intervals for low dimensional parameters in high
dimensional linear models.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 76(1):217–242.

[50]
Zhang, D. and Wu, W. B. (2021).
Convergence of covariance and spectral density estimates for
high-dimensional locally stationary processes.
The Annals of Statistics, 49(1):233–254.

[1]
Bach, F. R. and Jordan, M. I. (2004).
Learning graphical models for stationary time series.
IEEE Transactions on Signal Processing, 52(8):2189–2199.

[2]
Barber, R. F. and Candès, E. J. (2015).
Controlling the false discovery rate via knockoffs.
The Annals of Statistics, 43(5):2055–2085.

[3]
Barigozzi, M. and Farnè, M. (2022).
An algebraic estimator for large spectral density matrices.
Journal of the American Statistical Association.

[4]
Benjamini, Y. and Hochberg, Y. (1995).
Controlling the false discovery rate: a practical and powerful
approach to multiple testing.
Journal of the Royal Statistical Society: Series B
(Methodological), 57(1):289–300.

[5]
Benjamini, Y. and Yekutieli, D. (2001).
The control of the false discovery rate in multiple testing under
The Annals of Statistics, pages 1165–1188.

[6]
Brillinger, D. (2001).
Time Series: Data Analysis and Theory.
Classics in Applied Mathematics. Society for Industrial and Applied

[7]
Cai, T., Liu, W., and Luo, X. (2011).
A constrained l1 minimization approach to sparse precision matrix
Journal of the American Statistical Association,

[8]
Cai, T. T. and Liu, W. (2016).
Large-scale multiple testing of correlations.
Journal of the American Statistical Association,

[9]
Cai, T. T., Ren, Z., and Zhou, H. H. (2016).
Estimating structured high-dimensional covariance and precision
matrices: Optimal rates and adaptive estimation.
Electronic Journal of Statistics, 10(1):1–59.

[10]
Chatterjee, S. (2014).
Superconcentration and related topics, volume 15.

[11]
Chen, X., Xu, M., and Wu, W. B. (2013).
Covariance and precision matrix estimation for high-dimensional time
The Annals of Statistics, 41(6):2994–3021.

[12]
Dahlhaus, R. (2000).
Graphical interaction models for multivariate time series 1.
Metrika, 51(2):157–172.

[13]
Dette, H. and Paparoditis, E. (2009).
Bootstrapping frequency domain tests in multivariate time series with
an application to comparing spectral densities.
Journal of the Royal Statistical Society: Series B (Statistical
Methodology), 71(4):831–857.

[14]
Eichler, M. (2008).
Testing nonparametric and semiparametric hypotheses in vector
stationary processes.
Journal of Multivariate Analysis, 99(5):968–1009.

[15]
Eichler, M. (2012).
Graphical modelling of multivariate time series.
Probability Theory and Related Fields, 153(1):233–268.

[16]
Einmahl, U. and Mason, D. M. (1997).
Gaussian approximation of local empirical processes indexed by
Probability Theory and Related Fields, 107(3):283–311.

[17]
Fiecas, M., Leng, C., Liu, W., and Yu, Y. (2019).
Spectral analysis of high-dimensional time series.
Electronic Journal of Statistics, 13(2):4079–4101.

[18]
Fiecas, M. and Ombao, H. (2016).
Modeling the evolution of dynamic brain processes during an
associative learning experiment.
Journal of the American Statistical Association,

[19]
Gray, D. (2014).
Central european foreign exchange markets: a cross-spectral analysis
of the 2007 financial crisis.
The European Journal of Finance, 20(6):550–567.

[20]
Hannan, E. J. (1970).
Multiple time series.
John Wiley & Sons.

[21]
Javanmard, A. and Montanari, A. (2014).
Confidence intervals and hypothesis testing for high-dimensional
The Journal of Machine Learning Research, 15(1):2869–2909.

[22]
Jung, A. (2015).
Learning the conditional independence structure of stationary time
series: A multitask learning approach.
IEEE Transactions on Signal Processing, 63(21):5677–5690.

[23]
Koopmans, L. H. (1995).
The spectral analysis of time series.

[24]
Krampe, J. and Margaritella, L. (2021).
Dynamic factor models with sparse var idiosyncratic components.
arXiv preprint arXiv:2112.07149.

[25]
Krampe, J. and Paparoditis, E. (2021).
Sparsity concepts and estimation procedures for high-dimensional
vector autoregressive models.
Journal of Time Series Analysis, 42(5-6):554–579.

[26]
Krampe, J. and Subba Rao, S. (2022).
Inverse covariance operators of multivariate nonstationary time
arXiv preprint arXiv:2202.00933.

[27]
Li, S., Cai, T. T., and Li, H. (2023).
Transfer learning in large-scale gaussian graphical models with false
discovery rate control.
Journal of the American Statistical Association,

[28]
Liu, W. (2013).
Gaussian graphical model estimation with false discovery rate
The Annals of Statistics, pages 2948–2978.

[29]
Liu, W. and Wu, W. B. (2010).
Asymptotics of spectral density estimates.
Econometric Theory, pages 1218–1245.

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