\subsection{Proof of Theorem \ref{th: pop_sparsity gnar}.a}
We start by proving the first part (i.e., Theorem \ref{th: pop_sparsity gnar}.$a$).
\begin{proof}
Consider a stationary GNAR process $\boldsymbol{X}_t$, we can embed a latent graphical process $\tilde{\boldsymbol{X}}_t$ to this GNAR process by connecting all nodes which fail to satisfy $X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ at all lags } h$. By Lemma \ref{lma: conditional equivalences} and Theorem \ref{th: inv spec cross indentity} we have that 
\begin{equation} \label{eq: path length equivalence}
    X_{i, t} \perp X_{j, t + h} | \mathcal{A}^{(ij)} \textit{ for all } h \iff [\bmat{S}(\omega)]_{ij} \equiv 0.
\end{equation}
Now, denote by $\tilde{\mathcal{G}} = (\mathcal{K}, \tilde{\mathcal{E}})$ the corresponding partial correlation graph. So connecting (\ref{eq: path length equivalence}), Theorem \ref{th: inv spec cross indentity} and Proposition \ref{prop: conditional residual} we have that 
\begin{equation*} 
     (i, j) \notin \tilde{\mathcal{E}} \iff  [\bmat{S}(\omega)]_{ij} \equiv 0,
\end{equation*}
which by Proposition \ref{prop: var constrains} gives
\begin{align} \label{eq: end first part}
      (i, j) \notin \tilde{\mathcal{E}} &\iff (i, j) \notin \mathcal{A}_{r^*} \nonumber \\
      &\iff \delta (i, j) \geq 2 r^* + 1.
\end{align}
\end{proof}