## 3 Reconstructing Missing Blocks in Autocovariance Matrices
### Case I: \(r>q\)  
In this case, our main result will be Theorem 1. We previously introduced the necessary notation, the new conditions and some previous results (Hanzon's Theorem and Lemma 2). Let us denote the following matrices:  
\[\mathbf{G^{*}}= \begin{pmatrix}0\\ \vdots\\ 0\\ B_{0}\\ \vdots\\ B_{q}\end{pmatrix} \text{and}\ \mathbf{X}= \begin{pmatrix}C_{q}\\ C_{q-1}\\ \vdots\\ C_{q-r+1}\end{pmatrix} \text{are}\ nr\times n,\mathbf{F^{*}}= \begin{pmatrix}A_{1}&A_{2} &\cdots&A_{r}\\ I_{nt}&0&\cdots&0\\ 0&\ddots&\ddots&\ddots&\vdots\\ \vdots&&&0&0\\ 0&\cdots&0&I_{nt}&0\end{pmatrix} \text{is}\ nr\times nr,\]  
\[\mathbf{\theta}= \begin{pmatrix}H_{1}F^{*}\\ \vdots\\ H_{1}(F^{*})^{nr}\\ H_{2}(F^{*})^{kN-q}(F^{*})^{N}\\ \vdots\\ H_{2}(F^{*})^{kN-q}(F^{*})^{(nr-1)N}\end{pmatrix} \text{is}\ n^{2}r\times nr,J= \begin{pmatrix}C_{q+1}^{ff}&C_{q+1}^{fs}\\ &\vdots\\ C_{q+nr}^{ff}&C_{q+nr}^{fs}\\ C_{kN}^{sf}&C_{kN}^{ss}\\ C_{(k+1)N}^{sf}&C_{(k+1)N}^{ss}\\ \vdots\\ C_{(k+nr-1)N}^{sf}&C_{(k+nr-1)N}^{ss}\end{pmatrix} \text{is}\ n^{2}r\times n,\]  
where \(k\) is an integer such that \(kN>q\). \(\theta_{b}\) is the submatrix formed with columns of \(\theta\), such that the \(i\)th column of \(\theta\) is in\(\theta_{b}\), if \(I\
eq 1\), \(\ldots\), \(n_{1}\), \(n+1\), \(\ldots\), \(n+n_{1}\), \(2n+1\), \(\ldots\), \(2n+n_{1}\), \(\ldots\), \((r-1)n+1\), \(\ldots\), \((r-1)n+n_{1}\) and also the \(i\)th row of \(X= \begin{pmatrix}C_{q}\\ C_{q-1}\\ \vdots\\ C_{q-r+1}\end{pmatrix} \) is not a row of \(C_{j}\) or \(C_{-j}\) with \(j\)\(\in\)\(\{0,N,\,2N,\,3N\ldots\}\). \(\theta_{a}\) is the submatrix of \(\theta\) with the columns that are not in \(\theta_{b}\). Below we set out the new conditions. Note that although [1] refers to a state space model associated with the VARMA model, we only need to note the algebraic properties that hold in certain matrices constructed from the parameters of the VARMA model. **Condition ii:** The VARMA model is regular with \(B_{0}=\mathrm{I}\) and \(\sum\) positive definite. **Condition iv.1:** rank\(C_{nr}(F,\,G^{*})=nr\). **Condition v.1:** rank \(C_{nr}(F,\,V^{*}(F^{t})^{r-q-1}H_{1}^{t})=\mathrm{nr}\) where  
\(V^{*}=\sum_{k=0}^{\infty}F^{k}\,G^{*}\sum(G^{*})^{t}(F^{t})^{k}=[C_{mp}(F,\,G^ {*}\!\!\sum^{1/2})\ldots]\)\([C_{mp}(F,\,G^{*}\!\!\sum^{1/2})\ldots]\), which exists because the model is stationary. **Condition vi.\(\theta_{b}\)** is full column rank. We will also make use of [11] (Theorem 3.1.3.2-1 (iii), Theorem 3.1.2.3-29 (iii) and Corollary 2.4.3-25), which we summarize in the following theorem that we call Hanzon's Theorem. **Hanzon's Theorem:** Considering \(K_{-i}=0\) if \(i>0\), for \(i,j,h\in\mathbb{N}\), we denote  
\[ \begin{split} M_{i,j,h}&=\begin{pmatrix}K_{i}&K_{i+1} \ \cdots&K_{i+h-1}\\ K_{i+1}&K_{i+2}\ \ddots&K_{i+h}\\ \vdots&\vdots\ddots&\vdots\\ K_{i+j-1}&\cdots&K_{i+j+h-2}\end{pmatrix}\text{and}\\ \mathbf{Q}_{i,j,h}&=\begin{pmatrix}C_{i}&C_{i+1}\ \cdots&C_{i+h-1}\\ C_{i+1}&C_{i+2}\ \ddots&C_{i+h}\\ \vdots&\vdots\ddots&\vdots\\ C_{i+j-1}&\cdots&C_{j+j+h-2}\end{pmatrix}\end{split} \]  
If model (1) is stationary, regular and miniphase, the following conditions are satisfied for any orders \(r\) and \(q\):  
1. \(\mathrm{rank}\ M_{q-r+1,\rho,\omega_{0}}=\mathrm{rank}\ M_{q-r+1,\rho,nr}\)
2. \(\mathrm{rank}\ Q_{q-r+1,r,\omega_{0}}=\mathrm{rank}\ Q_{q-r+1,r,\mu r}\)
3. \(\mathrm{rank}\ M_{q-r+1,r,\mu r}=\mathrm{rank}\ Q_{q-r+1,r,\mu r}\)  
As a consequence, we can deduce the following Lemma. Lemma 2: _Suppose \(r>q\) and Conditions I, II or ii, and III hold. Therefore, Condition IV implies Condition iv.1._  
Proof: We can easily see that  
\[K_{i}=HF^{i}G\text{ for }i\geq 0\]  
and \(M_{0,r,\mu r}=\mathrm{O}_{r}(F,H)\ C_{nr}(F,G)\). Taking into account that \(O_{r}(F,H)\) is full column rank \(nr\) and, from Condition IV, \(C_{nr}(F,G)\) is full row rank \(nr\), then \(\mathrm{rank}\ M_{0,r,\mu r}=nr\). By Hanzon's Theorem, \(\mathrm{rank}\ M_{0,r,\mu r}=\mathrm{rank}\ M_{0,r,\omega}=nr\), \(\mathrm{rank}\ M_{q-r+1,r,\mu r}=\mathrm{rank}\ M_{q-r+1,r,\omega_{0}}\).