Nonsingularity of the reduced covariance-bias weight matrix

Prove that the reduced weight matrix \(\mathbf{W}^{*}_{\underline{n},\underline{m}}\), associated with the bias-corrected covariance estimator for a stationary spatial lattice process on an \(n_1\times n_2\) grid, is nonsingular whenever the cutoff parameters satisfy \(m_i\leq n_i-2\) for \(i\in\{1,2\}\).

Background

The paper constructs a bias-correction procedure for spatial covariance estimators by selecting a central set of lags $|h_i|\leq m_i$ and inverting the corresponding reduced weight matrix $\mathbf{W}^{*}_{\underline{n},\underline{m}}$. A sufficient condition established in Lemma $\ref{lemma:invertibility_W_nm}$ is $M_{\underline{m}}/N<1$, where $M_{\underline{m}}=\prod_{i=1}^2(2m_i+1)$ is the number of selected lags and $N=n_1n_2$ is the sample size.

The authors note that this sufficient condition is stricter than the admissible range $m_i\leq n_i-2$. Numerical evaluations for lattice dimensions $3\leq n_i\leq40$ found the reduced weight matrix nonsingular throughout the full admissible range, motivating the explicitly stated conjecture that nonsingularity holds whenever $m_i\leq n_i-2$ for both coordinate directions. Establishing this would remove the need for the stricter sufficient condition when applying the bias correction.