"Toeplitzification" or "redundancy (spatial) averaging", the well-known routine for deriving the Toeplitz covariance matrix estimate from the standard sample covariance matrix, recently regained new attention due to the important Random Matrix Theory (RMT) findings. The asymptotic consistency in the spectral norm was proven for the Kolmogorov's asymptotics when the matrix dimension N and independent identically distributed (i.i.d.) sample volume T both tended to infinity (N->inf, T->inf, T/N->c > 0). These novel RMT results encouraged us to reassess the well-known drawback of the redundancy averaging methodology, which is the generation of the negative minimal eigenvalues for covariance matrices with big eigenvalues spread, typical for most covariance matrices of interest. We demonstrate that for this type of Toeplitz covariance matrices, convergence in the spectral norm does not prevent the generation of negative eigenvalues, even for the sample volume T that significantly exceeds the covariance matrix dimension (T >> N). We demonstrate that the ad-hoc attempts to remove the negative eigenvalues by the proper diagonal loading result in solutions with the very low likelihood. We demonstrate that attempts to exploit Newton's type iterative algorithms, designed to produce a Hermitian Toeplitz matrix with the given eigenvalues lead to the very poor likelihood of the very slowly converging solution to the desired eigenvalues. Finally, we demonstrate that the proposed algorithm for restoration of a positive definite (p.d.) Hermitian Toeplitz matrix with the specified Maximum Entropy spectrum, allows for the transformation of the (unstructured) Hermitian maximum likelihood (ML) sample matrix estimate in a p.d. Toeplitz matrix with sufficiently high likelihood.

翻译：“托普利茨化”或“冗余（空间）平均”——从标准样本协方差矩阵推导托普利茨协方差矩阵估计的经典方法——近期因随机矩阵理论（RMT）的重要发现而重新受到关注。在柯尔莫哥洛夫渐近条件下，当矩阵维度N与独立同分布（i.i.d.）样本量T均趋于无穷大（N→∞, T→∞, T/N→c>0）时，该方法在谱范数下的渐近一致性已被证明。这些新的RMT结果促使我们重新审视冗余平均方法的已知缺陷：对于具有大特征值展宽（典型于大多数感兴趣的协方差矩阵）的协方差矩阵，该方法会产生负的最小特征值。我们证明，对于此类托普利茨协方差矩阵，即使在样本量T显著超过协方差矩阵维度（T>>N）的情况下，谱范数下的收敛性也无法阻止负特征值的产生。我们证明，通过适当对角加载来移除负特征值的临时性尝试会导致似然极低的解。我们进一步证明，利用牛顿型迭代算法（旨在生成具有指定特征值的埃尔米特托普利茨矩阵）会导致解收敛到期望特征值的过程极为缓慢，且似然很低。最后，我们证明，所提出的用于恢复具有指定最大熵谱的正定（p.d.）埃尔米特托普利茨矩阵的算法，能够将（无结构的）埃尔米特最大似然（ML）样本矩阵估计转化为具有足够高似然的p.d.托普利茨矩阵。