Given a random sample from a multivariate normal distribution whose covariance matrix is a Toeplitz matrix, we study the largest off-diagonal entry of the sample correlation matrix. Assuming the multivariate normal distribution has the covariance structure of an auto-regressive sequence, we establish a phase transition in the limiting distribution of the largest off-diagonal entry. We show that the limiting distributions are of Gumbel-type (with different parameters) depending on how large or small the parameter of the autoregressive sequence is. At the critical case, we obtain that the limiting distribution is the maximum of two independent random variables of Gumbel distributions. This phase transition establishes the exact threshold at which the auto-regressive covariance structure behaves differently than its counterpart with the covariance matrix equal to the identity. Assuming the covariance matrix is a general Toeplitz matrix, we obtain the limiting distribution of the largest entry under the ultra-high dimensional settings: it is a weighted sum of two independent random variables, one normal and the other following a Gumbel-type law. The counterpart of the non-Gaussian case is also discussed. As an application, we study a high-dimensional covariance testing problem.

翻译：给定来自协方差矩阵为Toeplitz矩阵的多元正态分布的随机样本，我们研究样本相关矩阵的最大非对角元。假设多元正态分布具有自回归序列的协方差结构，我们建立了最大非对角元极限分布中的相变现象。我们证明，极限分布为Gumbel型（具有不同参数），具体形式取决于自回归序列参数的大小。在临界情形下，我们得到极限分布为两个独立Gumbel分布随机变量的最大值。这一相变确立了自回归协方差结构与单位矩阵协方差结构行为差异的精确阈值。假设协方差矩阵为一般Toeplitz矩阵，我们在超高维设定下获得最大元的极限分布：它是两个独立随机变量的加权和，其中一个服从正态分布，另一个服从Gumbel型分布。同时讨论了非高斯情形的对应结果。作为应用，我们研究了一个高维协方差检验问题。