Consider a random vector $\mathbf{y}=\mathbf{\Sigma}^{1/2}\mathbf{x}$, where the $p$ elements of the vector $\mathbf{x}$ are i.i.d. real-valued random variables with zero mean and finite fourth moment, and $\mathbf{\Sigma}^{1/2}$ is a deterministic $p\times p$ matrix such that the spectral norm of the population correlation matrix $\mathbf{R}$ of $\mathbf{y}$ is uniformly bounded. In this paper, we find that the log determinant of the sample correlation matrix $\hat{\mathbf{R}}$ based on a sample of size $n$ from the distribution of $\mathbf{y}$ satisfies a CLT (central limit theorem) for $p/n\to \gamma\in (0, 1]$ and $p\leq n$. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of $\mathbf{y}$ is unknown, we show that after recentering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. At last, the obtained findings are applied for testing of uncorrelatedness of $p$ random variables. Surprisingly, in the null case $\mathbf{R}=\mathbf{I}$, the test statistic becomes completely pivotal and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.

翻译：考虑随机向量 $\mathbf{y}=\mathbf{\Sigma}^{1/2}\mathbf{x}$，其中向量 $\mathbf{x}$ 的 $p$ 个元素为独立同分布实值随机变量，均值为零且具有有限四阶矩，$\mathbf{\Sigma}^{1/2}$ 为确定性 $p\times p$ 矩阵，使得 $\mathbf{y}$ 的总体相关矩阵 $\mathbf{R}$ 的谱范数一致有界。本文发现，基于来自 $\mathbf{y}$ 分布的样本容量为 $n$ 的样本所得到的样本相关矩阵 $\hat{\mathbf{R}}$ 的对数行列式，在 $p/n\to \gamma\in (0, 1]$ 且 $p\leq n$ 时满足中心极限定理。提供了渐近期望与方差的显式表达式。当 $\mathbf{y}$ 的均值未知时，我们证明采用经验均值重新中心化后，所得中心极限定理在渐近期望中发生偏移依然成立。这一结果对大维随机矩阵理论以及非正态数据大样本相关矩阵的高维统计文献均具有独立意义。最后，将所得结论应用于 $p$ 个随机变量不相关性的检验。令人惊讶的是，在原假设 $\mathbf{R}=\mathbf{I}$ 的情形下，检验统计量完全成为枢轴量，大量模拟表明即使四阶矩不存在，所得中心极限定理依然成立，这预示着一种针对重尾高维数据具有稳健性的检验统计量。