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:考虑一个随机矢量 $\ mathbf{y\\ mathbtrial{ mathbf{x} 美元, 其中矢量 $\ mathbf{ 1/2} 美元是 i.d. 实际估值随机变量, 零平均值和有限的第四秒为 美元; $\ mathbff =Sigma{ {y\\ mathbf} p 美元是一个确定值 $p\ timeptime 美元, 这样人口相关矩阵的光谱标准 $\ mathflim{ 1/2\\ mathbf{x} 美元是统一约束的 $\ mathbfrim{x} 美元 。 在本文件中, 我们发现, 样本关联矩阵的 $\ mexcialflical 的 $ preal deal deal deal deal demogradeal 值的逻辑决定值 $ $\ pleq pral demoal demo deal demodal dreal exfral exfral * 。 也显示, 最近的数值测试显示, 最近的数值和变的数值显示, 数字的数值显示, 数字的数值显示, 和数值的数值的数值显示的数值显示的数值显示,</s>