In random matrix theory, the spectral distribution of the covariance matrix has been well studied under the large dimensional asymptotic regime when the dimensionality and the sample size tend to infinity at the same rate. However, most existing theories are built upon the assumption of independent and identically distributed samples, which may be violated in practice. For example, the observational data of continuous-time processes at discrete time points, namely, the high-frequency data. In this paper, we extend the classical spectral analysis for the covariance matrix in large dimensional random matrix to the spot volatility matrix by using the high-frequency data. We establish the first-order limiting spectral distribution and obtain a second-order result, that is, the central limit theorem for linear spectral statistics. Moreover, we apply the results to design some feasible tests for the spot volatility matrix, including the identity and sphericity tests. Simulation studies justify the finite sample performance of the test statistics and verify our established theory.

翻译：在随机矩阵理论中，当维度和样本量以相同速率趋于无穷时，协方差矩阵的谱分布在渐近意义下已得到充分研究。然而，现有理论大多建立在样本独立同分布的假设之上，这一假设在实践中可能不成立。例如，连续时间过程在离散时间点上的观测数据，即高频数据。本文利用高频数据，将经典的大维随机矩阵中协方差矩阵的谱分析推广至瞬时波动率矩阵。我们建立了一阶极限谱分布，并获得了二阶结果，即线性谱统计量的中心极限定理。此外，我们将结果应用于设计针对瞬时波动率矩阵的若干可行检验，包括单位阵检验和球性检验。模拟研究验证了检验统计量的有限样本表现，并证实了我们建立的理论。