Covariance Matrix Estimation in Time Series and Its Applications

这个项目的目标是建立一个系统的渐近理论，用于估计时间序列中的高维协方差矩阵；这是高维推理中的一个基本问题。特别是，研究者计划研究平稳过程的样本协方差和样本协方差矩阵的性质；处理平稳过程协方差矩阵的一致性估计及其在预测等问题中的应用；通过估计高阶累积量张量来探索随机过程的非高斯特征。协方差矩阵在环境科学、工程、经济和金融等各个领域中发挥着重要的作用。在对地震、经济、金融等时间序列进行分析、检验、监测和预测时，需要对协方差矩阵进行估计。这个项目的结果可以为协方差矩阵的估计提供理论基础，并可能改进在各种应用中使用的时间序列处理算法。

The goal of this project is to establish a systematic asymptotic theory for estimates of large dimensional covariance matrices in time series; a fundamental problem in high-dimensional inference. In particular, the investigator plans to study properties of sample covariances and sample covariance matrices for stationary processes; deal with consistent estimation of covariance matrices of stationary processes and its applications in prediction and other problems; and explore non-Gaussian features of random processes by estimating higher order cumulant tensors.Covariance matrices play a fundamental role in various fields including environmental science, engineering, economics and finance. Estimation of covariance matrices is needed in analyzing, testing, monitoring and predicting of seismic, economic and financial and other time series. Results developed from this project can provide a theoretical foundation for estimation of covariance matrices and can potentially improve time series processing algorithms that are used in various applications.