矩阵鞅的集中不等式及其应用

最近，关于随机矩阵的研究方兴未艾，随机矩阵最大奇异值是很多随机矩阵学者关心的课题。然而，关于随机矩阵最大奇异值的集中不等式却没有被充分研究。集中不等式是概率论，特别是概率极限理论中的重要研究课题。申请人前期研究发现，随时间变化随机系统可以被随机矩阵描述，因此本项目将研究矩阵鞅最大奇异值的集中不等式。由于连续时间纯断鞅不具有连续局部鞅部分，研究其指数型集中不等式有一定的意义。本项目主要将研究三类连续时间矩阵纯断鞅最大奇异值的集中不等式，并且将研究应用至矩阵值随机过程特征根的经验测度的胎紧性研究，在一定程度上促进随机矩阵的研究。

The field of random matrix theory has witnessed a remarkable development in the past decades, with abundant applications applied in mathematics, physics, statistics and engineering. This project mainly focuses on one particular issue-the maximum singular value of random matrix, which has received burgeoning interests among the scholars. Previous findings of applicants showed that random matrix can also be used to describe time-varying random system. Such results have enlightened us on a combination of probability limit theory with random matrices by exploring the concentration inequalities of the maximum singular value when the matrices is endowed with martingale property. Since continuous-time pure discontinuous martingale does not have continuous local martingale part, it is significant to study its exponential concentration inequality. In this project, three types of concentration inequalities for the maximum singular value of matrix pure discontinuous martingales will be investigated and applied to the tightness study of the empirical measure of eigenvalues of matrix-valued stochastic process.