Large Dimensional Matrix-Valued Stochastic Processes

This paper is a reading report on matrix-valued stochastic process,mainly including the collision properties of matrix eigenvalues and the stochastic differential equation satisfied,the law of large numbers of empirical measures of eigenvalues and the central limit theorem.After a brief introduction to the historical background and research status of random matrix theory and Matrix valued stochastic process,this paper discusses the stochastic process of large dimension matrix values through three parts.In the second chapter,we study Dyson’s Brownian motion in detail.We prove the eigenvalue processes never collide almost surely and are characterized by a system of stochastic differential equations(SDEs),we derive the weak convergence and limit characterization and the high-dimensional limit of empirical measures is extended to two different types of matrix-valued random processes.Finally,we show that the fluctuation of the empirical measure of Dyson’s Brownian motion around its limit measure can be characterized by a Gaussian process.In chapter 3,we give some properties of eigenvalues of Wishart process and Laguerre process.At the same time,we introduce a more general real symmetric matrixvalued random process and study the properties of eigenvalues and the high-dimensional limit of empirical measures,this real symmetric matrix is the general form of some matrix models listed previously.In chapter 4,we consider matrix-valued stochastic process driven by fractional Brownian motion,and give the properties of eigenvalues corresponding to the matrix fractional Brownian motion and fractional Wishart process with Hurst parameters,as well as their normalized empirical measures converge weakly to the semicircular Law and M-P law,respectively.