【作者】 张丽红；

【导师】 丁雪；

【作者基本信息】 吉林大学 ， 概率论与数理统计， 2024， 硕士

【摘要】 随机矩阵理论是概率统计研究领域的一个重要分支,不仅涉及到高维空间的几何和代数结构,还关联到概率论和统计学的基本问题,在物理学、统计学、工程学和生物学等多个领域都有广泛的应用.随机矩阵理论的起源和发展与量子力学的理论和实验密切相关.随机矩阵理论的研究可以追溯到量子力学在20世纪40年代和50年代初期的发展,在量子力学领域,系统的能级由希尔伯特空间中的埃尔米特算子A的特征值来描述,为了避免处理无穷维的算子,物理学家们需要对埃尔米特算子进行一些处理,使得A成为一个有限但高维的随机线性算子,即高维随机矩阵.因此,高维随机矩阵性质的研究引起了量子力学专家的关注,随后提出了一些极限定理.在20世纪50年代后期,概率论和统计学家们对高维随机矩阵的极限性质的研究产生了极大的兴趣,随机矩阵理论的研究开始蓬勃发展.随机矩阵是由随机变量构成的矩阵,这些随机变量作为矩阵的元素,可以是独立同分布的,或者是具有某种相关性的.在随机矩阵理论中,大维spiked模型是一个特别重要的模型,它在信号处理和经济学等多个学科领域都有广泛的应用.样本协方差(相关矩阵)的spiked模型是指总体的协方差矩阵或者相关矩阵的特征值除了少数非零非一的spiked特征值外,其余的均为1.spiked特征值个数的估计是许多学科领域一个基本而关键的问题.本文的主要研究内容是在大维随机矩阵理论的基础上,假设总体相关矩阵是符合大维spiked模型的情况下,考虑样本相关矩阵的特征值的收敛情况以及其应用,即在大维spiked模型下,根据样本相关矩阵的谱分布的性质来估计这个大维spiked模型中的spiked特征值的个数.当样本维数和样本容量都趋于无穷大,但其比值趋于一个常数时,本文给出了一个基于两个连续样本特征值之差的估计方法.然后,本文通过数值实验验证了文中给出的理论结果.更多还原

【Abstract】 Random matrix theory is an important branch of probability and statistics research,which not only involves the geometric and algebraic structures of high dimensional spaces,but also relates to the fundamental problems of probability theory and statistics.It has wide applications in many fields such as physics,statistics,engineering,and biology.The origin and development of random matrix theory are closely related to the theory and experiment of quantum mechanics.The research of random matrix theory can be traced back to the development of quantum mechanics in the 1940 s and early 1950 s.In the field of quantum mechanics,the energy levels of the system are described by the eigenvalues of the Hermitian operator A in the Hilbert space.In order to avoid dealing with infinite dimensional operators,physicists need to process the Hermitian operator,making A a finite but high dimensional random linear operator,that is,a high dimensional random matrix.Therefore,the study of the properties of high dimensional random matrices attracted the attention of quantum mechanics experts,and then some limit theorems were proposed.In the late 1950 s,probabilists and statisticians became very interested in the study of the limit properties of high dimensional random matrices,and random matrix theory began to flourish.Random matrices are matrices whose elements are random variables,which can be independently and identically distributed or possess some form of correlation.Within the theory of random matrices,the large dimensional spiked model is particularly significant and has found broad applications across various disciplines,including signal processing and economics.The spiked model for sample covariance(correlation matrix)refers to a situation where,apart from a few non-zero and non-unit ”spiked” eigenvalues,all other eigenvalues of the population covariance(or correlation)matrix are equal to 1.Estimating the number of spiked eigenvalues is a fundamental and critical issue in many fields of study.The main research content of this article is grounded in the theory of large dimensional random matrices.Assuming the population correlation matrix conforms to the large dimensional spiked model,the paper investigates the convergence of the eigenvalues of the sample correlation matrix and its application.Specifically,under the large dimensional spiked model,the number of spike eigenvalues is estimated based on the spectral distribution properties of the sample correlation matrix.As both the sample dimension and size approach infinity with a constant ratio,the paper provides an estimation method based on the differences between consecutive sample eigenvalues.The theoretical results are then verified through numerical experiments.更多还原