Asymptotic normality for traces of polynomials in independent complex Wishart matrices

An Introduction to Wishart Matrix Moments

These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matr…

正定矩阵锥上强混合时间序列的Wishart核密度估计

Wishart kernel density estimation for strongly mixing time series on the cone of positive definite matrices

提出Wishart核密度估计器用于正定矩阵锥上的密度估计，该估计器具有边界感知性，能缓解边界偏差，并在混合条件下建立了均方误差、一致强相合性和渐近正态性，模拟和实例表明其优于其他方法。

Partial transpose of random quantum states: exact formulas and meanders

We investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by approximating the matrix of trace 1 by the Wishart matrix of random trace) and shown to be the semicircular distribution or the free difference of two free Poisson distributions, depending on how dimensions of the concerned spaces grow. Our use of Wishart matrices gives exact combinatorial formulas for the moments of the partial transpose of the random state. We find …

嵌套复Wishart极值的单侧乘积尺度上界

One-Sided Product-Scale Upper Bounds for Nested Complex Wishart Extremes

本文针对嵌套复Wishart极值的最大特征值软边缘，证明了其上、下尾联合概率的有限N单侧乘积尺度上界，推导了分离稀疏网格的矩估计，适用于超临界分离场景。

Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices

Let $\mathbf{X}\in\mathbb{C}^{n\times m}$ ($m\geq n$) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and {\it single-spiked} covariance matrix $\mathbf{I}_n+ η\mathbf{u}\mathbf{u}^*$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, {\color{blue}$\mathbf{u}\in\mathbb{C}^{n\times 1}$} is an arbitrary vector with unit Euclidean norm, $η\geq 0$ is a non-random parameter, and $(\cdot)^*$ represents the conjugate-transpose. This paper investigates the distribution of the random quantity $κ_{\text{SC}}^2(\mathbf{X})=\sum_{k=1}^n λ…

具有奇异排斥作用的相互作用粒子系统的漂移参数估计

Drift parameter estimation for interacting particle systems with singular repulsion

针对奇异相互作用粒子系统，本文通过将粒子动力学约化为一维随机微分方程，构造了线性形式的显式漂移参数估计量，证明了其在不同相互作用核下的一致性与渐近正态性，并通过实例验证了方法的适用性。

Explicit Multi-Matrix Topological Expansion for Quaternionic Random Matrices

We present an explicit formula for the expected value of a product of several independent symplectically invariant matrices in which the trace and real part function may be applied, possibly to different subexpressions. This takes the form of a topological expansion; however, each term has two topologies: one for the trace, and another for the real part. The traces and real parts can always be written in terms of index contraction, but in some cases, it is possible to write the expression as a product in which the two functions are applied to bracketed intervals in a legal bracket diagram. We …

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line.…