Asymptotic Results of a Multiple-entry Reinforcement Process

高斯随机矩阵的渐近自由独立性与元素置换：第二部分：无穷小自由

Asymptotic free independence and entry permutations for Gaussian random matrices. Part II: Infinitesimal freeness

本文研究了具有置换元素的GUE矩阵的渐近无穷小分布，证明了随机均匀置换下GUE矩阵的零无穷小分布及不同置换间的无穷小自由性，并指出GUE矩阵与其转置非无穷小自由。

On Jiang's asymptotic distribution of the largest entry of a sample correlation matrix

Let $ \{X, X_{k,i}; i \geq 1, k \geq 1 \}$ be a double array of nondegenerate i.i.d. random variables and let $\{p_{n}; n \geq 1 \}$ be a sequence of positive integers such that $n/p_{n}$ is bounded away from $0$ and $\infty$. This paper is devoted to the solution to an open problem posed in Li, Liu, and Rosalsky (2010) on the asymptotic distribution of the largest entry $L_{n} = \max_{1 \leq i < j \leq p_{n}} \left | \hatρ^{(n)}_{i,j} \right |$ of the sample correlation matrix ${\bf Γ}_{n} = \left ( \hatρ_{i,j}^{(n)} \right )_{1 \leq i, j \leq p_{n}}$ where $\hatρ^{(n)}_{i,j}$ denotes the Pea…

Asymptotics of a two-dimensional sticky random walk

We study the asymptotic behavior of a Markov chain on $\mathbb{Z}^2$ that corresponds to the two-dimensional marginals of a reinforcement process on $\mathbb{Z}^{\mathbb{N}}$. Three distinct asymptotic regimes are identified, depending on the scaling of the reinforcement parameter $Δ$ with respect to the number of steps performed by the chain.

依赖结构下最大 Pearson 相关系数的渐近分布

Asymptotic Distributions of Largest Pearson Correlation Coefficients under Dependent Structures

本文研究 Toeplitz 依赖结构下样本相关矩阵最大非对角元的渐近分布，揭示自回归参数导致的 Gumbel 型相变，并推广至一般超高维与非高斯情形及协方差检验。

Extremal Laws for Laplacian Random Matrices

For an $n\times n$ Laplacian random matrix $L$ with Gaussian entries it is proven that the fluctuations of the largest eigenvalue and the largest diagonal entry of $L/\sqrt{n-1}$ are Gumbel. We first establish suitable non-asymptotic estimates and bounds for the largest eigenvalue of $L$ in terms of the largest diagonal element of $L$. An expository review of existing results for the asymptotic spectrum of a Laplacian random matrix is also presented, with the goal of noting the differences from the corresponding classical results for Wigner random matrices. Extensions to Laplacian block random…

Limiting behavior of largest entry of random tensor constructed by high-dimensional data

Let ${X}_{k}=(x_{k1}, \cdots, x_{kp})', k=1,\cdots,n$, be a random sample of size $n$ coming from a $p$-dimensional population. For a fixed integer $m\geq 2$, consider a hypercubic random tensor $\mathbf{T}$ of $m$-th order and rank $n$ with \begin{eqnarray*} \mathbf{T}= \sum_{k=1}^{n}\underbrace{{X}_{k}\otimes\cdots\otimes {X}_{k}}_{m~multiple}=\Big(\sum_{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\cdots x_{ki_{m}}\Big)_{1\leq i_{1},\cdots, i_{m}\leq p}. \end{eqnarray*} Let $W_n$ be the largest off-diagonal entry of $\mathbf{T}$. We derive the asymptotic distribution of $W_n$ under a suitable normalization…