The asymptotic trace norm of random circulants and the graph energy

大型随机矩形Toeplitz矩阵范数的精确渐近性

Precise asymptotics for the norm of large random rectangular Toeplitz matrices

该研究分析了独立元素的大型矩形随机Toeplitz和circulant矩阵的谱范数渐近行为，推导了对应极限结果，通过蒙特卡洛实验验证有限样本性能，并在高斯情形下开展高阶渐近分析提出极限分布猜想。

Dull cut off for circulants

Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with …

某些模式矩阵线性特征值统计量的时间相关涨落

Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices

本文研究元素为独立 Brown 运动的 RC_n(t) 与 SC_n(t) 矩阵，证明多项式检验函数下其线性特征值统计量的时间相关涨落随 n→∞ 具有过程收敛性。

大型随机矩形Toeplitz与循环矩阵的最小奇异值

The smallest singular value of large random rectangular Toeplitz and circulant matrices

本文研究大型随机矩形Toeplitz与循环矩阵的最小奇异值，证明其最小特征值收敛于零，并给出收敛速率的下界（快于多对数但慢于多项式）及矩形循环矩阵的多项式上界。

Asymptotic results for empirical measures of weighted sums of independent random variables

We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This implies almost sure convergence of empirical periodograms, almost sure convergence of spectra of circulant and reverse circulant matrices, and almost sure convergence of the CDF's generated from independent random variables by independent random orthogonal matrices. For special trigonometric …

高斯部分循环矩阵的改进RIP界

Improved RIP Bounds for Gaussian Partial Circulant Matrices

该研究针对高斯部分循环矩阵，改进了其受限等距（RIP）界的理论结果，通过优化证明步骤将相关界中的一个对数因子替换，提升了RIP界的紧度。

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one…

Large dimensional random k circulants

Consider random k-circulants A_{k,n} with n tends to infinity, k=k(n) and whose input sequence \{a_l\}_{l \ge 0} is independent with mean zero and variance one and \sup_n n^{-1}\sum_{l=1}^n \E |a_l|^{2+δ}< \infty for some δ> 0. Under suitable restrictions on the sequence \{k(n)\}_{n \ge 1}, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists and identify the limits. In particular, we prove the following: Suppose g \ge 1 is fixed and p_1 is the smallest prime divisor of g. Suppose P_g=\prod_{j=1}^g E_j where \{E_j\}_{1 \le j …

Asymptotic normality for traces of polynomials in independent complex Wishart matrices

We derive a non-asymptotic expression for the moments of traces of monomials in several independent complex Wishart matrices, extending some explicit formulas available in the literature. We then deduce the explicit expression for the cumulants. From the latter, we read out the multivariate normal approximation to the traces of finite families of polynomials in independent complex Wishart matrices.

Multivariate normal approximation for traces of random unitary matrices

In this article, we obtain a super-exponential rate of convergence in total variation between the traces of the first $m$ powers of an $n\times n$ random unitary matrices and a $2m$-dimensional Gaussian random variable. This generalizes previous results in the scalar case to the multivariate setting, and we also give the precise dependence on the dimensions $m$ and $n$ in the estimate with explicit constants. We are especially interested in the regime where $m$ grows with $n$ and our main result basically states that if $m\ll \sqrt{n}$, then the rate of convergence in the Gaussian approximatio…