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Wang Zhou

Publications and source records attributed to Wang Zhou.

At least 73 records · Page 4Linked to original sources
arXiv2014

Test of Independence for High-dimensional Random Vectors Based on Block Correlation Matrices

In this paper, we are concerned with the independence test for $k$ high-dimensional sub-vectors of a normal vector, with fixed positive integer $k$. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is raised for this purpose. We then construct the so-called Schott type statistic as our test statistic, which turns out to be a particular linear spectral statistic of the block correlation matrix. Interestingly, the limiting behavior of the Schott type statistic can be figured out with the aid of the Free Probability Theory and the Random Matrix Theory. Specifically, we will bring the so-called real second order freeness for Haar distributed orthogonal matrices, derived in \cite{MP2013}, into the framework of this high-dimensional testing problem. Our test does not require the sample size to be larger than the total or any partial sum of the dimensions of the $k$ sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those of the statistics proposed in \cite{JY2013} and \cite{JBZ2013}, is satisfactory. Real data analysis is also used to illustrate our method.

math.ST↗
arXiv2014

Canonical correlation coefficients of high-dimensional normal vectors: finite rank case

Consider a normal vector $\mathbf{z}=(\mathbf{x}',\mathbf{y}')'$, consisting of two sub-vectors $\mathbf{x}$ and $\mathbf{y}$ with dimensions $p$ and $q$ respectively. With $n$ independent observations of $\mathbf{z}$ at hand, we study the correlation between $\mathbf{x}$ and $\mathbf{y}$, from the perspective of the Canonical Correlation Analysis, under the high-dimensional setting: both $p$ and $q$ are proportional to the sample size $n$. In this paper, we focus on the case that $Σ_{\mathbf{x}\mathbf{y}}$ is of finite rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_1\geq\cdots\geq r_k>0$. Under the additional assumptions $(p+q)/n\to y\in (0,1)$ and $p/q\not\to 1$, we study the sample counterparts of $r_i,i=1,\ldots,k$, i.e. the largest k eigenvalues of the sample canonical correlation matrix $S_{\mathbf{x}\mathbf{x}}^{-1}S_{\mathbf{x}\mathbf{y}}S_{\mathbf{y}\mathbf{y}}^{-1}S_{\mathbf{y}\mathbf{x}}$, namely $λ_1\geq\cdots\geq λ_k$. We show that there exists a threshold $r_c\in(0,1)$, such that for each $i\in\{1,\ldots,k\}$, when $r_i\leq r_c$, $λ_i$ converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by $d_r$. When $r_i>r_c$, $λ_i$ possesses an almost sure limit in $(d_r,1]$, from which we can recover $r_i$ in turn, thus provide an estimate of the latter in the high-dimensional scenario.

math.ST↗
arXiv2013

Universality for a global property of the eigenvectors of Wigner matrices

Let $M_n$ be an $n\times n$ real (resp. complex) Wigner matrix and $U_nΛ_n U_n^*$ be its spectral decomposition. Set $(y_1,y_2...,y_n)^T=U_n^*x$, where $x=(x_1,x_2,...,$ $x_n)^T$ is a real (resp. complex) unit vector. Under the assumption that the elements of $M_n$ have 4 matching moments with those of GOE (resp. GUE), we show that the process $X_n(t)=\sqrt{\frac{βn}{2}}\sum_{i=1}^{\lfloor nt\rfloor}(|y_i|^2-\frac1n)$ converges weakly to the Brownian bridge for any $\mathbf{x}$ such that $||x||_\infty\rightarrow 0$ as $n\rightarrow \infty$, where $β=1$ for the real case and $β=2$ for the complex case. Such a result indicates that the othorgonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthorgonal (resp. unitary) group from a certain perspective.

math.PR↗
arXiv2013

SLE curves and natural parametrization

Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_κ$ curves is well defined for all $κ<8$. Our proof uses a two-interior-point local martingale.

math.PR↗
arXiv2013

Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice

In this paper we present the proof of the convergence of the critical bond percolation exploration process on the square lattice to the trace of SLE$_{6}$. This is an important conjecture in mathematical physics and probability. The case of critical site percolation on the hexagonal lattice was established in the seminal work of Smirnov via proving Cardy's formula. Our proof uses a series of transformations and conditioning to construct a pair of paths: the $+\partial$CBP and the $-\partial$CBP. The convergence in the site percolation case on the hexagonal lattice allows us to obtain certain estimates on the scaling limit of the $+\partial$CBP and the $-\partial$CBP. By considering a path which is the concatenation of $+\partial$CBPs and $-\partial$CBPs in an alternating manner, we can prove the convergence in the case of bond percolation on the square lattice.

math.PR↗
arXiv2012

Central limit theorem for partial linear eigenvalue statistics of Wigner matrices

In this paper, we study the complex Wigner matrices $M_n=\frac{1}{\sqrt{n}}W_n$ whose eigenvalues are typically in the interval $[-2,2]$. Let $λ_1\leq λ_2...\leqλ_n$ be the ordered eigenvalues of $M_n$. Under the assumption of four matching moments with the Gaussian Unitary Ensemble(GUE), for test function $f$ 4-times continuously differentiable on an open interval including $[-2,2]$, we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold $u$ in the bulk of the Wigner semicircle law as $\mathcal{A}_n[f; u]=\sum_{l=1}^nf(λ_l)\mathbf{1}_{\{λ_l\leq u\}}$. And the second one is $\mathcal{B}_n[f; k]=\sum_{l=1}^{k}f(λ_l)$ with positive integer $k=k_n$ such that $k/n\rightarrow y\in (0,1)$ as $n$ tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from $\mathcal{B}_n[f; \lfloor nt\rfloor]$.

math.PR↗
arXiv2011

Tracy-Widom law for the extreme eigenvalues of sample correlation matrices

Let the sample correlation matrix be $W=YY^T$, where $Y=(y_{ij})_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{\sum_{j=1}^nx_{ij}^2}$. We assume $\{x_{ij}: 1\leq i\leq p, 1\leq j\leq n\}$ to be a collection of independent symmetric distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij}, 1\leq j\leq n$ to be identically distributed. We assume $0<p<n$ and $p/n\rightarrow y$ with some $y\in(0,1)$ as $p,n\rightarrow\infty$. In this paper, we provide the Tracy-Widom law ($TW_1$) for both the largest and smallest eigenvalues of $W$. If $x_{ij}$ are i.i.d. standard normal, we can derive the $TW_1$ for both the largest and smallest eigenvalues of the matrix $\mathcal{R}=RR^T$, where $R=(r_{ij})_{p,n}$ with $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.

math.ST↗
arXiv2011

Generalized four-point characterization method for resistive and capacitive contacts

In this paper, a four-point characterization method is developed for resistive samples connected to either resistive or capacitive contacts. Provided the circuit equivalent of the complete measurement system is known including coaxial cable and connector capacitances as well as source output and amplifier input impedances, a frequency range and capacitive scaling factor can be determined, whereby four-point characterization can be performed. The technique is demonstrated with a discrete element test sample over a wide frequency range using lock-in measurement techniques from 1 Hz - 100 kHz. The data fit well with a circuit simulation of the entire measurement system. A high impedance preamplifier input stage gives best results, since lock-in input impedances may differ from manufacturer specifications. The analysis presented here establishes the utility of capacitive contacts for four-point characterizations at low frequency.

cond-mat.other↗
arXiv2011

New estimators of spectral distributions of Wigner matrices

We introduce kernel estimators for the semicircle law. In this first part of our general theory on the estimators, we prove the consistency and conduct simulation study to show the performance of the estimators. We also point out that Wigner's semicircle law for our new estimators and the classical empirical spectral distributions is still true when the elements of Wigner matrices don't have finite variances but are in the domain of attraction of normal law.

math-ph↗
arXiv2011

A Note on Rate of Convergence in Probability to Semicircular Law

In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability is $O(n^{-1/2})$ when the dimension $n$ tends to infinity.

math.PR↗
arXiv2010

Functional CLT for sample covariance matrices

Using Bernstein polynomial approximations, we prove the central limit theorem for linear spectral statistics of sample covariance matrices, indexed by a set of functions with continuous fourth order derivatives on an open interval including $[(1-\sqrt{y})^2,(1+\sqrt{y})^2]$, the support of the Marucenko--Pastur law. We also derive the explicit expressions for asymptotic mean and covariance functions.

math.ST↗
arXiv2009

On normal approximations to $U$-statistics

Let ${X_1,...,X_n}$ be i.i.d. random observations. Let $\mathbb{S}=\mathbb{L}+\mathbb{T}$ be a $U$-statistic of order $k\ge2$ where $\mathbb{L}$ is a linear statistic having asymptotic normal distribution, and $\mathbb{T}$ is a stochastically smaller statistic. We show that the rate of convergence to normality for $\mathbb{S}$ can be simply expressed as the rate of convergence to normality for the linear part $\mathbb{L}$ plus a correction term, $(\operatorname {var}\mathbb{T})\ln^2(\operatorname {var}\mathbb{T})$, under the condition ${\mathbb{E}\mathbb{T}^2<\infty}$. An optimal bound without this $\log$ factor is obtained under a lower moment assumption ${\mathbb {E}|\mathbb{T}|^α<\infty}$ for ${α<2}$. Some other related results are also obtained in the paper. Our results extend, refine and yield a number of related-known results in the literature.

math.PR↗
arXiv2007

Boundary proximity of SLE

This paper examines how close the chordal $\SLE_κ$ curve gets to the real line asymptotically far away from its starting point. In particular, when $κ\in(0,4)$, it is shown that if $β>β_κ:=1/(8/κ-2)$, then the intersection of the $\SLE_κ$ curve with the graph of the function $y=x/(\log x)^β$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $β=β_κ$. The critical $\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\log\log x)^α}$, $x>e^e$, in an unbounded set if $α\le 1$, but not if $α>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient integrability condition for the intersection of the $\SLE_κ$ path with the graph of $y$ to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the $\SLE_κ$ curve and real axis is $2-8/κ$ when $4<κ<8$.

math.PR↗
arXiv2007

Circular law, Extreme Singular values and Potential theory

Consider the empirical spectral distribution of complex random $n\times n$ matrix whose entries are independent and identically distributed random variables with mean zero and variance $1/n$. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.

math.PR↗
arXiv2007

Uniformly bounded components of normality

Suppose that $f(z)$ is a transcendental entire function and that the Fatou set $F(f)\neq\emptyset$. Set $$B_1(f):=\sup_{U}\frac{\sup_{z\in U}\log(|z|+3)}{\inf_{w\in U}\log(|w|+3)}$$ and $$B_2(f):=\sup_{U}\frac{\sup_{z\in U}\log\log(|z|+30)}{\inf_{w\in U}\log(|w|+3)},$$ where the supremum $\sup_{U}$ is taken over all components of $F(f)$. If $B_1(f)<\infty$ or $B_2(f)<\infty$, then we say $F(f)$ is strongly uniformly bounded or uniformly bounded respectively. In this article, we will show that, under some conditions, $F(f)$ is (strongly) uniformly bounded.

math.CV↗