Universality for the largest eigenvalue of sample covariance matrices with general population

This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form $\mathcal{W}_N=\Sigma^{1/2}XX^*\Sigma ^{1/2}$. Here, $X=(x_{ij})_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij},1\leq i\leq M,1\leq j\leq N$ such that $\mathbb{E}x_{ij}=0$, $\mathbb{E}|x_{ij}|^2=1/N$. On dimensionality, we assume that $M=M(N)$ and $N/M\rightarrow d\in(0,\infty)$ as $N\rightarrow\infty$. For a class of general deterministic positive-definite $M\times M$ matrices $\Sigma$, under some additional assumptions on the distribution of $x_{ij}$'s, we show that the limiting behavior of the largest eigenvalue of $\mathcal{W}_N$ is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd\H{o}s, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case ($\Sigma=I$). Consequently, in the standard complex case ($\mathbb{E}x_{ij}^2=0$), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of $\mathcal{W}_N$ converges weakly to the type 2 Tracy-Widom distribution $\mathrm{TW}_2$. Moreover, in the real case, we show that when $\Sigma$ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit $\mathrm{TW}_1$ holds for the normalized largest eigenvalue of $\mathcal {W}_N$, which extends a result of F\'{e}ral and P\'{e}ch\'{e} in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1281 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 $U_n\Lambda_n U_n^*$ 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 $X_n(t)=\sqrt{\frac{\beta n}{2}}\sum_{i=1}^{\lfloor nt\rfloor}(|y_i|^2-\frac1n)$ converges weakly to the Brownian bridge for any $\mathbf{x}$ such that $||x||_\infty\rightarrow 0$ as $n\rightarrow \infty$, where $\beta=1$ for the real case and $\beta=2$ 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.Comment: typos correcte

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 $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.Comment: 35 pages, a major revisio

Modeling the Resupply, Diffusion, and Evaporation of Cesium on the Surface of Controlled Porosity Dispenser Photocathodes

High quantum efficiency (QE) photocathodes are useful for many accelerator applications requiring high brightness electron beams, but suffer from short operational lifetime due to QE decay. For most photocathodes, the decrease in QE is primarily attributed to the loss of a cesium layer at the photocathode surface during operation. The development of robust, long life, high QE photoemitters is critically needed for applications demanding high brightness electron sources. To that end, a controlled porosity dispenser (CPD) photocathode is currently being explored and developed to replace the cesium during operation and increase photocathode lifetime. A theoretical model of cesium resupply, diffusion, and evaporation on the surface of a sintered wire CPD photocathode is developed to understand and optimize the performance of future controlled porosity photocathodes. For typical activation temperatures within the range of 500K--750K, simulation found differences of less than 5 % between the quantum efficiency (QE) maximum and minimum over ideal homogenous surfaces. Simulations suggest more variation for real cases to include real surface non uniformity. The evaporation of cesium from a tungsten surface is modeled using an effective one-dimensional potential well representation of the binding energy. The model accounts for both local and global interactions of cesium with the surface metal as well as with other cesium atoms. The theory is compared with the data of Taylor and Langmuir comparing evaporation rates to sub-monolayer surface coverage of cesium, gives good agreement, and reproduces the nonlinear behavior of evaporation with varying coverage and temperature

Spectral statistics of large dimensional Spearman's rank correlation matrix and its application

Let $\mathbf{Q}=(Q_1,\ldots,Q_n)$ be a random vector drawn from the uniform distribution on the set of all $n!$ permutations of $\{1,2,\ldots,n\}$. Let $\mathbf{Z}=(Z_1,\ldots,Z_n)$, where $Z_j$ is the mean zero variance one random variable obtained by centralizing and normalizing $Q_j$, $j=1,\ldots,n$. Assume that $\mathbf {X}_i,i=1,\ldots ,p$ are i.i.d. copies of $\frac{1}{\sqrt{p}}\mathbf{Z}$ and $X=X_{p,n}$ is the $p\times n$ random matrix with $\mathbf{X}_i$ as its $i$th row. Then $S_n=XX^*$ is called the $p\times n$ Spearman's rank correlation matrix which can be regarded as a high dimensional extension of the classical nonparametric statistic Spearman's rank correlation coefficient between two independent random variables. In this paper, we establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension, namely, $p=p(n)$ and $p/n\to c\in(0,\infty)$ as $n\to\infty$. We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni's cumulant method in [Ann. Statist. 36 (2008) 2553-2576] to bypass the so-called joint cumulant summability. In addition, we raise a two-step comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT, we then construct a distribution-free statistic to test complete independence for components of random vectors. Owing to the nonparametric property, we can use this test on generally distributed random variables including the heavy-tailed ones.Comment: Published at http://dx.doi.org/10.1214/15-AOS1353 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Engaging Central Banks in Climate Change? The Mix of Monetary and Climate Policy

Given the recent debate on central banks’ role under climate change, this research theoretically investigates the mix of monetary and climate policy and provides some insights for central banks who are considering their engagement in the climate change issue. The “climate-augmented” monetary policy is pioneeringly proposed and studied. We build an extended Environmental Dynamic Stochastic General Equilibrium (E-DSGE) model as the method. By this model, we find the following results. First, the making process of monetary policy should consider the existing climate policy and environmental regulation. Second, the coefficients in traditional monetary policy can be better set to enhance welfare when climate policy is given. This provides a way to optimise the policy mix. Third, if a typical form climate target is augmented into the monetary policy rule, a dilemma could be created. This means that it has some risks for central banks to care for the climate proactively by using the narrow monetary policy. At the current stage, central banks could and should use other measures to help the climate and the financial stability

Safety Evaluation of Highway Tunnel-Entrance Illuminance Transition Based on Eye-Pupil Changes

Utilizing the EMR-8B eye-tracker system, the pupil changes of eight drivers were monitored when they drove through 26 typical highway tunnels. Based on the test results, the driver’s pupil areas and pupil illuminance were found to be in a power function relationship at tunnel entrances. Furthermore, a quantitative relationship between the pupil area and its critical velocity was established, and the ratio of pupil area’s velocity in relation to its critical velocity was used to evaluate the lighting transitions and to establish the ideal curve of pupil illuminance at tunnel entrances. The results demonstrated that the relationship between the pupil illuminance of the tunnel entrance and the driver’s pupil areas conforms to the Stevens law found in experimental psychology; severe pupil illuminance transition within the range of 10 metres of the existing highway tunnel entrances, which results in great visual load, is in urgent need of improvement.</p