631 research outputs found
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 . Here, is
an random matrix with independent entries such that , . On
dimensionality, we assume that and as
. For a class of general deterministic positive-definite
matrices , under some additional assumptions on the
distribution of 's, we show that the limiting behavior of the largest
eigenvalue of 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 (). Consequently, in
the standard complex case (), 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 converges weakly to the
type 2 Tracy-Widom distribution . Moreover, in the real case, we
show that when is spiked with a fixed number of subcritical spikes,
the type 1 Tracy-Widom limit holds for the normalized largest
eigenvalue of , 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 and more generally distributed .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 be an real (resp. complex) Wigner matrix and
be its spectral decomposition. Set
, where is a real (resp.
complex) unit vector. Under the assumption that the elements of have 4
matching moments with those of GOE (resp. GUE), we show that the process
converges weakly to the Brownian bridge for any
such that as ,
where for the real case and 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 , where with
. We assume to be a collection of independent symmetric distributed
random variables with sub-exponential tails. Moreover, for any , we assume
to be identically distributed. We assume and
with some as . In this
paper, we provide the Tracy-Widom law () for both the largest and
smallest eigenvalues of . If are i.i.d. standard normal, we can
derive the for both the largest and smallest eigenvalues of the matrix
, where with , .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 be a random vector drawn from the uniform
distribution on the set of all permutations of . Let
, where is the mean zero variance one random
variable obtained by centralizing and normalizing , . Assume
that are i.i.d. copies of
and is the random matrix
with as its th row. Then is called the
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, and as . 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
- …