Clt Of Linear Spectral Statistics For Large Dimensional Sample Covariance Matrices

In last two or three decades,with the rapid development of the modern science and technology,more and more scientists are facing to analyze large dimensional and massive data,calculations and inferences on large-scale science and engineering. For large-scale massive data sets,to analyze the interior relationship of data, naturally,one can use parametric or nonparametric regression and canonical correlation analysis.Furthermore,the space-time variety of data can be explored and forecasted by multivariate time series analysis and neural network.On the other hand,principal component analysis and factor analysis can be used to reduce calculating times,to finish the visualization of large-scale data.However,it is found that when dealing with large dimensional data,many classical statistical procedures induce large,even intolerable errors.Thus,it is important to develop new approaches to deal with large dimensional data,for example,forecasting long-time climates,performing visualization of the aero-exploring data,evaluating the earthquake data,making digitalization of space probes and so on.For these important and practical problems,sample covariance matrix is an important statistic because many important statistics are functionals of it.When we make statistical inferences,such as estimations and/or hypothesis tests,the sample covariance matrices must be investigated.However,for the large dimensional data.the sample covariance matrix is not a good estimator of the population counterpart.Many methods of parametric estimations and tests for multivariate statistical problems can not be simply used to deal with large dimensional data. In contrast,the theory of large dimensional random matrices,especially,that of the large sample covariance matrices become a very significant and powerful mathematical topic.When we deal with the large dimensional and massive data,Spectral analysis of large dimensional random matrices(LDRM) plays an important role in large dimensional data analysis.After finding the limiting spectral distribution (LSD) F(x) of the empirical spectral distribution F_n(x)(ESD i.e.the empirical distribution of the eigenvalues) of LDRM,one can derive the limit of the corre-sponding linear spectral statistics(LSS)∫f(x)dF_n(x).Then,in order to conduct further statistical inference,it is important to find the limiting distribution of LSS of LDRM.A general conjecture about the convergence rate of ESD to LSD puts it at the order of O(n^-1).If this is true,then it seems natural to consider the asymptotic properties of the empirical process G_n(x)=n(F_n(x)-F(x)).Unfortunately,many lines of evidence show that the process G_n(x) can not converge in any metric space. As an alternative,we turned back to find the limiting distribution of the LSS G_n(f). In this thesis,using the Bernstein polynomial approximation,Stieltjes transform method and the central limit theorem(CLT) of martingale,under suitable moment conditions,we prove the CLT of LSS G_n(f) with a generalized regular class C⁴ of the kernel functions for large dimensional sample covariance matrices.These asymptotic properties of empirical statistics suggest that more efficient statistical inferences,such as hypothesis testing,constructing confidence intervals or regions, etc.,on a class of population parameters,are possible.The improved criteria on the constraint conditions of kernel functions in our results should also provide a better understanding of the asymptotic properties of the ESD of large dimensional sample covariance matrices.