arXiv:1305.1376v1 math.ST 7 May 2013CLT for linear spectral statistics of randommatrix S1TShurong Zheng, Zhidong Bai and Jianfeng YaoSchool of Mathematics and Statistics and KLAS, Northeast Normal University,Changchun City 130024, P. R. ChinaDepartment of Statistics and Actuarial Science, Hong Kong University, P. R. ChinaMay 8, 2013AbstractAs a generalization of the univariate Fisher statistic, random Fisher matricesare widely-used in multivariate statistical analysis, e.g. for testing the equality oftwo multivariate population covariance matrices. The asymptotic distributions ofseveral meaningful test statistics depend on the related Fisher matrices. Such Fishermatrices have the form F = SyMS1x M where M is a non-negative and non-randomHermitian matrix, and Sx and Sy are p p sample covariance matrices from twoindependent samples where the populations are assumed centred and normalized(i.e. mean 0, variance 1 and with independent components). In the large-dimensionalcontext, Zheng (2012) establishes a central limit theorem for linear spectral statisticsof a standard Fisher matrix where the two population covariance matrices are equal,i.e. the matrix M is the identity matrix and F = SyS1x . It is however of significantimportance to obtain a CLT for general Fisher matrices F with an arbitrary Mmatrix. For the mentioned test of equality, null distributions of test statistics rely ona standard Fisher matrix with M = Ip while under the alternative hypothesis, thesedistributions depends on a general Fisher matrix with arbitrary M. As a first stepto this goal, we propose in this paper a CLT for spectral statistics of the randommatrix S1x T for a general non-negative definite and non-random Hermitian matrixT (note that T plays the role of MM). When T is inversible, such a CLT can bedirectly derived using the CLT of Bai and Silverstein (2004) for the matrix T1Sx.However, in many large-dimensional statistic problems, the deterministic matrix T1is usually not inversible or has eigenvalues close to zero. The CLT from this papercovers this general situation.1 IntroductionFor a p p random matrix An with eigenvalues (j), linear spectral statistics (LSS) oftype 1psummationtextj f(j) for various test functions f are of central importance in the theory ofrandom matrices and its applications Central limit theorems (CLT) for such LSS of largedimensional random matrices have a long history, and received considerable attention inrecent years. They have important applications in various domains like number theory,high-dimensional multivariate statistics and wireless communication networks; for moreinformation, the readers are referred to the recent survey paper Johnstone (2007). Tomention a few, in an early work, Jonsson (1982) gave a CLT for (tr(An), ,tr(Akn)for a sequence of Wishart matrices (An), where k is a fixed number, and the dimensionp of the matrices grows proportionally to the sample size n. Subsequent works includeCostin and Lebowitz (1995), Johansson (1998) which considered extensions of classicalGaussian ensembles, and Sina and Soshnikov (1998a,b) where Gaussian fluctuations areidentified for LSS of Wigner matrices with a class of more general test functions. A gen-eral CLT for LSS of Wigner matrices was given in Bai and Yao (2005) where in partiular,the limiting mean and covariance functions are identified. Similarly, Bai and Silverstein(2004) established a CLT for general sample covariance matrices with explicit limitingparameters. In Lytova and Pastur (2009), the authors reconsider such CLTs but with anew idea of interpolation that allows the generalisation from Gaussian matrix ensemblesto matrix ensembles with general entries satisfying a moment condtiion. Recent improv-ments are proposed in Pan and Zhou (2008) that propose a generalization of the CLTin Bai and Silverstein (2004) (see also Wang and Yao (2013) for a complement on theseCLTs). Finally, Pan (2012) and Bai and Zheng (2013) extend Bai and Silverstein (2004)sCLT to biased and unbiased sample covariance matrices, respectively.Random Fisher matrices are widely-used in multivariate statistical analysis, e.g. fortesting the equality of two multivariate population covariance matrices. The asymptoticdistributions of several meaningful test statistics depend on the related Fisher matrices.Such Fisher matrices have the form F = SyMS1x M where M is a non-negative deter-ministic Hermitian matrix, and Sx and Sy are p p sample covariance matrices fromtwo independent samples where the populations are assumed centred and normalized (i.e.mean 0, variance 1 and with independent components). In the large-dimesional context,2Zheng (2012) establishes a CLT for linear spectral statistics of a standard Fisher matrixwhere the two population covariance matrices are equal, i.e. the matrix M is the identitymatrix and F = SyS1x . It is however of significant importance to obtain a CLT for generalFisher matrices F with an arbitrary