Central limit theorem for traces of large random symmetric matrices with independent matrix elements

Abstract

We study Wigner ensembles of symmetric random matrices A = (aij), i,j = 1,... ,n with matrix elements aij, i < j being independent symmetrically distributed random variables 4u aij = aJ i --I " ngWe assume that Var~ij = 1, for i < j, Var~ii <const and that all higher moments of ~ij also exist and grow not faster than the Gaussian ones. Under fornmlated conditions we prove the central limit theorem for the traces of powers of A growing with n more slowly than x/~. The limit of Var(Trace AP), 1 << p << ~, does not depend on the fourth and higher moments of ~ij and the rate of growth of p, and 1 As a corollary we improve the estimates on the rate of convergence of equals to ~. the maximal eigenvalue to 1 and prove central limit theorem for a general class of linear statistics of the spectra.

Universality results for the largest eigenvalues of some sample covariance matrix ensembles

“…We say that such paths are correlated. To estimate the number of correlated paths and their contribution to the variance, we use the construction procedure defined in Section 3 of [27]. This construction associates a path of length 4s N − 2 to a pair of correlated paths.…”

“…In this case, the limiting Gaussian distribution does depend on the fourth moment of the law of the entries. The above CLT is also stated but not proved in Remark 6 of [27] (a factor 1/β is missing) in the case where γ = 1.…”

“…The terminology we use is close to the one used in [29], [27], [28] and [30]. We recall the main definitions that will be needed here and also assume that the reader is acquainted with most of the techniques used in the above papers.…”

“…We denote by k 1 the number of the odd up steps of T 1 . As the trajectory of P 1 ∨ P 2 is obtained by inserting T 2 at the instant t e in T 1 , and using the fact that P 1 ∨ P 2 and P 1 ∪ P 2 have all the same edges but one, one can then deduce (see [27], p. 11-13, for the detail) that the contribution of correlated pairs is at most of order…”

Definitions and Basic Properties

“…Heuristical derivations of Gaussian limit for linear statistics may be found in Politzer [95], Beenakker [24,25], Costin, Lebowitz [36] while some of the references where rigorous analysis is performed include Johansson [71], Basor [22], Sinai, Soshnikov [100], Soshnikov [103], Bai, Silverstein [13] and Dumitriu, Edelman [43]. It should be noticed that similar results are obtained for spectrum fluctuations of matrices from classical compact groups, but they will be discussed in detail in Chapter 5.…”