TY - JOUR

T1 - Central limit theorem for traces of large random symmetric matrices with independent matrix elements

AU - Sinai, Yakov G.

AU - Soshnikov, A.

PY - 1998/1/1

Y1 - 1998/1/1

N2 - 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 aij = aji = ξij/n1/2. We assume that Var ξij = 1/4, for i < j, Var ξii ≤const and that all higher moments of ξij also exist and grow not faster than the Gaussian ones. Under formulated conditions we prove the central limit theorem for the traces of powers of A growing with n more slowly than √n. The limit of Var(Trace Ap), l ≪ p ≪ √n, does not depend on the fourth and higher moments of ξij and the rate of growth of p, and equals to 1/π. As a corollary we improve the estimates on the rate of convergence of the maximal eigenvalue to 1 and prove central limit theorem for a general class of linear statistics of the spectra.

AB - 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 aij = aji = ξij/n1/2. We assume that Var ξij = 1/4, for i < j, Var ξii ≤const and that all higher moments of ξij also exist and grow not faster than the Gaussian ones. Under formulated conditions we prove the central limit theorem for the traces of powers of A growing with n more slowly than √n. The limit of Var(Trace Ap), l ≪ p ≪ √n, does not depend on the fourth and higher moments of ξij and the rate of growth of p, and equals to 1/π. As a corollary we improve the estimates on the rate of convergence of the maximal eigenvalue to 1 and prove central limit theorem for a general class of linear statistics of the spectra.

UR - http://www.scopus.com/inward/record.url?scp=27144550468&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27144550468&partnerID=8YFLogxK

U2 - 10.1007/BF01245866

DO - 10.1007/BF01245866

M3 - Article

AN - SCOPUS:27144550468

VL - 29

SP - 1

EP - 24

JO - Bulletin of the Brazilian Mathematical Society

JF - Bulletin of the Brazilian Mathematical Society

SN - 1678-7544

IS - 1

ER -