This paper focuses on the dilute real symmetric Wigner matrix Mn=1/√n（aij）n×n, whose offdiagonal entries aij （1 ≤ em ≠ j ≤ n） have mean zero and unit variance, Eaij4 =θnα （θ 〉 0） and the fifth moments of aij satisfy a Lindeberg type condition. When the dilute parameter 0 〈 α ≤ 1/3 and the test function satisfies some regular conditions, it proves that the centered linear eigenvalue statistics of Mn obey the central limit theorem.
The paper focuses on the largest eigenvalues of theβ-Hermite ensemble and theβ-Laguerre ensemble. In particular, we obtain the precise moment convergence rates of their largest eigenvalues. The results are motivated by the complete convergence for partial sums of i.i.d, random variables, and the proofs depend on the small deviations for largest eigenvalues of the β ensembles and tail inequalities of the generalβ Tracy-Widom law.
In this paper,we consider the limiting spectral distribution of the information-plus-noise type sample covariance matrices Cn=1/N（Rn＋σXn）（Rn＋σXn）,under the assumption that the entries of Xn are independent but non-identically distributed random variables.It is proved that,almost surely,the empirical spectral distribution of Cn converges weakly to a non-random distribution whose Stieltjes transform satisfies a certain equation.Our result extends the previous one with the entries of Xn are i.i.d.random varibles to a more general case.The proof of the result mainly employs the Stein equation and the cumulant expansion formula of independent random variables.
