A result on the limiting spectral distribution of random matrices with unequal variance entries

A classical result in random matrix theory reveals that the limiting spectral distribution of a Wigner matrix whose entries have a common variance and satisfy other regular assumptions almost surely converges to the semicircular law. In the paper, we will relax the assumption of uniform covariance of each entry, when the average of the normalized sums of the variances in each row of the data matrix converges to a constant, we prove that the same limiting spectral distribution holds. A similar result on a sample covariance matrix is also established. The proofs mainly depend on the Stein equation and the generalized Stein equation of independent random variables.