Partial linear eigenvalue statistics for non-Hermitian random matrices

DOI10.1137/S0040585X97T991179arXiv1912.08856MaRDI QIDQ6331381

Publication date: 18 December 2019

Abstract: For an

n i m e s n

independent-entry random matrix

X_{n}

with eigenvalues

l a m b d a_{1}, l d o t s, l a m b d a_{n}

, the seminal work of Rider and Silverstein asserts that the fluctuations of the linear eigenvalue statistics

s u m_{i = 1}^{n} f (l a m b d a_{i})

converge to a Gaussian distribution for sufficiently nice test functions

f

. We study the fluctuations of

s u m_{i = 1}^{n - K} f (l a m b d a_{i})

, where

K

randomly chosen eigenvalues have been removed from the sum. In this case, we identify the limiting distribution and show that it need not be Gaussian. Our results hold for the case when

K

is fixed as well as the case when

K

tends to infinity with

n

. The proof utilizes the predicted locations of the eigenvalues introduced by E. Meckes and M. Meckes. As a consequence of our methods, we obtain a rate of convergence for the empirical spectral distribution of

X_{n}

to the circular law in Wasserstein distance, which may be of independent interest.

Mathematics Subject Classification ID

Statistics (62-XX) Probability theory and stochastic processes (60-XX)