Partial linear eigenvalue statistics for non-hermitian random matrices

For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues $\lambda_1, \dots, \lambda_n$, the seminal work of Rider and Silverstein [Ann. Probab., 34 (2006), pp. 2118–2143] asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n f(\lambda_i)$ converge to a Gaussian distribution for sufficiently nice test functions $f$. We study the fluctuations of $\sum_{i=1}^{n-K} f(\lambda_i)$, 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 for 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, [Ann. Fac. Sci. Toulouse Math. (6), 24 (2015), pp. 93–117]. 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.