Large $H$-selfadjoint random matrices are considered. The matrix $H$ is assumed to have one negative eigenvalue, hence the matrix in question has precisely one eigenvalue of nonpositive type. It is showed that this eigenvalue converges in probability to a deterministic limit. The weak limit of distribution of the real eigenvalues is investigated as well.
"On a class of $H$-selfadjont random matrices with one eigenvalue of nonpositive type." Electron. Commun. Probab. 17 1 - 14, 2012. https://doi.org/10.1214/ECP.v17-2148