On a class of $H$-selfadjont random matrices with one eigenvalue of nonpositive type

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.