Abstract
This article studies the limiting spectral distributions of random birth-death Q matrices. Under the strictly stationary ergodic condition, we prove that the empirical spectral distribution converges weakly to a non- random probability distribution. Furthermore, in the situations without strictly stationary ergodic condition, we study a class of random birth-death Q matrices, corresponding to generalizations of the Beta-Hermite ensembles, and establish the existences as well as convolution formulations for their limiting spectral distributions. In particular, for the random birth-death Q matrices corresponding to the Beta-Hermite ensembles, the limiting spectral distribution is the convolution of the semi-circle law and Dirac measure δ -2.
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