随机生灭Q矩阵的极限谱分布

Limiting spectral distribution of random birth-death Q matrices

本文主要研究随机生灭Q矩阵的极限谱分布.在严平稳遍历的情形下,本文证明随机生灭Q矩阵的经验谱分布弱收敛于某个非随机概率分布.进一步,在非严平稳遍历情形下,本文研究了比BetaHermite系综更广的一类随机矩阵模型,建立了与之相应的随机生灭Q矩阵的极限谱分布存在性,并且证明它的极限谱分布具有卷积表达式.特别地,Beta-Hermite系综所对应的随机生灭Q矩阵的极限谱分布是经典半圆率与Dirac测度δ-2的卷积.

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.