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Block Toeplitz and Hankel matrices arise in many aspects of applications. Inthis paper, we will research the distributions of eigenvalues for some modelsand get the semicircle law. Firstly we will give trace formulae of blockToeplitz and Hankel matrix. Then we will prove that the almost sure limit$\gamma_{_T}^{(m)}$ $(\gamma_{_H}^{(m)})$ of eigenvalue distributions of randomblock Toeplitz (Hankel) matrices exist and give the moments of the limitdistributions where $m$ is the order of the blocks. Then we will prove theexistence of almost sure limit of eigenvalue distributions of random blockToeplitz and Hankel band matrices and give the moments of the limitdistributions. Finally we will prove that $\gamma_{_T}^{(m)}$$(\gamma_{_H}^{(m)})$ converges weakly to the semicircle law as $m\ra\iy$.

Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices