Consider real symmetric, complex Hermitian Toeplitz, and real symmetric Hankel band matrix models where the bandwidth b N →∞ but b N /N→b∈[0,1] as N→∞. We prove that the distributions of eigenvalues converge weakly to universal symmetric distributions γ T (b) and γ H (b). In the case b>0 or b=0 but with the addition of bN ³ CNfrac12+e0b_{N}geq CN^{frac{1}{2}+epsilon_{0}} for some positive constants ε 0 and C, we prove the almost sure convergence. The even moments of these distributions are the sums of some integrals related to certain pair partitions. In particular, when the bandwidth grows slowly, i.e., b=0, γ T (0) is the standard Gaussian distribution, and γ H (0) is the distribution |x|exp (−x 2). In addition, from the fourth moments, we know that γ T (b) are different for different b, γ H (b) different for different b Î [0,frac12]bin[0,frac{1}{2}] , and γ H (b) different for different b Î [frac12,1]bin [frac{1}{2},1] .
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机译：考虑实对称，复Hermitian Toeplitz和实对称汉克尔带矩阵模型，其中带宽b N →∞但b N / N→b∈[0,1]为N→∞。我们证明了特征值的分布弱收敛于通用对称分布γ T （b）和γ H （b）。在b> 0或b = 0的情况下，但加上b N ³CN frac12 + e 0 b_ {N} geq CN对于一些正常数ε 0 和C，^ {frac {1} {2} + epsilon_ {0}}，我们证明了几乎确定的收敛性。这些分布的偶数矩是与某些对分区有关的一些积分的总和。特别地，当带宽缓慢增长时，即b = 0，γ T （0）是标准的高斯分布，而γ H （0）是||的分布。 x | exp（-x 2 ）。另外，从第四时刻开始，我们知道对于不同的bγ T （b）是不同的，对于不同的bγ H （b）是不同的Î[0，frac12 ] bin [0，frac {1} {2}]和γ H （b）对于不同的bδ[frac12,1] bin [frac {1} {2}，1]是不同的。
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