Szegő’s First Limit Theorem provides the limiting statistical distribution of the eigenvalues of large Toeplitz matrices. Szegő’s Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large Toeplitz matrices. In this paper we survey results extending the First and Second Limit Theorems to Kac–Murdock–Szegő (KMS) matrices. These are matrices whose entries along the diagonals are not necessarily constants, but modeled by functions. We clarify and extend some existing results, and explain some apparently contradictory results in the literature.

Szegő 的第一极限定理提供了大型 Toeplitz 矩阵特征值的极限统计分布。 Szegő 托普利茨矩阵的第二（或强）极限定理给出了第一极限定理的二阶修正，并允许计算大型托普利茨矩阵行列式的渐近。在本文中，我们调查了将第一和第二极限定理扩展到 Kac-Murdock-Szegő (KMS) 矩阵的结果。这些矩阵的对角线条目不一定是常数，而是由函数建模。我们澄清和扩展了一些现有的结果，并解释了文献中一些明显矛盾的结果。