Asymptotic Distribution of Eigenvalues of Block Toeplitz Matrices
Richard A. Davis, Keh‐Shin Lii, Dimitris N. Politis · 2011
Let g(λ), —π∆λ∆π, be a p×p (p — 1, 2, …) matrix-valued Hermitian function. Further g(λ) is bounded on [—π, π], that is, its elements are bounded on [—π, π]. The Fourier coefficients 1 $$\begin{array}{lll}{*{20}{c}} {{\alpha _k} = \frac{1}{{2\pi }}\int_{ - \pi }^{ - \pi } {{e^{ik{\rm{\lambda }}}}g(\lambda )d\lambda,} } & {k = 0, \pm 1, \cdot \cdot \cdot } \\\end{array}$$ are then bounded in k. We call the np×np matrix $${A_n} = ({a_{j - k}};j,k = 1, \cdot \cdot \cdot,n)$$ (an n×n matrix of the p×p blocks aj-k) the nth section block Toeplitz matrix generated by g(λ). Notice that the block Toeplitz matrix An is generally not Toeplitz. Our interest is in obtaining the asymptotic distribution of eigenvalues of A „ as n→∞. The proof is suggested by an argument given in the one-dimensional case (p=1) (see [3]) and is based on results in the multidimensional prediction problem [5].