Spectral Norm of Circulant-Type Matrices

We first discuss the convergence in probability and in distribution of the spectral norm of scaled Toeplitz, circulant, reverse circulant, symmetric circulant, and a class of k-circulant matrices when the input sequence is independent and identically distributed with finite moments of suitable order and the dimension of the matrix tends to infinity. When the input sequence is a stationary two-sided moving average process of infinite order, it is difficult to derive the limiting distribution of the spectral norm, but if the eigenvalues are scaled by the spectral density, then the limits of the maximum of modulus of these scaled eigenvalues can be derived in most of the cases.