Explicit formulas for limiting spectral distributions of Toeplitz and Hankel matrices

Derive explicit formulas for the limiting spectral distributions of normalized random Toeplitz and Hankel matrices generated by independent, mean-zero, variance-one random variables with all moments finite.

Background

The paper contrasts Wigner matrices, which contain on the order of n^2 independent random variables and have the semicircular limiting spectral distribution, with patterned matrices such as Toeplitz and Hankel matrices, which contain only on the order of n independent random variables. For Toeplitz and Hankel matrices, the limiting spectral distribution is known to exist and to have unbounded support, but the paper identifies the absence of explicit formulas for these limiting laws as an unresolved issue.

This problem is distinct from the paper’s principal result, which constructs a different random-matrix model with O(n) independent variables whose empirical spectral distribution converges to the semicircular law.