Limiting eigenvalue distribution of heavy-tailed Toeplitz matrices

Abstract

We consider an $N\times N$ random symmetric Toeplitz matrix with an i.i.d. input sequence drawn from a distribution that lies in the domain of attraction of an $\alpha$-stable law for $0<\alpha <2$. We show that under an appropriate scaling, its empirical eigenvalue distribution, as $N\to \infty$, converges weakly to a random symmetric probability distribution on $\mathbb{R}$, which can be described as the expected spectral measure of a certain random unbounded self-adjoint operator on ${\ell }^2(\mathbb{Z})$. The limiting distribution turns out to be almost surely subgaussian. Furthermore, the support of the limiting distribution is bounded almost surely if $0<\alpha <1$ and is unbounded almost surely if $1\le \alpha <2$.

33 pages