Limiting spectral distribution of Toeplitz and Hankel matrices with dependent entries

Abstract: This article deals with the limiting spectral distributions (LSD) of symmetric Toeplitz and Hankel matrices with dependent entries. For any fixed positive integer $m$, we consider these $n \times n$ matrices with entries ${Y^{(m)}_j / \sqrt n \ ;{j\geq 0}}$, where $Y^{(m)}_j= \sum_{r=-m}^{m} X_{j+r}$ and ${X_k}$ are i.i.d. with mean zero and variance one. We provide an explicit expression for the moment sequences of the LSDs. As a special case, this article provides an alternate proof for the LSDs of these matrices when the entries are i.i.d. with mean zero and variance one. The method is based on the moment method. The idea of proof can also be applied to other patterned random matrices, namely reverse circulant and symmetric circulant matrices.