Limiting Spectral Distribution of Random k-Circulants

Consider random k-circulants Ak,n with n→∞,k=k(n) and whose input sequence {al}l≥0 is independent with mean zero and variance one and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sup_{n}n^{-1}\sum_{l=1}^{n}\mathbb{E}|a_{l}|^{2+\delta}<\infty$\end{document} for some δ>0. Under suitable restrictions on the sequence {k(n)}n≥1, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists, and we identify the limits. In particular, we prove the following: Suppose g≥1 is fixed and p1 is the smallest prime divisor of g. Suppose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_{g}=\prod_{j=1}^{g}E_{j}$\end{document} where {Ej}1≤j≤g are i.i.d. exponential random variables with mean one.