Limiting Spectral Distribution of Random k-Circulants

Abstract

Consider random k-circulants A _k,n with n→∞,k=k(n) and whose input sequence {a _l } _l≥0 is independent with mean zero and variance one and $\sup_{n}n^{-1}\sum_{l=1}^{n}\mathbb{E}|a_{l}|^{2+\delta}<\infty$ 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 p ₁ is the smallest prime divisor of g. Suppose $P_{g}=\prod_{j=1}^{g}E_{j}$ where {E _j }_1≤j≤g are i.i.d. exponential random variables with mean one.

(i) If k ^g=−1+sn where s=1 if g=1 and $s=o(n^{p_{1}-1})$ if g>1, then the empirical spectral distribution of n ^−1/2 A _k,n converges weakly in probability to $U_{1}P_{g}^{1/(2g)}$ where U ₁ is uniformly distributed over the (2g)th roots of unity, independent of P _g .

(ii) If g≥2 and k ^g=1+sn with $s=o(n^{p_{1}-1})$, then the empirical spectral distribution of n ^−1/2 A _k,n converges weakly in probability to $U_{2}P_{g}^{1/(2g)}$ where U ₂ is uniformly distributed over the unit circle in ℝ², independent of P _g .

On the other hand, if k≥2, k=n ^o(1) with gcd (n,k)=1, and the input is i.i.d. standard normal variables, then $F_{n^{-1/2}A_{k,n}}$ converges weakly in probability to the uniform distribution over the circle with center at (0,0) and radius $r=\exp(\mathbb{E}[\log\sqrt{E}_{1}])$.