Limiting Spectral Distribution of Random k-Circulants

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 ??>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 ?², 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 $r=\exp(\mathbb{E}\log\sqrt{E}_{1}])$ .