Consider random k-circulants A_{ k,n} with n tends to infinity, k=k(n) and whose input sequence \{ a_l\}_{ l \ge 0} is independent with mean zero and variance one and \sup_n n^{ -1}\sum_{ l=1}^n \E |a_l|^{ 2+\delta}< \infty for some \delta > 0. Under suitable restrictions on the sequence \{ k(n)\}_{ n \ge 1}, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists and identify the limits. In particular, we prove the following: Suppose g \ge 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 \le j \le g} are i.i.d. exponential random variables with mean one. (i) If k^g = -1+ s n 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_1P_g^{ 1/2g} where U_1 is uniformly distributed over the (2g)th roots of unity, independent of P_g. (ii) If g \ge 2 and k^g = 1+ s n 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_2P_g^{ 1/2g} where U_2 is uniformly distributed over the unit circle in \mathbb R^2, independent of P_g. On the other hand, if k \ge 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(\E [ \log \sqrt E_1]). We also show that when n=k^2+1\to \infty, and the input is i.i.d. with finite (2+\delta) moment, then the spectral radius, with appropriate scaling and centering, converges to the Gumbel distribution.
Categories: math.PR