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 _nn ^-1 Σ ⁿ _l=1 E{pipe}al{pipe} ^2+δ < ∞ 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 = Π ^g _j=1E _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 ^p1-1) if g>1, then the empirical spectral distribution of n ^-1/2A _k,n converges weakly in probability to U ₁ P ^1/(2g) _g 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 ^p1-1), then the empirical spectral distribution of n ^-1/2A _k,n converges weakly in probability to U ₁ P ^1/(2g) _g where U ₂ is uniformly distributed over the unit circle in ℝ 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 Ak,n} converges weakly in probability to the uniform distribution over the circle with center at (0,0) and radius r=exp(E[log√E ₁]).