  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.
