Large dimensional random k circulants

Circulant matrices with general shift by k places have been studied in the literature and formula for their eigenvalues is known. We first reestablish this formula and some further properties of these eigenvalues in a manner suitable for our use. We then consider random k = k(n) circulants Ak,n with n → ∞ and whose input sequence {ai} is independent with mean zero and variance one and supn n −1 ∑n i=1 E|ai| < ∞ for some δ > 0. Under suitable restrictions on {k(n)}, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists and identify the limits. As examples, (i) if kg = −1 + sn where g ≥ 1 fixed and s = o(n1/3), then the LSD is U1( ∏g i=1 Ei) 1/2g where Ei are i.i.d. Exp(1) and U1 is uniformly distributed over the (2g)th roots of unity, independent of the {Ei}, and (ii) if kg = 1 + sn where g ≥ 2 is fixed and s = o(n g+1 g−1 ) or s = o(n) according as g ≥ 2 is odd or even, then the LSD is U2( ∏g i=1 Ei) 1/2g where {Ei} are i.i.d. Exp(1) and U2 is uniformly distributed over the unit circle, independent of the {Ei}. We then consider the limit distribution of the spectral norm of Ak,n. We show that when n = k2 +1 → ∞, the spectral norm, with appropriate scaling and centering, which we provide explicitly, converges to the Gumbel distribution.