In this presentation we study the limiting distribution for the joint-law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition.
Under an appropriated normalization, the joint-law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws.
The latter implies that a normalized condition number converges in distribution to a Fréchet law as the dimension of the matrix increases. Roughly speaking, the condition number measures how much the output value of a linear system can change by a small perturbation in the input argument.
The proof relies on the celebrated Einmahl--Komlós--Major--Tusnády coupling.
This is a joint work with Paulo Manrique, National Polytechnic Institute, Mexico.
