In this presentation we study the limiting distribution of the largest singular value, the smallest singular value, and the so-called condition number for random circulant matrices, where the generating sequence consists of independent and identically distributed (i.i.d.) random elements satisfying the Lyapunov condition.

Under an appropriate normalization, the joint distribution of the extremal (minimum and maximum) singular values converges in distribution, as the matrix dimension approaches infinity, to an independent product of Rayleigh and Gumbel laws. As a consequence, the condition number (properly normalized) converges in distribution to a Fréchet law in the large-dimensional limit. Broadly speaking, the condition number quantifies the sensitivity of the output of a linear system to small perturbations in its input. The proof is based on the celebrated Einmahl–Komlós–Major–Tusnády coupling. This work is based on a joint paper with Paulo Manrique (IPN, Mexico), in Extremes 2022.