Probability and Stochastic Analysis Seminar  RSS

04/12/2024, 17:00 — 18:00 — Online
Gerardo Barrera Vargas, IST Lisbon

The asymptotic distribution of the condition number for random circulant matrices

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 based in a paper with Paulo Manrique, Extremes 2022.

link

Except for a few of the oldest sessions these are from the Seminário de Probabilidade e Mecânica Estatística at IMPA which is co-sponsored by several institutions, in particular Instituto Superior Técnico.