Another Marčenko-Pastur law for Kendall's tau
Bandeira *et al.* (2017) show that the eigenvalues of the Kendall correlation matrix of n i.i.d. random vectors in ℝᵖ are asymptotically distributed like 1/3 + (2/3)·Y₍q₎, where Y₍q₎ has a Marčenko–Pastur law with parameter q = lim(p/n) if p, n → ∞ proportionately to one another. Here we show that another Marčenko–Pastur law emerges in the \"ultra-high dimensional\" scaling limit where p ∼ q′·n²⁄2 for some q′ > 0. In this quadratic scaling regime, Kendall correlation eigenvalues converge weakly almost surely to (1/3)·Y₍q′₎.
Mar 1, 2025
