Another Marcenko-Pastur law for Kendall's tau

Abstract

Bandeira et al. (2017) show that the eigenvalues of the Kendall correlation matrix of $n$ i.i.d. random vectors in ${\mathbb{R}}^p$ 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 $p,n\to \infty$ proportionately to one another. Here we show that another Marčenko-Pastur law emerges in the "ultra-high dimensional" scaling limit where $p\sim q^{\prime }{n}^2/2$ for some $q^{\prime }>0$: in this quadratic scaling regime, Kendall correlation eigenvalues converge weakly almost surely to $(1/3)Y_{q^{\prime }}$.