Another Marcenko-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 ${\mathbb{R}}^p$ are asymptotically distributed like $1\mathrm{/}3+(2\mathrm{/}3)Y_q$, where $Y_q$ has a Marčenko-Pastur law with parameter $q=\lim (p\mathrm{/}n)$ if $p,n\to \mathrm{\infty }$ proportionately to one another. Here we show that another Marčenko-Pastur law emerges in the “ultra-high dimensional” scaling limit where $p\sim q^{\mathrm{\prime }}{n}^2\mathrm{/}2$ for some $q^{\mathrm{\prime }}>0$: in this quadratic scaling regime, Kendall correlation eigenvalues converge weakly almost surely to $(1\mathrm{/}3)Y_{q^{\mathrm{\prime }}}$.