Another Marčenko-Pastur law for Kendall's tau

2025/03/24 by Bousseyroux, Pierre, Espana, Tomas, Smerlak, Matteo
#FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2503.18645

Abstract

Bandeira et al. (2017) show that the eigenvalues of the Kendall correlation matrix of n i.i.d. random vectors in 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→∞ 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'.

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