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Another Marčenko-Pastur law for Kendall's tau

Pierre Bousseyroux, Tomas Espana, Matteo Smerlak

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 $p\sim q'\, n^2/2$ for some $q'>0$: in this quadratic scaling regime, Kendall correlation eigenvalues converge weakly almost surely to $(1/3)Y_{q'}$.

Another Marčenko-Pastur law for Kendall's tau

Abstract

Bandeira et al. (2017) show that the eigenvalues of the Kendall correlation matrix of i.i.d. random vectors in are asymptotically distributed like , where has a Marčenko-Pastur law with parameter if proportionately to one another. Here we show that another Marčenko-Pastur law emerges in the "ultra-high dimensional" scaling limit where for some : in this quadratic scaling regime, Kendall correlation eigenvalues converge weakly almost surely to .

Paper Structure

This paper contains 7 sections, 5 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of Theorem \ref{['theo:kendall_quad']}
  4. Numerics
  5. RMT Reminders
  6. Proof of Proposition \ref{['prop:prop1']}
  7. Proof of step 1 \ref{['step1']}

Key Result

Theorem 1

Under the assumptions of Def. def:kendall, and if $q_n = p/n \rightarrow q >0$ when $n \rightarrow \infty$, the sequence $(F^{\boldsymbol{\tau}_n})_n$ converges weakly in probability to the distribution associated with the random variable $(1/3) + (2/3) Y_{q}$, where $Y_{q}$ has a MP law with parame

Figures (2)

  • Figure 1: Histogram of the eigenvalues of $\mathbf{H}$ for $n=70$ and $p=1225$ (${q'}\approx0.4$ and $q \approx 17.5$). The superimposed red line is the density function of $(1/3)Y_{q'}$.
  • Figure 2: Histogram of the eigenvalues of $\boldsymbol{\tau}$ for $n=70$ (from left to right and top to bottom, $p \in \{300, 400, 500, 600\}$). The superimposed red line is the density function of $(1/3)Y_{q'}$. The vertical black dashed line corresponds to its mean $(1/3)$. The two vertical red dashed lines correspond to the theoretical minimum and maximum eigenvalues of the density of $(2/3) Y_q$.

Theorems & Definitions (8)

  • Definition 1: Kendall Correlation Matrix
  • Theorem 1: Theorem 1 of bandeira2017marvcenkopastur
  • Theorem 2
  • Proposition 1
  • Definition 2: marchenko1967distribution
  • Proposition 1
  • proof
  • Lemma 1: Theorem A.43 of bai2010spectral