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Maximum interpoint distance of high-dimensional random vectors

Johannes Heiny, Carolin Kleemann

Abstract

A limit theorem for the largest interpoint distance of $p$ independent and identically distributed points in $\mathbb{R}^n$ to the Gumbel distribution is proved, where the number of points $p=p_n$ tends to infinity as the dimension of the points $n\to\infty$. The theorem holds under moment assumptions and corresponding conditions on the growth rate of $p$. We obtain a plethora of ancillary results such as the joint convergence of maximum and minimum interpoint distances. Using the inherent sum structure of interpoint distances, our result is generalized to maxima of dependent random walks with non-decaying correlations and we also derive point process convergence. An application of the maximum interpoint distance to testing the equality of means for high-dimensional random vectors is presented. Moreover, we study the largest off-diagonal entry of a sample covariance matrix. The proofs are based on the Chen-Stein Poisson approximation method and Gaussian approximation to large deviation probabilities.

Maximum interpoint distance of high-dimensional random vectors

Abstract

A limit theorem for the largest interpoint distance of independent and identically distributed points in to the Gumbel distribution is proved, where the number of points tends to infinity as the dimension of the points . The theorem holds under moment assumptions and corresponding conditions on the growth rate of . We obtain a plethora of ancillary results such as the joint convergence of maximum and minimum interpoint distances. Using the inherent sum structure of interpoint distances, our result is generalized to maxima of dependent random walks with non-decaying correlations and we also derive point process convergence. An application of the maximum interpoint distance to testing the equality of means for high-dimensional random vectors is presented. Moreover, we study the largest off-diagonal entry of a sample covariance matrix. The proofs are based on the Chen-Stein Poisson approximation method and Gaussian approximation to large deviation probabilities.
Paper Structure (30 sections, 23 theorems, 218 equations, 2 figures)

This paper contains 30 sections, 23 theorems, 218 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Objective and structure of this paper
  3. Notation
  4. Main results: convergence of the maximum
  5. Maximum interpoint distance
  6. Maximum of dependent random walks
  7. Maximum of dependent random walks in the case of strong correlation
  8. More applications
  9. -norms
  10. Testing the equality of means for high-dimensional vectors
  11. Largest off-diagonal entry of a sample covariance matrix
  12. Extension to point process convergence
  13. Proof of Theorem \ref{['thm:summax']}
  14. Preliminaries
  15. Proof under condition (C1)
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $(\mathbf{x}_i)_{i\le p}$ be iid $\mathbb{R}^n$-valued random vectors, whose components fulfill the standard conditions. Assume one of the conditions (B1) -- (B4) on $X$ and that $p=p_n\to\infty$ satisfies Then we have where $G$ is standard Gumbel distributed. The sequences $(b_n^{(2)})$ and $(c_n^{(2)})$ are given by where with

Figures (2)

  • Figure 1: Uniformly distributed points on the two-dimensional unit ball
  • Figure 2: 500 normal distributed points on $\mathbb{R}^2$

Theorems & Definitions (37)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof : Sketch of the proof
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Corollary 2.8
  • proof
  • ...and 27 more