Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Point process convergence for symmetric functions of high-dimensional random vectors

Johannes Heiny, Carolin Kleemann

TL;DR

The convergence of a sequence of point processes with dependent points to a Poisson random measure is proved and a generalization of maximum convergence to point process convergence is given for simple linear rank statistics, rank-type U-statistics and the entries of sample covariance matrices.

Abstract

The convergence of a sequence of point processes with dependent points, defined by a symmetric function of iid high-dimensional random vectors, to a Poisson random measure is proved. This also implies the convergence of the joint distribution of a fixed number of upper order statistics. As applications of the result a generalization of maximum convergence to point process convergence is given for simple linear rank statistics, rank-type U-statistics and the entries of sample covariance matrices.

Point process convergence for symmetric functions of high-dimensional random vectors

TL;DR

The convergence of a sequence of point processes with dependent points to a Poisson random measure is proved and a generalization of maximum convergence to point process convergence is given for simple linear rank statistics, rank-type U-statistics and the entries of sample covariance matrices.

Abstract

The convergence of a sequence of point processes with dependent points, defined by a symmetric function of iid high-dimensional random vectors, to a Poisson random measure is proved. This also implies the convergence of the joint distribution of a fixed number of upper order statistics. As applications of the result a generalization of maximum convergence to point process convergence is given for simple linear rank statistics, rank-type U-statistics and the entries of sample covariance matrices.
Paper Structure (13 sections, 16 theorems, 141 equations, 1 figure)

This paper contains 13 sections, 16 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of this paper
  3. Notation
  4. Point process convergence
  5. Applications
  6. Relative ranks
  7. Kendall's tau
  8. Spearman's rho
  9. Sample covariances
  10. Proofs of the Results
  11. Proofs of the results in Section \ref{['sec1']}
  12. Proof of Proposition \ref{['prop:tau']}
  13. Proof of Proposition \ref{['prop:ext']}

Key Result

Theorem 2.1

Let $\mathbf{x}_1,\ldots, \mathbf{x}_p$ be $n$-dimensional, independent and identically distributed random vectors and $p=p_n$ is some sequence of positive integers tending to infinity as $n\to\infty$. Additionally, let $g=g_n:\mathbb{R}^{mn}\to (v,w)$ be a measurable and symmetric function, where $ Then we have $M_n\stackrel{d}{\rightarrow} M$.

Figures (1)

  • Figure 1: Four largest distances between 500 normal distributed points

Theorems & Definitions (28)

  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • Example 2.4: Interpoint distances
  • Proposition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • proof
  • Lemma 3.1: Lemma C4 in han:chen:liu:2017
  • ...and 18 more