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A remark on moment-dependent phase transitions in high-dimensional Gaussian approximations

Anders Bredahl Kock, David Preinerstorfer

TL;DR

This article studies the critical growth rates of dimension $d$ below which Gaussian approximations are asymptotically valid but beyond which they are not, and how these thresholds depend on the number of moments $m$ that the observations possess.

Abstract

In this article, we study the critical growth rates of dimension below which Gaussian critical values can be used for hypothesis testing but beyond which they cannot. We are particularly interested in how these growth rates depend on the number of moments that the observations possess.

A remark on moment-dependent phase transitions in high-dimensional Gaussian approximations

TL;DR

This article studies the critical growth rates of dimension below which Gaussian approximations are asymptotically valid but beyond which they are not, and how these thresholds depend on the number of moments that the observations possess.

Abstract

In this article, we study the critical growth rates of dimension below which Gaussian critical values can be used for hypothesis testing but beyond which they cannot. We are particularly interested in how these growth rates depend on the number of moments that the observations possess.
Paper Structure (7 sections, 2 theorems, 31 equations)

This paper contains 7 sections, 2 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. The phase transition at Gaussian critical values
  3. Consequences for high-dimensional hypothesis testing
  4. Appendix
  5. The family of distributions used in establishing the phase transition
  6. Proof of Theorem \ref{['thm:LB']}
  7. Proof of Theorem \ref{['thm:cherno']}

Key Result

Theorem 2.1

Let $m\in(2,\infty)$, $\alpha\in(0,1)$, and $c_d(\alpha)$ be a sequence that satisfies eq:critvalmaxtest. There exist i.i.d. random vectors $\bm{X}_1,\hdots,\bm{X}_n$ with independent entries $\bm{X}_{ij}\sim P_m$, and $P_m$ depending neither on $n$ nor $d$, having mean zero, variance one, and finit then

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 3.1