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A new fine-scale Berry-Esseen-type Gumbel-limit theorem for multivariate maxima

James Allen Fill

TL;DR

The paper addresses the fine-scale distributional behavior of the minimum $\ell^1$-norm among multivariate maxima in a high-dimensional Pareto-record setting, establishing a Berry–Esseen-type limit for the centered and rescaled $\varphi_n$. The authors develop a multi-layer Poisson-approximation framework, employing boundary refinements, Poissonization, conditioning, discretization, and Arratia–Chen–Stein techniques to achieve a sharp $O((\mathop{\mathrm{L}}_2 n)^{-1/2}\mathop{\mathrm{L}}_3 n)$ rate in the Kolmogorov distance to a Gumbel limit for $-G$, where $G$ has a location and scale tied to $d$ via $- \frac{\mathop{\mathrm{L}}[(d-1)!]}{d-1}$ and $\frac{1}{d-1}$ respectively. The method yields a detailed, scale-sensitive description of the near-boundary maxima, culminating in a rigorous Poisson-process-based account of the smallest maximum and its potential asymptotic independence from the rest of the maxima. This sharp analysis refines prior results on the maximum's location and supports precise probabilistic inference for multivariate Pareto records in high dimensions.

Abstract

For $d \geq 2$ and i.i.d. $d$-dimensional observations $\mathbf{X}^{(1)}, \mathbf{X}^{(2)}, \ldots$ with independent Exponential$(1)$ coordinates, let $\varphi_n$ denote the minimum $\ell^1$-norm among the maxima of $\{\mathbf{X}^{(1)}, \ldots, \mathbf{X}^{(n)}\}$. (A _maximum_ from this set is an observation $\mathbf{X}^{(k)}$ with $1 \leq k \leq n$ such that $\mathbf{X}^{(k)} \not\prec \mathbf{X}^{(i)}$ for all $1 \leq i \leq n$, where $\mathbf{x} \prec \mathbf{y}$ means that $x_j < y_j$ for $1 \leq j \leq d$.) Key roles in the study of multivariate Pareto records are played by $\varphi_n$ and by the more easily handled maximum with the maximum $\ell^1$-norm. Fill, Naiman, and Sun (2024) proved that \[ \varphi_n = \ln n - \ln \ln \ln n - \ln(d - 1) + O_{\mathrm{p}}\!\left( \frac{1}{\ln \ln n} \right), \] where $Z_n = O_{\mathrm{p}}(a_n)$ means that $Z_n / a_n$ is bounded in probability, and conjectured that \[ (\ln \ln n) \left(\varphi_n - [\ln n - \ln \ln \ln n - \ln(d - 1)] \right) \] has a nondegenerate limiting distribution, suggesting that the limiting distribution might be that of $ - G$, where $G$ has a Gumbel distribution with location $ - \frac{\ln[(d - 1)!]}{d - 1}$ and scale $\frac{1}{d - 1}$. In the present paper we prove a Berry-Esseen-type theorem for this convergence in distribution, thereby establishing a very sharp result for $\varphi_n$.

A new fine-scale Berry-Esseen-type Gumbel-limit theorem for multivariate maxima

TL;DR

The paper addresses the fine-scale distributional behavior of the minimum -norm among multivariate maxima in a high-dimensional Pareto-record setting, establishing a Berry–Esseen-type limit for the centered and rescaled . The authors develop a multi-layer Poisson-approximation framework, employing boundary refinements, Poissonization, conditioning, discretization, and Arratia–Chen–Stein techniques to achieve a sharp rate in the Kolmogorov distance to a Gumbel limit for , where has a location and scale tied to via and respectively. The method yields a detailed, scale-sensitive description of the near-boundary maxima, culminating in a rigorous Poisson-process-based account of the smallest maximum and its potential asymptotic independence from the rest of the maxima. This sharp analysis refines prior results on the maximum's location and supports precise probabilistic inference for multivariate Pareto records in high dimensions.

Abstract

For and i.i.d. -dimensional observations with independent Exponential coordinates, let denote the minimum -norm among the maxima of . (A _maximum_ from this set is an observation with such that for all , where means that for .) Key roles in the study of multivariate Pareto records are played by and by the more easily handled maximum with the maximum -norm. Fill, Naiman, and Sun (2024) proved that where means that is bounded in probability, and conjectured that \[ (\ln \ln n) \left(\varphi_n - [\ln n - \ln \ln \ln n - \ln(d - 1)] \right) \] has a nondegenerate limiting distribution, suggesting that the limiting distribution might be that of , where has a Gumbel distribution with location and scale . In the present paper we prove a Berry-Esseen-type theorem for this convergence in distribution, thereby establishing a very sharp result for .
Paper Structure (13 sections, 11 theorems, 167 equations)

This paper contains 13 sections, 11 theorems, 167 equations.

Table of Contents

  1. Introduction, background, assumptions, notation, and statement of main results
  2. Distribution tails
  3. Slight modification $(b_n)$ of the boundary $(b^*_n)$
  4. A boundary $(\underline{b}_n)$ that is appreciably smaller than $(b_n)$
  5. Poissonization
  6. Conditioning
  7. Discretization
  8. A sequence of Poisson distributions indexed by $m$
  9. A sequence of Poisson distributions indexed by $n$
  10. Poisson approximation for $Z_{n, m}$
  11. The quantity $\mathfrak{b}_1(n, m)$
  12. Distribution of the smallest maximum
  13. Collected results from Fill et al. FNS_2024, and closely related results

Key Result

Theorem 1.1

Fix $d \geq 2$. Let $\mathbf{X}^{(1)}, \mathbf{X}^{(2)}, \ldots$, $\mathbf{X}^{(n)}$ be independent random $d$-dimensional vectors each having independent Exponential$(1)$ coordinates, let $\varphi_n$ denote the minimum $\ell^1$-norm of any maximum of these vectors, and define $\varphi_n^{\circ}$ as

Theorems & Definitions (27)

  • Theorem 1.1: main theorem: Gumbel limit
  • Theorem 1.2: Poisson approximation
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Remark 4.1
  • proof : Proof of \ref{['TV22']}
  • Proposition 6.1
  • ...and 17 more