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

Abstract

For $d\ge 2$ and i.i.d. $d$-dimensional observations ${\mathbf{X}}^{(1)},{\mathbf{X}}^{(2)},\dots$ with independent Exponential$(1)$ coordinates, let $\ddot{I}{†}_n$ denote the minimum ${\ell }^1$-norm among the maxima of $\{{\mathbf{X}}^{(1)},\dots ,{\mathbf{X}}^{(n)}\}$. (A _maximum_ from this set is an observation ${\mathbf{X}}^{(k)}$ with $1\le k\le n$ such that ${\mathbf{X}}^{(k)}\nprec {\mathbf{X}}^{(i)}$ for all $1\le i\le n$, where $\mathbf{x}\prec \mathbf{y}$ means that $x_j<y_j$ for $1\le j\le d$.) Key roles in the study of multivariate Pareto records are played by $\ddot{I}{†}_n$ and by the more easily handled maximum with the maximum ${\ell }^1$-norm. Fill, Naiman, and Sun (2024) proved that \[ φ_n = \ln n - \ln \ln \ln n - \ln(d - 1) + O_{\mathrm{p}}\!\left( \frac{1}{\ln \ln n} \right), \] where $Z_n=O_{\mathrm{p}}(a_n)$ means that $Z_n/a_n$ is bounded in probability, and conjectured that \[ (\ln \ln n) \left(φ_n - [\ln n - \ln \ln \ln n - \ln(d - 1)] \right) \] 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 $\ddot{I}{†}_n$.

34 pages