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

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 $\varphi_n$.