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

For $d\ge 2$ and i.i.d. $d$-dimensional observations ${\mathbf{X}}^{(1)},{\mathbf{X}}^{(2)},\dots$ with independent Exponential$(1)$ coordinates, let ${\phi }_n$ denote the minimum ${\mathrm{\ell }}^1$-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 ${\mathbf{X}}^{(k)}\prec ̸{\mathbf{X}}^{(i)}$ 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 ${\phi }_n$ and by the more easily handled maximum with the maximum ${\mathrm{\ell }}^1$-norm. Fill, Naiman, and Sun (2024) proved that $${\phi }_n=\ln n-\ln \ln \ln n-\ln (d-1)+O_{\mathrm{p}}\text{\hspace{0.17em}⁣}\left(\frac{1}{\ln \ln n}\right),$$ where $Z_n=O_{\mathrm{p}}(a_n)$ means that $Z_n\mathrm{/}{a}_n$ is bounded in probability, and conjectured that $$(\ln \ln n)({\phi }_n-[\ln n-\ln \ln \ln n-\ln (d-1)])$$ 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 ${\phi }_n$.