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 $\varphi_n$.
Categories: math.PR
