A new fine-scale Berry-Esseen-type Gumbel-limit theorem for multivariate maxima
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914280338620416 |
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| author | Fill, James Allen |
| author_facet | Fill, James Allen |
| contents | For $d \geq 2$ and i.i.d. $d$-dimensional observations $\mathbf{X}^{(1)}, \mathbf{X}^{(2)}, \ldots$ with independent Exponential$(1)$ coordinates, let $φ_n$ 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 $φ_n$ 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 $φ_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18170 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new fine-scale Berry-Esseen-type Gumbel-limit theorem for multivariate maxima Fill, James Allen Probability 60D05 (Primary) 60F05 (Secondary) For $d \geq 2$ and i.i.d. $d$-dimensional observations $\mathbf{X}^{(1)}, \mathbf{X}^{(2)}, \ldots$ with independent Exponential$(1)$ coordinates, let $φ_n$ 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 $φ_n$ 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 $φ_n$. |
| title | A new fine-scale Berry-Esseen-type Gumbel-limit theorem for multivariate maxima |
| topic | Probability 60D05 (Primary) 60F05 (Secondary) |
| url | https://arxiv.org/abs/2601.18170 |