Sharpened localization of the trailing point of the Pareto record frontier
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2024
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| author | Fill, James Allen Naiman, Daniel Sun, Ao |
| author_facet | Fill, James Allen Naiman, Daniel Sun, Ao |
| contents | For $d\ge2$ and iid $d$-dimensional observations $X^{(1)},X^{(2)},\dots$ with independent Exponential$(1)$ coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 2020) of the boundary (relative to the closed positive orthant), or "frontier", $F_n$ of the closed Pareto record-setting (RS) region \[ \mbox{RS}_n:=\{0\le x\in{\mathbb R}^d:x\not\prec X^{(i)}\mbox{\ for all $1\le i\le n$}\} \] at time $n$, where $0\le x$ means that $0\le x_j$ for $1\le j\le d$ and $x\prec y$ means that $x_j<y_j$ for $1\le j\le d$. With $x_+:=\sum_{j=1}^d x_j$, let \[ F_n^-:=\min\{x_+:x\in F_n\}\quad\mbox{and}\quad F_n^+:=\max\{x_+:x\in F_n\}. \] Almost surely, there are for each $n$ unique vectors $λ_n\in F_n$ and $τ_n\in F_n$ such that $F_n^+=(λ_n)_+$ and $F_n^-=(τ_n)_+$; we refer to $λ_n$ and $τ_n$ as the leading and trailing points, respectively, of the frontier. Fill and Naiman provided rather sharp information about the typical and almost sure behavior of $F^+$, but somewhat crude information about $F^-$, namely, that for any $\varepsilon >0$ and $c_n\to\infty$ we have \[ {\mathbb P}(F_n^- -\ln n\in (-(2+\varepsilon)\ln\ln\ln n,c_n))\to 1 \] (describing typical behavior) and almost surely \[ \limsup \frac{F_n^- - \ln n}{\ln \ln n} \le 0 \quad \mbox{and} \quad \liminf \frac{F_n^- - \ln n}{\ln \ln \ln n} \in [-2, -1]. \]
In this paper we use the theory of generators (minima of $F_n$) together with the first- and second-moment methods to improve considerably the trailing-point location results to \[ F_n^- - (\ln n - \ln \ln \ln n) \overset{\mathrm{P}}{\longrightarrow} - \ln(d - 1) \] (describing typical behavior) and, for $d \ge 3$, almost surely \begin{align*} &\limsup [F_n^- - (\ln n - \ln \ln \ln n)] \leq -\ln(d - 2) + \ln 2 \\ \mbox{and }&\liminf [F_n^- - (\ln n - \ln \ln \ln n)] \ge - \ln d - \ln 2. \end{align*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharpened localization of the trailing point of the Pareto record frontier Fill, James Allen Naiman, Daniel Sun, Ao Probability Data Structures and Algorithms 60D05 (Primary) 60F05, 60F15, G0G70, 60G17 (Secondary) For $d\ge2$ and iid $d$-dimensional observations $X^{(1)},X^{(2)},\dots$ with independent Exponential$(1)$ coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 2020) of the boundary (relative to the closed positive orthant), or "frontier", $F_n$ of the closed Pareto record-setting (RS) region \[ \mbox{RS}_n:=\{0\le x\in{\mathbb R}^d:x\not\prec X^{(i)}\mbox{\ for all $1\le i\le n$}\} \] at time $n$, where $0\le x$ means that $0\le x_j$ for $1\le j\le d$ and $x\prec y$ means that $x_j<y_j$ for $1\le j\le d$. With $x_+:=\sum_{j=1}^d x_j$, let \[ F_n^-:=\min\{x_+:x\in F_n\}\quad\mbox{and}\quad F_n^+:=\max\{x_+:x\in F_n\}. \] Almost surely, there are for each $n$ unique vectors $λ_n\in F_n$ and $τ_n\in F_n$ such that $F_n^+=(λ_n)_+$ and $F_n^-=(τ_n)_+$; we refer to $λ_n$ and $τ_n$ as the leading and trailing points, respectively, of the frontier. Fill and Naiman provided rather sharp information about the typical and almost sure behavior of $F^+$, but somewhat crude information about $F^-$, namely, that for any $\varepsilon >0$ and $c_n\to\infty$ we have \[ {\mathbb P}(F_n^- -\ln n\in (-(2+\varepsilon)\ln\ln\ln n,c_n))\to 1 \] (describing typical behavior) and almost surely \[ \limsup \frac{F_n^- - \ln n}{\ln \ln n} \le 0 \quad \mbox{and} \quad \liminf \frac{F_n^- - \ln n}{\ln \ln \ln n} \in [-2, -1]. \] In this paper we use the theory of generators (minima of $F_n$) together with the first- and second-moment methods to improve considerably the trailing-point location results to \[ F_n^- - (\ln n - \ln \ln \ln n) \overset{\mathrm{P}}{\longrightarrow} - \ln(d - 1) \] (describing typical behavior) and, for $d \ge 3$, almost surely \begin{align*} &\limsup [F_n^- - (\ln n - \ln \ln \ln n)] \leq -\ln(d - 2) + \ln 2 \\ \mbox{and }&\liminf [F_n^- - (\ln n - \ln \ln \ln n)] \ge - \ln d - \ln 2. \end{align*} |
| title | Sharpened localization of the trailing point of the Pareto record frontier |
| topic | Probability Data Structures and Algorithms 60D05 (Primary) 60F05, 60F15, G0G70, 60G17 (Secondary) |
| url | https://arxiv.org/abs/2402.17221 |