Sharpened localization of the trailing point of the Pareto record frontier

TL;DR

The paper sharpens the trailing-point localization for the Pareto record frontier in $d\ge2$ dimensions with i.i.d. Exponential$(1)$ coordinates. The authors prove that the trailing point satisfies $F_n^- - (\ln n - \ln\ln\ln n) \overset{P}{\longrightarrow} -\ln(d-1)$, and establish almost-sure limsup/liminf bounds: $\limsup [F_n^- - (\ln n - \ln\ln\ln n)] \le -\ln(d-2) + \ln 2$ and $\liminf [F_n^- - (\ln n - \ln\ln\ln n)] \ge - \ln d - \ln 2$. Extending these results to $\widehat{F}^-_n$ (the minimum coordinate-sum among current records), the paper shows analogous typical behavior with a sharper remainder, proves $\widehat{F}^-_n - F^-_n \xrightarrow{P}0$, and provides a conjecture on a nondegenerate limiting distribution for a suitable scaling. Central to the analysis are the generator framework (minima of $F_n$) and first- and second-moment methods for counts of remaining records and generators, yielding precise mean/variance bounds and enabling tight probabilistic control of the trailing boundary. The results give a deeper, more precise description of the Pareto frontier geometry, with potential implications for simulations and extreme-value understanding in higher dimensions.

Abstract

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*}

Figures (2)

• Figure 1: Record frontier $F_n$ based on $n$ observations (for some $n \geq 10$) resulting in 10 current records (shown as solid points), with the three hyperplanes $x_+ = F_n^+$, $x_+ = F_n^-$, and $x_+ = \widehat{F}_n^-$, the leading point $\lambda_n$ and the trailing point $\tau_n$. Concerning the three hyperplanes, see Definition and \ref{['hFn-def']}.
• Figure 2: Example of a record frontier in dimension $d=3$ with $\rho=8$ remaining records shown in red and the resulting $\gamma = 17$ generators: three one-dimensional generators shown in violet, eight two-dimensional generators shown in blue, and six three-dimensional (interior) generators shown in green. The lower boundary of one of the orthants $O^+_g$ is shown using dashed lines.

Theorems & Definitions (52)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Definition 1.4
• Lemma 1.5: nondecreasing sample paths for $F_n^-$
• proof
• Theorem 1.6: Kiefer Kiefer(1972)
• Theorem 1.7: Fillboundary(2020), Theorem 1.12
• Theorem 1.8
• Corollary 1.9