The Pareto Record Frontier

Abstract

For iid $d$-dimensional observations $X^{(1)}, X^{(2)}, \ldots$ with independent Exponential$(1)$ coordinates, consider the boundary (relative to the closed positive orthant), or "frontier", $F_n$ of the closed Pareto record-setting (RS) region \[ \mbox{RS}_n := \{0 \leq x \in {\mathbb R}^d: x \not\prec X^{(i)}\ \mbox{for all $1 \leq i \leq n$}\} \] at time $n$, where $0 \leq x$ means that $0 \leq x_j$ for $1 \leq j \leq d$ and $x \prec y$ means that $x_j < y_j$ for $1 \leq j \leq d$. With $x_+ := \sum_{j = 1}^d x_j$, let \[ F_n^- := \min\{x_+: x \in F_n\} \quad \mbox{and} \quad F_n^+ := \max\{x_+: x \in F_n\}, \] and define the width of $F_n$ as \[ W_n := F_n^+ - F_n^-. \] We describe typical and almost sure behavior of the processes $F^+$, $F^-$, and $W$. In particular, we show that $F^+_n \sim \ln n \sim F^-_n$ almost surely and that $W_n / \ln \ln n$ converges in probability to $d - 1$; and for $d \geq 2$ we show that, almost surely, the set of limit points of the sequence $W_n / \ln \ln n$ is the interval $[d - 1, d]$. We also obtain modifications of our results that are important in connection with efficient simulation of Pareto records. Let $T_m$ denote the time that the $m$th record is set. We show that $F^+_{T_m} \sim (d! m)^{1/d} \sim F^-_{T_m}$ almost surely and that $W_{T_m} / \ln m$ converges in probability to $1 - d^{-1}$; and for $d \geq 2$ we show that, almost surely, the sequence $W_{T_m} / \ln m$ has $\liminf$ equal to $1 - d^{-1}$ and $\limsup$ equal to $1$.