The Pareto Record Frontier

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", $Fn$ 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 $0 \leq xj$ for $1 \leq j \leq d$ and $x \prec y$ means that $xj < yj$ for $1 \leq j \leq d$. With $x+ := \sum{j = 1}^d xj$, let [ Fn^- := \min{x+: x \in Fn} \quad \mbox{and} \quad Fn⁺ := \max{x+: x \in Fn}, ] and define the width of $Fn$ as [ Wn := Fn⁺ - Fn^-. ] 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 $Wn / \ln \ln n$ converges in probability to $d - 1$; and for $d \geq 2$ we show that, almost surely, the set of limit points of the sequence $Wn / \ln \ln n$ 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 $Tm$ denote the time that the $m$th record is set. We show that $F⁺{Tm} \sim (d! m)^{1/d} \sim F^-{Tm}$ almost surely and that $W{Tm} / \ln m$ 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 $1$.