The Pareto record frontier

Abstract

For i.i.d. $d$-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 \[ \mbox {RS}_{n} := \{0 \leq x \in \mathbb {R}^{d}: x \not \prec X^{(i)} \ \text {for all}\ 1 \leq i \leq n\} \] 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 \[ F_{n}^{-} := \min \{x_{+}: x \in F_{n}\} \quad \text {and} \quad F_{n}^{+} := \max \{x_{+}: x \in F_{n}\}, \] 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 $1$.