Processes of rth Largest

TL;DR

This paper studies the stochastic process formed by the r-th largest sample value, proving its Markov property, analyzing its asymptotic behavior, and exploring continuous-time analogs with Poisson processes, revealing unique features.

Contribution

It introduces a Markov process framework for the r-th largest sample, analyzes its asymptotics, and extends the theory to continuous-time Poisson process models with new insights.

Findings

The r-th largest process is Markovian.

Weak limits are established under tail domain conditions.

Continuous-time processes exhibit infinitely many points below the r-th highest.

Abstract

For integers $n\geq r$, we treat the $r$th largest of a sample of size $n$ as an $\mathbb{R}^\infty$-valued stochastic process in $r$ which we denote $\mathbf{M}^{(r)}$. We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behaviour of $\mathbf{M}^{(r)}$ as $r\to\infty$, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of $\mathbf{M}^{(r)}$ and $\mathbf{M}^{(r)}$ itself, after norming and centering. In continuous time, an analogous process $\mathbf{Y}^{(r)}r$ based on a two-dimensional Poisson process on $\mathbb{R}_+\times \mathbb{R}$ is treated similarly, but we find that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the $r$th highest point up to time $t$ for any $t>0$. This necessitates a different approach to the asymptotics in this case.