  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.
