Processes of r^th largest

Abstract

For integers n ≥ r, we treat the rth largest of a sample of size n as an $\mathbb {R}^{\infty }$-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, 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 M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on $\mathbb {R}_{+}\times \mathbb {R}$ is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t > 0. This necessitates a different approach to the asymptotics in this case.