, Volume 21, Issue 4, pp 485–508 | Cite as

Processes of rth largest

  • Boris Buchmann
  • Ross Maller
  • Sidney I. ResnickEmail author


For integers nr, 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.


Extremes Domain of attraction Markov property Extremal process 

AMS 2000 Subject Classifications

Primary—60F05, 60G70, 60K99 Secondary—60G55, 60G51 


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Authors and Affiliations

  1. 1.Research School of Finance, Actuarial Studies & StatisticsAustralian National UniversityCanberraAustralia
  2. 2.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

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