Advertisement

Extremes

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

Processes of rth largest

  • Boris Buchmann
  • Ross Maller
  • Sidney I. Resnick
Article

Abstract

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.

Keywords

Extremes Domain of attraction Markov property Extremal process 

AMS 2000 Subject Classifications

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold, B.C., Becker, A., Gather, U., Zahedi, H.: On the Markov property of order statistics. J. Statist. Plann. Inference 9(2), 147–154 (1984)MathSciNetCrossRefGoogle Scholar
  2. Cramer, E., Tran, T.H.: Generalized order statistics from arbitrary distributions and the Markov chain property. J. Statist. Plann. Inference 139(12), 4064–4071 (2009)MathSciNetCrossRefGoogle Scholar
  3. de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer, New York (2006)CrossRefGoogle Scholar
  4. Deheuvels, P.: A Construction of Extremal Processes. In: Probability and Statistical Inference (Bad Tatzmannsdorf, 1981), pp. 53–57. Reidel, Dordrecht (1982)Google Scholar
  5. Deheuvels, P.: The strong approximation of extremal processes. II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 62(1), 7–15 (1983)MathSciNetCrossRefGoogle Scholar
  6. Dwass, M.: Extremal processes. Ann. Math. Statist 35, 1718–1725 (1964)MathSciNetCrossRefGoogle Scholar
  7. Dwass, M.: Extremal processes. II. Illinois. J. Math. 10, 381–391 (1966)MathSciNetzbMATHGoogle Scholar
  8. Dwass, M.: Extremal processes. III. Bull. Inst. Math. Acad. Sinica 2, 255–265 (1974). Collection of articles in celebration of the sixtieth birthday of Ky FanMathSciNetzbMATHGoogle Scholar
  9. Engelen, R., Tommassen, P., Vervaat, W.: Ignatov’s theorem: a new and short proof. J. Appl. Probab. 25A, 229–236 (1988). A celebration of applied probabilityMathSciNetCrossRefGoogle Scholar
  10. Goldie, C.M., Rogers, L.C.G.: The k-record processes are i.i.d. Z. Wahrsch. Verw. Gebiete 67(2), 197–211 (1984)MathSciNetCrossRefGoogle Scholar
  11. Goldie, C.M., Maller, R.A.: Generalized densities of order statistics Statist. Neerlandica 53(2), 222–246 (1999)MathSciNetCrossRefGoogle Scholar
  12. Ignatov, Z.: Ein von der Variationsreihe erzeugter Poissonscher Punktprozeß. Annuaire Univ. Sofia Fac. Math. Méc. 71(2), 79–94 (1986). 1976/77MathSciNetzbMATHGoogle Scholar
  13. Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975). With a foreword by G.S. Watson, Wiley Series in Probability and Mathematical StatisticszbMATHGoogle Scholar
  14. Molchanov, I.: Theory of Random Sets. Probability and its Applications (New York). Springer, London (2005)Google Scholar
  15. Rényi, A.: Théorie des éléments saillants d’une suite d’observations. Ann. Fac. Sci. Univ. Clermont-Ferrand No. 8, 7–13 (1962)MathSciNetzbMATHGoogle Scholar
  16. Resnick, S.I.: Tail equivalence and its applications. J. Appl. Probab. 8, 136–156 (1971)MathSciNetCrossRefGoogle Scholar
  17. Resnick, S.I.: Limit laws for record values. Stochastic Process. Appl. 1, 67–82 (1973)MathSciNetCrossRefGoogle Scholar
  18. Resnick, S.I.: Inverses of extremal processes. Adv. Appl. Probab. 6, 392–406 (1974)MathSciNetCrossRefGoogle Scholar
  19. Resnick, S.I.: Weak convergence to extremal processes. Ann. Probab. 3(6), 951–960 (1975)MathSciNetCrossRefGoogle Scholar
  20. Resnick, S.I.: Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007). ISBN: 0-387-24272-4Google Scholar
  21. Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer, New York (2008). Reprint of the 1987 originalzbMATHGoogle Scholar
  22. Resnick, S.I., Rubinovitch, M.: The structure of extremal processes. Adv. Appl. Probab. 5, 287–307 (1973)MathSciNetCrossRefGoogle Scholar
  23. Rüschendorf, L.: Two remarks on order statistics. J. Statist. Plann. Inference 11(1), 71–74 (1985)MathSciNetCrossRefGoogle Scholar
  24. Shorrock, R.W.: On discrete time extremal processes. Adv. Appl. Probab. 6, 580–592 (1974)MathSciNetCrossRefGoogle Scholar
  25. Shorrock, R.W.: Extremal processes and random measures. J. Appl. Probab. 12, 316–323 (1975)MathSciNetCrossRefGoogle Scholar
  26. Stam, A.J.: Independent Poisson processes generated by record values and inter-record times. Stochastic Process. Appl. 19(2), 315–325 (1985)MathSciNetCrossRefGoogle Scholar
  27. Vervaat, W., Holwerda, H. (eds.): Probability and Lattices, Volume 110 of CWI Tract. Stichting Mathematisch Centrum. Centrum voor Wiskunde en Informatica, Amsterdam (1997)Google Scholar
  28. Weissman, I.: Extremal processes generated by independent nonidentically distributed random variables. Ann. Probab. 3, 172–177 (1975)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

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

Personalised recommendations