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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
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)
de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer, New York (2006)
Deheuvels, P.: A Construction of Extremal Processes. In: Probability and Statistical Inference (Bad Tatzmannsdorf, 1981), pp. 53–57. Reidel, Dordrecht (1982)
Deheuvels, P.: The strong approximation of extremal processes. II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 62(1), 7–15 (1983)
Dwass, M.: Extremal processes. Ann. Math. Statist 35, 1718–1725 (1964)
Dwass, M.: Extremal processes. II. Illinois. J. Math. 10, 381–391 (1966)
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 Fan
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 probability
Goldie, C.M., Rogers, L.C.G.: The k-record processes are i.i.d. Z. Wahrsch. Verw. Gebiete 67(2), 197–211 (1984)
Goldie, C.M., Maller, R.A.: Generalized densities of order statistics Statist. Neerlandica 53(2), 222–246 (1999)
Ignatov, Z.: Ein von der Variationsreihe erzeugter Poissonscher Punktprozeß. Annuaire Univ. Sofia Fac. Math. Méc. 71(2), 79–94 (1986). 1976/77
Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975). With a foreword by G.S. Watson, Wiley Series in Probability and Mathematical Statistics
Molchanov, I.: Theory of Random Sets. Probability and its Applications (New York). Springer, London (2005)
Rényi, A.: Théorie des éléments saillants d’une suite d’observations. Ann. Fac. Sci. Univ. Clermont-Ferrand No. 8, 7–13 (1962)
Resnick, S.I.: Tail equivalence and its applications. J. Appl. Probab. 8, 136–156 (1971)
Resnick, S.I.: Limit laws for record values. Stochastic Process. Appl. 1, 67–82 (1973)
Resnick, S.I.: Inverses of extremal processes. Adv. Appl. Probab. 6, 392–406 (1974)
Resnick, S.I.: Weak convergence to extremal processes. Ann. Probab. 3(6), 951–960 (1975)
Resnick, S.I.: Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007). ISBN: 0-387-24272-4
Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer, New York (2008). Reprint of the 1987 original
Resnick, S.I., Rubinovitch, M.: The structure of extremal processes. Adv. Appl. Probab. 5, 287–307 (1973)
Rüschendorf, L.: Two remarks on order statistics. J. Statist. Plann. Inference 11(1), 71–74 (1985)
Shorrock, R.W.: On discrete time extremal processes. Adv. Appl. Probab. 6, 580–592 (1974)
Shorrock, R.W.: Extremal processes and random measures. J. Appl. Probab. 12, 316–323 (1975)
Stam, A.J.: Independent Poisson processes generated by record values and inter-record times. Stochastic Process. Appl. 19(2), 315–325 (1985)
Vervaat, W., Holwerda, H. (eds.): Probability and Lattices, Volume 110 of CWI Tract. Stichting Mathematisch Centrum. Centrum voor Wiskunde en Informatica, Amsterdam (1997)
Weissman, I.: Extremal processes generated by independent nonidentically distributed random variables. Ann. Probab. 3, 172–177 (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was initiated and partially supported by ARC grants DP1092502 and DP160104737. S. Resnick also received significant support from US Army MURI grant W911NF-12-1-0385 to Cornell University; Resnick gratefully acknowledges hospitality, administrative support and space during several visits to the Research School of Finance, Actuarial Studies & Statistics, Australian National University.
Rights and permissions
About this article
Cite this article
Buchmann, B., Maller, R. & Resnick, S.I. Processes of rth largest. Extremes 21, 485–508 (2018). https://doi.org/10.1007/s10687-018-0308-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-018-0308-x