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Processes of rth largest

Buchmann, Boris; Maller, Ross; Resnick, Sidney I.

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For integers n ≥ r, we treat the rth largest of a sample of size n as an R∞ -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...[Show more]

dc.contributor.authorBuchmann, Boris
dc.contributor.authorMaller, Ross
dc.contributor.authorResnick, Sidney I.
dc.date.accessioned2021-01-14T00:10:03Z
dc.identifier.issn1386-1999
dc.identifier.urihttp://hdl.handle.net/1885/219353
dc.description.abstractFor integers n ≥ r, we treat the rth largest of a sample of size n as an R∞ -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 R+×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.
dc.description.sponsorshipThis 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
dc.format.mimetypeapplication/pdf
dc.language.isoen_AU
dc.publisherKluwer Academic Publishers
dc.rights© Springer Science+Business Media, LLC, part of Springer Nature 2018
dc.sourceExtremes
dc.subjectExtremes
dc.subjectDomain of attraction
dc.subjectMarkov property
dc.subjectExtremal process
dc.titleProcesses of rth largest
dc.typeJournal article
local.description.notesImported from ARIES
local.identifier.citationvolume21
dc.date.issued2018
local.identifier.absfor010104 - Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)
local.identifier.ariespublicationa383154xPUB9458
local.publisher.urlhttps://link.springer.com
local.type.statusAccepted Version
local.contributor.affiliationBuchmann, Boris, College of Business and Economics, ANU
local.contributor.affiliationMaller, Ross, College of Business and Economics, ANU
local.contributor.affiliationResnick, Sidney I., Cornell University
dc.relationhttp://purl.org/au-research/grants/arc/DP1092502
dc.relationhttp://purl.org/au-research/grants/arc/DP160104737
local.bibliographicCitation.issue4
local.bibliographicCitation.startpage485
local.bibliographicCitation.lastpage508
local.identifier.doi10.1007/s10687-018-0308-x
dc.date.updated2020-11-02T04:18:40Z
local.identifier.scopusID2-s2.0-85042377313
dcterms.accessRightsOpen Access
dc.provenancehttps://v2.sherpa.ac.uk/id/publication/17259..."The Accepted Version can be archived in Institutional Repository. 12 months embargo" from SHERPA/RoMEO site (as at 4/02/2021).
CollectionsANU Research Publications

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