On extreme values in stationary sequences View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

1974-12

AUTHORS

M. R. Leadbetter

ABSTRACT

In this paper, extreme value theory is considered for stationary sequences ζn satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are:To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(ξ1...ξn), for such sequences.To obtain limiting laws of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathop {\lim }\limits_{n \to \infty } \Pr \{ M_n^{(r)} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } u\} = e^{ - \tau } \sum\limits_{s = 0}^{r - 1} {\tau ^s /S!} $$ \end{document} where Mn(r)is the r-th largest of ξ1...ξn, and Prξ1>un∼Τ/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results.As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {ξn} were i.i.d.To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions. More... »

PAGES

289-303

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf00532947

DOI

http://dx.doi.org/10.1007/bf00532947

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1006825626


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "University of North Carolina, USA", 
          "id": "http://www.grid.ac/institutes/grid.410711.2", 
          "name": [
            "University of North Carolina, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Leadbetter", 
        "givenName": "M. R.", 
        "id": "sg:person.010453124705.04", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010453124705.04"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "sg:pub.10.1007/bf02590962", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1030972034", 
          "https://doi.org/10.1007/bf02590962"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "1974-12", 
    "datePublishedReg": "1974-12-01", 
    "description": "In this paper, extreme value theory is considered for stationary sequences \u03b6n satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are:To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(\u03be1...\u03ben), for such sequences.To obtain limiting laws of the form \\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n$$\\mathop {\\lim }\\limits_{n \\to \\infty } \\Pr \\{ M_n^{(r)} \\underset{\\raise0.3em\\hbox{$\\smash{\\scriptscriptstyle-}$}}{ \\leqslant } u\\}  = e^{ - \\tau } \\sum\\limits_{s = 0}^{r - 1} {\\tau ^s /S!} $$\n\\end{document} where Mn(r)is the r-th largest of \u03be1...\u03ben, and Pr\u03be1>un\u223c\u03a4/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results.As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {\u03ben} were i.i.d.To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions.", 
    "genre": "article", 
    "id": "sg:pub.10.1007/bf00532947", 
    "inLanguage": "en", 
    "isAccessibleForFree": true, 
    "isPartOf": [
      {
        "id": "sg:journal.1053886", 
        "issn": [
          "0178-8051", 
          "1432-2064"
        ], 
        "name": "Probability Theory and Related Fields", 
        "publisher": "Springer Nature", 
        "type": "Periodical"
      }, 
      {
        "issueNumber": "4", 
        "type": "PublicationIssue"
      }, 
      {
        "type": "PublicationVolume", 
        "volumeNumber": "28"
      }
    ], 
    "keywords": [
      "stationary sequence", 
      "stationary normal sequences", 
      "extreme value theory", 
      "basic theorems", 
      "Poisson property", 
      "dependence restrictions", 
      "covariance condition", 
      "asymptotic distribution", 
      "value theory", 
      "asymptotic form", 
      "\u03ben", 
      "extreme values", 
      "maximum Mn", 
      "such sequences", 
      "theorem", 
      "Gnedenko", 
      "\u03be1", 
      "theory", 
      "strong mixing", 
      "existence", 
      "assumption", 
      "law", 
      "distribution", 
      "form", 
      "sequence", 
      "restriction", 
      "conditions", 
      "properties", 
      "results", 
      "normal sequence", 
      "values", 
      "consequences", 
      "mixing", 
      "aim", 
      "Th", 
      "Mn", 
      "paper", 
      "possible non-degenerate asymptotic forms", 
      "non-degenerate asymptotic forms", 
      "mild covariance conditions"
    ], 
    "name": "On extreme values in stationary sequences", 
    "pagination": "289-303", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1006825626"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/bf00532947"
        ]
      }
    ], 
    "sameAs": [
      "https://doi.org/10.1007/bf00532947", 
      "https://app.dimensions.ai/details/publication/pub.1006825626"
    ], 
    "sdDataset": "articles", 
    "sdDatePublished": "2022-01-01T18:01", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20220101/entities/gbq_results/article/article_150.jsonl", 
    "type": "ScholarlyArticle", 
    "url": "https://doi.org/10.1007/bf00532947"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/bf00532947'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/bf00532947'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bf00532947'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bf00532947'


 

This table displays all metadata directly associated to this object as RDF triples.

102 TRIPLES      22 PREDICATES      67 URIs      58 LITERALS      6 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/bf00532947 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author Nec63271f0df5430ca25e2b962dffa66c
4 schema:citation sg:pub.10.1007/bf02590962
5 schema:datePublished 1974-12
6 schema:datePublishedReg 1974-12-01
7 schema:description In this paper, extreme value theory is considered for stationary sequences ζn satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are:To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(ξ1...ξn), for such sequences.To obtain limiting laws of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathop {\lim }\limits_{n \to \infty } \Pr \{ M_n^{(r)} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } u\} = e^{ - \tau } \sum\limits_{s = 0}^{r - 1} {\tau ^s /S!} $$ \end{document} where Mn(r)is the r-th largest of ξ1...ξn, and Prξ1>un∼Τ/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results.As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {ξn} were i.i.d.To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions.
8 schema:genre article
9 schema:inLanguage en
10 schema:isAccessibleForFree true
11 schema:isPartOf N1f36863d256649c6a5cdacfce3011428
12 N8dff119a17704e3eb66f0edba8fcff34
13 sg:journal.1053886
14 schema:keywords Gnedenko
15 Mn
16 Poisson property
17 Th
18 aim
19 assumption
20 asymptotic distribution
21 asymptotic form
22 basic theorems
23 conditions
24 consequences
25 covariance condition
26 dependence restrictions
27 distribution
28 existence
29 extreme value theory
30 extreme values
31 form
32 law
33 maximum Mn
34 mild covariance conditions
35 mixing
36 non-degenerate asymptotic forms
37 normal sequence
38 paper
39 possible non-degenerate asymptotic forms
40 properties
41 restriction
42 results
43 sequence
44 stationary normal sequences
45 stationary sequence
46 strong mixing
47 such sequences
48 theorem
49 theory
50 value theory
51 values
52 ξ1
53 ξn
54 schema:name On extreme values in stationary sequences
55 schema:pagination 289-303
56 schema:productId Ne1042a21bbb345af99f53ae0d852735f
57 Neca4dba54da7441f9869d23d5114f5c1
58 schema:sameAs https://app.dimensions.ai/details/publication/pub.1006825626
59 https://doi.org/10.1007/bf00532947
60 schema:sdDatePublished 2022-01-01T18:01
61 schema:sdLicense https://scigraph.springernature.com/explorer/license/
62 schema:sdPublisher Ncea7e552392f449d92d5963941404906
63 schema:url https://doi.org/10.1007/bf00532947
64 sgo:license sg:explorer/license/
65 sgo:sdDataset articles
66 rdf:type schema:ScholarlyArticle
67 N1f36863d256649c6a5cdacfce3011428 schema:volumeNumber 28
68 rdf:type schema:PublicationVolume
69 N8dff119a17704e3eb66f0edba8fcff34 schema:issueNumber 4
70 rdf:type schema:PublicationIssue
71 Ncea7e552392f449d92d5963941404906 schema:name Springer Nature - SN SciGraph project
72 rdf:type schema:Organization
73 Ne1042a21bbb345af99f53ae0d852735f schema:name doi
74 schema:value 10.1007/bf00532947
75 rdf:type schema:PropertyValue
76 Nec63271f0df5430ca25e2b962dffa66c rdf:first sg:person.010453124705.04
77 rdf:rest rdf:nil
78 Neca4dba54da7441f9869d23d5114f5c1 schema:name dimensions_id
79 schema:value pub.1006825626
80 rdf:type schema:PropertyValue
81 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
82 schema:name Mathematical Sciences
83 rdf:type schema:DefinedTerm
84 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
85 schema:name Pure Mathematics
86 rdf:type schema:DefinedTerm
87 sg:journal.1053886 schema:issn 0178-8051
88 1432-2064
89 schema:name Probability Theory and Related Fields
90 schema:publisher Springer Nature
91 rdf:type schema:Periodical
92 sg:person.010453124705.04 schema:affiliation grid-institutes:grid.410711.2
93 schema:familyName Leadbetter
94 schema:givenName M. R.
95 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010453124705.04
96 rdf:type schema:Person
97 sg:pub.10.1007/bf02590962 schema:sameAs https://app.dimensions.ai/details/publication/pub.1030972034
98 https://doi.org/10.1007/bf02590962
99 rdf:type schema:CreativeWork
100 grid-institutes:grid.410711.2 schema:alternateName University of North Carolina, USA
101 schema:name University of North Carolina, USA
102 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...