Almost sure convergence in extreme value theory

Shihong Cheng, Liang Peng, Yongcheng Qi

Research output: Contribution to journalArticle

72 Scopus citations

Abstract

Let X1 , . . . , Xn be independent random variables with common distribution function F. Define Mn := max Xi1≤i≤n and let G(cursive Greek chi) be one of the extreme - value distributions. Assume F ∈ D(G), i.e., there exist an > 0 and bn ∈ double-struck R sign such that P{(Mn - bn)/an ≤ cursive Greek chi} → G(cursive Greek chi), for cursive Greek chi ∈ double-struck R sign . Let 1(-∞,cursive Greek chi](·) denote the indicator function of the set (-∞, cursive Greek chi] and S(G) =: {cursive Greek chi : 0 < G(cursive Greek chi) < 1}. Obviously, 1(-∞,cursive Greek chi]((Mn - bn)/an) does not converge almost surely for any cursive Greek chi ∈ S(G). But we shall prove (equation presented).

Original languageEnglish (US)
Pages (from-to)43-50
Number of pages8
JournalMathematische Nachrichten
Volume190
DOIs
StatePublished - Jan 1 1998

Keywords

  • Almost sure convergence
  • Arithmetic means
  • Extreme value distribution
  • Logarithmetic means

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