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Convergence in distribution of quotients of order statistics

Authors
Journal
Stochastic Processes and their Applications
0304-4149
Publisher
Elsevier
Publication Date
Volume
3
Issue
3
Identifiers
DOI: 10.1016/0304-4149(75)90027-7
Keywords
  • Order Statistics
  • Limit Theorem
  • Wiener-Taubner Theorem
  • Partial Maxima
  • Regular Variation

Abstract

Abstract Let X 1, X 2,… be i.i.d. random variables with continuous distribution function F < 1. It is known that if 1 - F( x) varies regularly of order - p, the successive quotients of the order statistics in decreasing order of X 1,…, X n are asymptotically independent, as n→∞, with distribution functions x kp , k = 1, 2, …. A strong converse is proved, viz. convergence in distribution of this type of one of the quotients implies regular varation of 1 - F( x).

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