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December 1997 The domain of attraction of the α-sun operator for type II and type III distributions
Gerard Hooghiemstra, Priscilla E. Greenwood
Bernoulli 3(4): 479-489 (December 1997).


Let ( Y n) be a sequence of independent random variables with common distribution F and define the iteration X 0 =x 0 , X n :=X n -1(αX n -1+Y n) , α [0,1) . We denote by D (Φ γ) the domain of maximal attraction of Φ γ , the extreme value distribution of the first type. Greenwood and Hooghiemstra showed in 1991 that for F D(Φ γ) there exist norming constants a n >0 and b n R such that a n - 1 {X n-b n/(1-α)} has a non-degenerate (distributional) limit. In this paper we show that the same is true for F D(Ψ γ)D(Λ) , the type II and type III domains. The method of proof is entirely different from the method in the aforementioned paper. After a proof of tightness of the involved sequences we apply (modify) a result of Donnelly concerning weak convergence of Markov chains with an entrance boundary.


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Gerard Hooghiemstra. Priscilla E. Greenwood. "The domain of attraction of the α-sun operator for type II and type III distributions." Bernoulli 3 (4) 479 - 489, December 1997.


Published: December 1997
First available in Project Euclid: 6 April 2007

zbMATH: 0899.60018
MathSciNet: MR1483700

Keywords: extremal limits , Self-similar Markov processes , weak convergence

Rights: Copyright © 1997 Bernoulli Society for Mathematical Statistics and Probability

Vol.3 • No. 4 • December 1997
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