The domain of attraction of the α-sun operator for type II and type III distributions

Abstract

Let $\left(Y_n\right)$ be a sequence of independent random variables with common distribution $F$ and define the iteration $X_0=x_0$, $X_n:=X_{n-1}\vee (\alpha X_{n-1}+Y_n)$, $\alpha \in [0,1)$. We denote by $D\left({\Phi }_{\gamma }\right)$ the domain of maximal attraction of ${\Phi }_{\gamma }$, the extreme value distribution of the first type. Greenwood and Hooghiemstra showed in 1991 that for $F\in D\left({\Phi }_{\gamma }\right)$ there exist norming constants $a_n>0$ and $b_n\in R$ such that $a_n^{-1}\{X_n-b_n/(1-\alpha \left)\right\}$ has a non-degenerate (distributional) limit. In this paper we show that the same is true for $F\in D\left({\Psi }_{\gamma }\right)\cup D\left(\Lambda \right)$, 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.