Let [math] be a sequence of independent random variables with common distribution [math] and define the iteration [math] , [math] , [math] . We denote by [math] the domain of maximal attraction of [math] , the extreme value distribution of the first type. Greenwood and Hooghiemstra showed in 1991 that for [math] there exist norming constants [math] and [math] such that [math] has a non-degenerate (distributional) limit. In this paper we show that the same is true for [math] , 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.