On the domain of attraction of an operator between supremum and sum

Summary Between the operations which produce partial maxima and partial sums of a sequenceY ₁,Y ₂, ..., lies the inductive operation:X _n =X _n-1(X _n-1+Y _n ),n1, for 0<<1. If theY _n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X _n ,n1}, is, for 0<<1, parallel to the extreme value case =0. ForFD() we give a full proof of the convergence, whereas forFD()D(), we only succeeded in proving tightness of the involved sequence. The processX _n is interesting for some applied probability models.