On the domain of attraction of an operator between supremum and sum (Q756265)

Between the operations which produce partial maxima and partial sums of a sequence \(Y_ 1,Y_ 2,...\), lies the inductive operation: \(X_ n=X_{n-1}\vee (\alpha X_{n-1}+Y_ n),\) \(n\geq 1\), for \(0<\alpha <1\). If the \(Y_ n\) are independent random variables with common distribution F, we show that the limiting behavior of normed sequences formed from \(\{X_ n,\quad n\geq 1\},\) is, for \(0<\alpha <1\), parallel to the extreme value case \(\alpha =0\). For \(F\in D(\Phi_{\gamma})\) we give a full proof of the convergence, whereas for \(F\in D(\Psi_{\gamma})\cup D(\Lambda)\), we only succeeded in proving tightness of the involved sequence. The process \(X_ n\) is interesting for some applied probability models.