Limit laws for the maximum of weighted and shifted i.i.d. random variables
Let \(X_ 0,X_ 1,X_ 2,..\). be i.i.d. r.v.'s with d.f. F and define a sequence by \(M_ n=\max(X_ 0,...,X_{n-1})\). Classical results by \textit{B. Gnedenko} [Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423-453 (1943)] exhibited the class of possible nondegenerate limit laws that can arise for such sequences and discussed the domain of attraction of this class. This work has later been extended considerably. In the present paper extensions to weighted or translated sequences are studied; special cases are \(\{a^ nX_ n\}\), \(0<a<1\), and \(\{X_ n-nd\}\), \(d>0\). Depending on the weights or translates the limit laws may be both inside and outside Gnedenko's classical class.
scientific article; zbMATH DE number 3856098