Gnedenko's (1943) study of the class $\mathscr{G}$ of limit laws for the sequence of maxima $M_n \equiv \max\{X_0, \cdots, X_{n - 1}\}$ of independent identically distributed random variables $X_0, X_1, \cdots$ is extended to limit laws for weighted sequences $\{w_n(\gamma)X_n\}$ (the simplest case $\{\gamma^nX_n\}$ has geometric weights $(0 \leq \gamma < 1))$ and translated sequences $\{X_n - v_n(\delta)\}$ (the simplest case is $\{X_n - n\delta\} (\delta > 0))$. Limit laws for these simplest cases belong to the family $\mathscr{G}$ characterized by Gnedenko; with more general weights or translates, limit laws outside $\mathscr{G}$ may arise.
Publié le : 1984-05-14
Classification:
Domain of attraction,
extreme value distribution,
extreme value theory,
maxima,
regular variation,
shifted sequence,
weighted sequence,
60F05,
62G30
@article{1176993306,
author = {Daley, D. J. and Hall, Peter},
title = {Limit Laws for the Maximum of Weighted and Shifted I.I.D. Random Variables},
journal = {Ann. Probab.},
volume = {12},
number = {4},
year = {1984},
pages = { 571-587},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993306}
}
Daley, D. J.; Hall, Peter. Limit Laws for the Maximum of Weighted and Shifted I.I.D. Random Variables. Ann. Probab., Tome 12 (1984) no. 4, pp. 571-587. http://gdmltest.u-ga.fr/item/1176993306/