The Annals of Probability

Limit Laws for the Maximum of Weighted and Shifted I.I.D. Random Variables

D. J. Daley and Peter Hall

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Probab., Volume 12, Number 2 (1984), 571-587.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993306

Digital Object Identifier
doi:10.1214/aop/1176993306

Mathematical Reviews number (MathSciNet)
MR735854

Zentralblatt MATH identifier
0538.60024

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62G30: Order statistics; empirical distribution functions

Keywords
Domain of attraction extreme value distribution extreme value theory maxima regular variation shifted sequence weighted sequence

Citation

Daley, D. J.; Hall, Peter. Limit Laws for the Maximum of Weighted and Shifted I.I.D. Random Variables. Ann. Probab. 12 (1984), no. 2, 571--587. doi:10.1214/aop/1176993306. https://projecteuclid.org/euclid.aop/1176993306


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