## The Annals of Probability

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

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

#### 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