## The Annals of Mathematical Statistics

### Moment Convergence of Sample Extremes

James Pickands III

#### Abstract

Let $Z_n$ be the maximum of $n$ independent identically distributed random variables each having the distribution function $F(x)$. If there exists a non-degenerate distribution function (df) $\Lambda(x)$, and a pair of sequence $a_n, b_n$, with $a_n > 0$, such that \begin{equation*}\tag{1.1}\lim_{n\rightarrow\infty}P\{a_n^{-1}(Z_n - b_n) \leqq x\} = \lim_{n\rightarrow\infty} F^n (a_nx + b_n) = \Lambda(x)\end{equation*} on all points in the continuity set of $\Lambda(x)$, we say that $\Lambda(x)$ is an extremal distribution, and that $F(x)$ lies in its domain of attraction. The possible forms of $\Lambda(x)$ have been completely specified, and their domains of attraction characterized by Gnedenko . These results and their applications are contained in the book by Gumbel . A natural question is whether the various moments of $a_n^{-1} (Z_n - b_n)$ converge to the corresponding moments of the limiting extremal distribution. Sen  and McCord  have shown that they do for certain distribution functions $F(x)$, satisfying (1.1). Von Mises ( pages 271-294) has shown that they do for a wide class of distribution functions having two derivatives for all sufficiently large $x$. In Section 2, the question is answered affirmatively for all distribution functions $F(x)$ in the domain of attraction of any extremal distribution provided the moments are finite for sufficiently large $n$. If there exists a sequence $a_n$ such that \begin{equation*}\tag{1.2}Z_n - a_n \rightarrow 0, \text{i.p.}\end{equation*} we say that $Z_n$ is stable in probability. If \begin{equation*}\tag{1.3}Z_n/a_n \rightarrow 1, \text{i.p.}\end{equation*} we say that $Z_n$ is relatively stable in probability. Necessary and sufficient conditions are well known for stability and relative stability both in probability (see Gnedenko ) and with probability one (see Geffroy , and Barndorff-Nielsen ). In Section 3 necessary and sufficient conditions are found for $m$th absolute mean stability and relative stability. The results of this work are valid for smallest values as well as for largest values.

#### Article information

Source
Ann. Math. Statist., Volume 39, Number 3 (1968), 881-889.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177698320

Digital Object Identifier
doi:10.1214/aoms/1177698320

Mathematical Reviews number (MathSciNet)
MR224231

Zentralblatt MATH identifier
0176.48804

JSTOR
links.jstor.org

#### Citation

III, James Pickands. Moment Convergence of Sample Extremes. Ann. Math. Statist. 39 (1968), no. 3, 881--889. doi:10.1214/aoms/1177698320. https://projecteuclid.org/euclid.aoms/1177698320