The Annals of Mathematical Statistics

Moment Convergence of Sample Extremes

James Pickands III

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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 [5]. These results and their applications are contained in the book by Gumbel [6]. 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 [9] and McCord [8] have shown that they do for certain distribution functions $F(x)$, satisfying (1.1). Von Mises ([10] 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 [5]) and with probability one (see Geffroy [4], and Barndorff-Nielsen [1]). 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
http://projecteuclid.org/euclid.aoms/1177698320

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aoms/1177698320

Mathematical Reviews number (MathSciNet)
MR224231

Zentralblatt MATH identifier
0176.48804

Citation

III, James Pickands. Moment Convergence of Sample Extremes. The Annals of Mathematical Statistics 39 (1968), no. 3, 881--889. doi:10.1214/aoms/1177698320. http://projecteuclid.org/euclid.aoms/1177698320.


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