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Moment Convergence of Sample Extremes

James Pickands III
The Annals of Mathematical Statistics
Vol. 39, No. 3 (Jun., 1968), pp. 881-889
Stable URL: http://www.jstor.org/stable/2239764
Page Count: 9
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Moment Convergence of Sample Extremes
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Abstract

Let Zn 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) Λ(x), and a pair of sequence an, bn, 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 Λ(x), we say that Λ(x) is an extremal distribution, and that F(x) lies in its domain of attraction. The possible forms of Λ(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 an -1 (Zn - bn) 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 an such that \begin{equation*}\tag{1.2}Z_n - a_n \rightarrow 0, \text{i.p.}\end{equation*} we say that Zn is stable in probability. If \begin{equation*}\tag{1.3}Z_n/a_n \rightarrow 1, \text{i.p.}\end{equation*} we say that Zn 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 mth absolute mean stability and relative stability. The results of this work are valid for smallest values as well as for largest values.

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