Moment Convergence of Sample Extremes

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.