Moment Convergence of Sample Extremes

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 aₙ > 0, such that {equation*}{1.1}limn→∞P\{a_n⁻¹(Z_n - b_n) x\} = limn→∞ F^n (a_nx + b_n) = Λ(x){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 {equation*}{1.2}Z_n - a_n → 0, i.p.{equation*} we say that Zₙ is stable in probability. If {equation*}{1.3}Z_n/a_n → 1, i.p.{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.