Expansions for Quantiles and Multivariate Moments of Extremes for Distributions of Pareto Type

Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - sum_{i = 0}^infty c_i x^{-alpha - i beta}$ for $alpha > 0$ and $beta > 0$. An inversion theorem is proved and used to derive an expansion for the quantile $F^{-1} (u)$ and powers of it. From this an expansion in powers of $(n^{-1}, n^{-beta/alpha})$ is given for the multivariate moments of the extremes ${X_{n, n - s_i}, 1 leq i leq k }/n^{1/alpha}$ for fixed ${bf s} = (s_1, ..., s_k)$, where $k geq 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $alpha < 1$.