  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 $\alpha < 1$.
