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 $F(x)=1-{\sum }_{i=0}^{\mathrm{\infty }}{c}_i{x}^{-\alpha -i\beta }$ 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 $(n^{-1},n^{-\beta \mathrm{/}\alpha })$ is given for the multivariate moments of the extremes ${X_{n, n - s_i}, 1 \leq i \leq k }/n^{1/\alpha}$forfixed${\bf s} = (s_1, …, s_k)$,where$k \geq 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $\alpha <1$.