  Multivariate distributions are explored using the joint distributions of
marginal sample quantiles. Limit theory for the mean of a function of order
statistics is presented. The results include a multivariate central limit
theorem and a strong law of large numbers. A result similar to Bahadur's
representation of quantiles is established for the mean of a function of the
marginal quantiles. In particular, it is shown that
\[\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^n\phi\bigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}\bigr)-\bar{\gamma}\Biggr)=\frac{1}{\sqrt{n}}\sum_{i=1}^nZ_{n,i}+\mathrm{o}_P(1)\]
as $n\rightarrow\infty$, where $\bar{\gamma}$ is a constant and $Z_{n,i}$ are
i.i.d. random variables for each $n$. This leads to the central limit theorem.
Weak convergence to a Gaussian process using equicontinuity of functions is
indicated. The results are established under very general conditions. These
conditions are shown to be satisfied in many commonly occurring situations.
