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
\[\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^{n}\phi\bigl(X_{n\dvtx i}^{(1)},\ldots,X_{n\dvtx i}^{(d)}\bigr)-\bar{\gamma}\Biggr)=\frac{1}{\sqrt{n}}\sum _{i=1}^{n}Z_{n,i}+\mathrm{o}_{P}(1)\]
¶
as n → ∞, where γ̄ is a constant and Zn,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.