Limit theorems for functions of marginal quantiles

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 √ n(1/nσ ⁿ_i=1φ(X⁽¹⁾_{n : i}, ⋯ , X^(d)_{n : i}) - ȳ)=1/√nσⁿ _i=1 Z_n,i + oP (1) as n→ ∞, where ȳ 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.