Substantial difficulties are encountered in derivation of exact distributional and inferential properties for the estimators of process capability indices (PCI). A study of asymptotic behavior of these estimators will thus be valuable. In this article, the asymptotic distributions of the estimators of a rather wide class of PCI's are investigated in a unified approach. The results obtained include the asymptotic distributions of the estimators of the early forms of capability indices which were recently studied by some other researchers. As an application of the asymptotic results, approximations for the means, variances and mean square error of the estimated PCI's are also discussed and derived.