In this paper, we develop a random-matrix formalism that enables analysis of a variety of polarization-mode dispersion (PMD) related problems. In particular, we address the problems of higher order error in a discrete fiber model and limit of multistaged PMD compensation schemes. Our solution to the first problem leads to a simple condition for the validity of the model, which is often overlooked in PMD simulations. For the second issue, we have found an asymptotic bound on the limit of a multistaged PMD compensation scheme. The theory is confirmed by numerical simulations, and future work is suggested.