Asymptotic methods and functional central limit theorems

Abstract

This chapter sketches the fundamentals of asymptotic distribution theory, and applies these speciÖcally to questions relating to weak convergence on function spaces. These results have important applications in the analysis of nonstationary time series models. A simple case of the functional central limit theorem for processes with independent increments is stated and proved, after detailing the necessary results relating to the topology of spaces of functions and of probability measures. The concepts of weak convergence, tightness and stochastic equicontinuity, and their roles in the derivation of functional central limit theorems, are deÖned and reviewed. It is also shown how to extend the analysis to the vector case, and to various functionals of Brownian motion arising in nonstationary regression theory. The analysis is then widened to consider the problem of dependent increments, contrasting linear and nonparametric representations of dependence. The properties of Brownian motion and related Gaussian processes are examined, including variance-transformed processes, the Ornstein-Uhlenbeck process and fractional Brownian motion. Next, the case of functionals whose limits are characterized stochastic integrals is considered. This theory is essential to (for example) the analysis of multiple regression in integrated processes. The derivation of the ItÙ integral is summarized, followed by application to the weak convergence of covariances. The Önal section of the chapter considers increment distributions with inÖnite variance, and shows how weak convergence to a LÈvy process generalizes the usual case of the FCLT, having a Gaussian limit.