Asymptotic methods and functional central limit theorems
Davidson, James
Date: 21 February 2006
Book chapter
Publisher
Palgrave
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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 ...
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.
Economics
Faculty of Environment, Science and Economy
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