Time Series Analysis Nonstationary and Noninvertible Distribution Theory

Weak convergence of a stochastic process defined on a function space is discussed. In doing so we explore various weak convergence results generically called functional central limit theorems or invariance principles. Emphasis is placed on how to apply those theorems to deal with statistics arising from nonstationary linear time series models. It turns out that, in most cases, the continuous mapping theorem is quite powerful for obtaining limiting random variables of statistics in the sense of weak convergence. In some cases, however, the continuous mapping theorem does not apply. In those cases limiting forms involve the Ito integral.