Asymptotics for nonlinear transform at ions of integrated time series

An asymptotic theory for stochastic processes generated from nonlinear tran sformations of nonstationary integrated time series is developed. Various n onlinear functions of integrated series such as ARIMA time series are studi ed, and the asymptotic distributions of sample moments of such functions ar e obtained and analyzed. The transformations considered in the paper includ e a variety of functions that are used in practical nonlinear statistical a nalysis. It is shown that their asymptotic theory is quite different from t hat of integrated processes and stationary time series. When the transforma tion function is exponentially explosive, for instance, the convergence rat e of sample functions is path dependent. In particular, the convergence rat e depends not only on the size of the sample but also on the realized sampl e path. Some brief applications of these asymptotics are given to illustrat e the effects of nonlinearly transformed integrated processes on regression . The methods developed in the paper are useful in a project of greater sco pe concerned with the development of a general theory of nonlinear regressi on for nonstationary time series.