Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes.
The functional central limit theorem (FCLT) says that under some conditions the stochastic process \(X_n: [0,1]\to\mathbb R\) defined by \(X_n(\xi)=\sigma_n^{-1} \sum_{t=1}^{[n\xi]} (x_t-Ex_t)\), \(0<\xi \leq 1\), converges in distribution to standard Brownian motion on \([0,1]\). One of the conditions ensuring this result is that \(x_t\) is either \(\alpha\)-mixing or \(\varphi\)-mixing. The paper considers alternatives to this condition that are either weaker than mixing, or more easily verifiable, or both. The main tool is the concept of near-epoch dependence (NED) on a mixing process. This approach is applied to ARMA processes, bilinear models, GARCH models, and threshold autoregressive models. It is shown that SETAR processes with a unit root regime have short memory under some conditions. In the case of the ESTAR model the analytic solution is not available, but the relevant moments can be computed using simulations. A new FCLT is given for semiparametric linear processes, where the forcing processes are of the NED-on-mixing type.
scientific article; zbMATH DE number 1869170
scientific article