Limit Theory for Moderate Deviations from a Unit Root under Weak Dependence

An asymptotic theory is given for autoregressive time series with weakly dependent innovations and a root of the form rho_{n} = 1+c/n^{alpha}, involving moderate deviations from unity when alpha in (0,1) and c in R are constant parameters. The limit theory combines a functional law to a diffusion on D[0,infinity) and a central limit theorem. For c > 0, the limit theory of the first order serial correlation coefficient is Cauchy and is invariant to both the distribution and the dependence structure of the innovations. To our knowledge, this is the first invariance principle of its kind for explosive processes. The rate of convergence is found to be n^{alpha}rho_{n}^{n}, which bridges asymptotic rate results for conventional local to unity cases (n) and explosive autoregressions ((1 + c)^{n}). For c < 0, we provide results for alpha in (0,1) that give an n^{(1+alpha)/2} rate of convergence and lead to asymptotic normality for the first order serial correlation, bridging the /n and n convergence rates for the stationary and conventional local to unity cases. Weakly dependent errors are shown to induce a bias in the limit distribution, analogous to that of the local to unity case. Linkages to the limit theory in the stationary and explosive cases are established.