Abstract: Limit distributions of the least squares estimate of the autoregressive coefficient of a nearly nonstationary autoregressive model with strong dependent and infinite variance innovations are established in this paper. It is shown that under some regularity conditions, the ordinary least squares estimator of the autoregressive parameter converges to a functional of a fractional Ornstein-Uhlenbeck stable process. This paper not only generalizes the recent results of Buchmann and Chan to models with long-memory finite variance innovations, but also demonstrates the subtlety involved in the asymptotics when jumps are present. To this end, some newly established weak convergence theory involving so-calledconvergence is employed to handle these subtleties. Results of this paper work toward a better understanding of inference for jump processes that are commonly encountered in finance and related fields.
Key words and phrases: Autoregressive process, infinite variance, least squares, fractional Ornstein-Uhlenbeck processes, long-range dependence, nearly nonstationary processes, stochastic integrals, unit-root problem.