Barely-stationary AR(1) sequences near random walk

Abstract

This article investigates a sequence of barely-stationary AR(1) processes near unit root time series, or random walk (RW). Autoregressive coefficient is allowed to depend on the sample size and we are concerned with the case that stationary AR(1) processes are converging to RW at various rates as the sample size tends to infinity. In particular, barely-stationary sequence is newly suggested for which the increasing rate in variance is specified between certain power rates. Some relevant asymptotic results are reported including the limit of the least squares estimator as a functional of fractional Brownian motion. As an application, two-sided test for RW is briefly discussed and local limiting power is obtained.