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.

本文研究了单位根时间序列或随机游走（RW）附近的几乎平稳的AR（1）过程序列。自回归系数取决于样本量，并且我们关注的情况是，随着样本量趋于无穷大，固定AR（1）过程以各种速率收敛到RW。尤其是，新近提出了一种非平稳序列，对于该序列，在某些电价之间指定了方差的增加速率。报告了一些相关的渐近结果，包括最小二乘估计的极限作为分数布朗运动的函数。作为应用，简要讨论了RW的双面测试并获得了局部极限功率。