  This paper examines the asymptotic inference for AR(1) models with a possible
structural break in the AR parameter $\beta $ near the unity at an unknown time
$k_{0}$. Consider the model $y_{t}=\beta_{1}y_{t-1}I\{t\leq k_{0}\}+\beta
_{2}y_{t-1}I\{t>k_{0}\}+\varepsilon_{t},~t=1,2,\cdots ,T,$ where $I\{\cdot \}$
denotes the indicator function. We examine two cases: Case (I)
$|\beta_{1}|<1,\beta_{2}=\beta_{2T}=1-c/T$; and case
(II)~$\beta_{1}=\beta_{1T}=1-c/T,|\beta _{2}|<1$, where $c$\ is a fixed
constant, and $\{\varepsilon_{t},t\geq 1\}$\ is a sequence of i.i.d. random
variables which are in the domain of attraction of the normal law with zero
means and possibly infinite variances. We derive the limiting distributions of
the least squares estimators of $\beta_{1}$ and $\beta_{2} $, and that of the
break-point estimator for shrinking break for the aforementioned cases. Monte
Carlo simulations are conducted to demonstrate the finite sample properties of
the estimators. Our theoretical results are supported by Monte Carlo
simulations.\newline
