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This paper examines the asymptotic inference for AR(1) models with a possiblestructural 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 fixedconstant, and $\{\varepsilon_{t},t\geq 1\}$\ is a sequence of i.i.d. randomvariables which are in the domain of attraction of the normal law with zeromeans and possibly infinite variances. We derive the limiting distributions ofthe least squares estimators of $\beta_{1}$ and $\beta_{2} $, and that of thebreak-point estimator for shrinking break for the aforementioned cases. MonteCarlo simulations are conducted to demonstrate the finite sample properties ofthe estimators. Our theoretical results are supported by Monte Carlosimulations.\newline

Asymptotic inferences for an AR(1) model with a change point: stationary and nearly non-stationary cases