This article examines the asymptotic inference for AR(1) models with a possible structural break in the AR parameter β near the unity at an unknown time k0. Consider the model yt = β1yt − 1I{t ≤ k0} + β2yt − 1I{t > k0} + ϵt, t = 1,2, … ,T, where I{ ⋅ } denotes the indicator function. We examine two cases: case I | β1 | < 1,β2 = β2T = 1 − c ∕ T; and case II β1 = β1T = 1 − c ∕ T, | β2 | < 1, where c is a fixed constant, and {ϵt,t ≥ 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 β1 and β2 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.