Asymptotic behaviour of the least squares estimator of the mean of AR(1) models (Q1882114)

The shifted AR(1) model started at zero: \[ X_k^{(n)}=\alpha^{(n)}X_{k-1}^{(n)} + \varepsilon_k^{(n)},\;X_0^{(n)} =0,\;Z_k^{(n)}=X_k^{(n)} + m^{(n)}h\big(k/n\big),\quad k=1,2,\dots,\,n=1,2,\dots, \] where \(m^{(n)} \in R\) is an unknown parameter and \(h\) is a known real function, is considered. It is shown that the limit distribution of the MLS estimator of \(m^{(n)}\) based on \(\{Z_k^{(n)}: [T_1n] \leq k \leq [T_2n]\}\), where \(0 \leq T_1 <T_2\) is normal. The speed of convergence is different if \(n(1-\alpha^{(n)}) \to \gamma \in R\) or if \(n(1-\alpha^{(n)}) \to \pm \infty\). The stationary case of the shifted AR(1) model is also treated. Connection with the maximum likelihood estimator of the shift parameter of continuous AR(1) processes is discussed.