Consider autoregression AR(1): $$u_t=\beta u_{t-1}+ \varepsilon_t, \quad t \in \mathbb{Z}.$$
$\{\varepsilon_t\}$ - i.i.d. random variables with $E\varepsilon_1 = 0,$ $E\varepsilon_1^2<\infty$
We consider case $|\beta| <1.$
Let $\{u_t^{0}\}$ be the stationary solution of this equation when $\beta=\beta_0$ and $\{u_t^\ast\}$ - stationary solution when $\beta=\beta_n=\beta_0+n^{-1/2}\tau, \quad\tau=const<\infty.$
Thus $u_i^0 = \sum_{j=0}^{\infty}{\beta^j_0\varepsilon_{i-j}}$ and $u_i^\ast = \sum_{j=0}^{\infty}{\beta^j_n\varepsilon_{i-j}}.$
Is it true that sum $$n^{-1/2}\sum_{t=1}^{n}{[u_t^\ast-u_t^0]} \stackrel{P} \rightarrow 0, \quad n\to \infty \,?$$
I know how to prove that $u_t \stackrel{P} \rightarrow u_t^0$ but can I say smth about the sum?
Thanks in advance!