Consider autoregression AR(1): $$u_t=\beta u_{t-1}+ \varepsilon_t, \quad t \in \mathbb{Z}.$$ Let $\{u_t^{0}\}$ be the stationary solution of this equation when $\beta=\beta_0$ and $\{u_t\}$ - stationary solution when $\beta=\beta_n=\beta_0+n^{-1/2}\tau.$
Let function $\varphi$ be continuous a.e.
Is it true that sum $$n^{-1/2}\sum_{t=1}^{n}{[\varphi(u_t)-\varphi(u_t^0)]} \stackrel{P} \rightarrow 0, \quad n\to \infty \,?$$
I know how to prove that $u_t \stackrel{P} \rightarrow u_t^0$ and hence by Mann-Wald theorem $\varphi(u_t) \stackrel{P} \rightarrow \varphi(u_t^0).$ But can I say smth about the sum?