Given the model:
\begin{aligned}
Y_t &= \delta Y_{t-1}+u_t, \\
u_t &= \rho u_{t-1}+\epsilon_t,
\end{aligned} 
where $\epsilon_t\sim i.i.d. (0,\sigma^2)$, $|\delta|,|\rho|<1$. Then how to find the prob. limit of the OLS estimator $\hat\delta$?

I have tried to use WLLN, since the $Y_t$ is not i.i.d., however the variance of $Y_t$ doesn't converge to 0. WLLN doesn't applies. So how to find the prob. limit of the OLS estimator $\hat\delta$?