given the models as following:
$Y_t=\delta Y_{t-1}+u_t$,and $u_t=\rho u_{t-1}+\epsilon_t$,where $\epsilon_t\sim i.i.d. (0,\sigma^2)$, $|\delta|,|\rho|<0$. Then how to find the prob. limit of the ols estimator $\delta$ hat?

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 $\delta$ hat, thx!