Assume that a discrete-time process is given as below:
$$X_t=\alpha +\beta X_{t-1} + N_t$$
where $N_t$ is an i.i.d process with mean 0 and variance $\sigma^2$ and $\alpha$ and $\beta$ are the parameters of the process.
(a) What conditions should be sufficed for $\alpha$ and $\beta$ so that the given process becomes wide-sense stationary?
(b) Assuming that conditions defined in part (a) are sufficed, will this be a mean-ergodic process?
Note 1: An i.i.d process is a process like $\{X(t)\}$ where $X(t)\sim F(x)$ and for each $n\in \mathbb{N}$ and each $t_1,\dots,t_n$ we have $F(x_1,\dots,x_n;t_1,\dots,t_n)=F(x_1)\dots F(x_n)$. We also know that an i.i.d process is strict-sense stationary (and hence, weak-sense stationary).
Note 2: For a process to be mean ergodic, we should have that the time average of the process ($\frac{1}{2T} \int_{-T}^{T} X(t)dt$) should converge to the mean of the process (ensemble average) as $t$ goes to infinity.
My try:
I wrote the mean of the process like this:
$E(X_t)=\alpha + \beta E(X_{t-1})+E(N_t)$
Since $E(N_t)=0$, we will have:
$E(X_t)=\alpha + \beta E(X_{t-1})$
Now, what should I do with this? How should I process further? We do not know anything else about the process $X_t$.
