I'm trying to show that a certain process is not ergodic, but as I don't have much experience, I would first like to learn how to show simple cases.
We know that if a discrete stochastic process is i.i.d. so this is ergodic. So let's try to give an example of a stochastic process that is not independent to give an example of a stationary process that is not ergodic.
Let $X_t \sim Bernoulli(p)$ taking values $0$ or $1$ for all $t \geq 0$.
We say that $(X_t)_{t \in \mathbb{N}}$ is mean-ergodic if: $$ \ \frac{1}{T} \sum_{t=1}^T X_t \overset{prob}{\rightarrow} p $$ Since the process is not independent, showing this I think gets a little more complicated. However, I would like to know how to show for this simple case and to know if there are other procedures to show non-ergodicity.