The book "asymptotic theory for econometricians" ststes the theory that if a stationary sequence is alpha or phi mixing, it is also ergodic, but not the other way around. However, when I look at the definitions they seem intuititely to me to capture the same idea.
I cannot think of an example of a probability distribution over a stationary time series of random variables that is ergodic but not alpha or phi mixing.
One of the most classical example is the linear process $$ X_n:=\sum_{i=0}^{+\infty} 2^{-i}\xi_{n-i}, $$ where $\left(\xi_j\right)_{j\in\mathbb Z} $ is an i.i.d. sequence where $\xi_0$ takes the values $0$ and $1$ with probability $1/2$. Ergodicity follows from the fact that $X_n$ is a functional of i.i.d. and the sequence $\left(X_n\right)_{n\geqslant 1}$ is not mixing because for all $n$, $$\left\{X_n\geqslant 2^{-n} \right\}=\left\{\xi_0=1 \right\}$$ up to a set of measure $0$.
