From page 3 of a note:

A stationary process is ergodicif any two variables positioned far apart in the sequence are almost independently distributed.A formal definition is the following: $\{w_t\}$ is ergodic if for any two bounded functions $f$ in $(k + 1)$ variables and $g$ in $(l + 1)$ variables: $$\lim_{N \to \infty} |E(f(w_t, ..., w_{t+k})g(W_{t+N}, ..., w_{t+N+l}))|- |E(f(w_t, ..., w_{t+k}))| |E(g(W_{t+N}, ..., w_{t+N+l}))| =0.$$

This definition of ergodicity can also apply to wide-sense stationary processes, does it?

This definition of ergodicity seems more similar to mixing than to ergodicity. So is it actually a type of mixing for (wide-sense) stationary processes?

Is this definition of ergodicity for a stationary process, same as the mean-square ergodicity in the first and second moment for wide-sense stationary processes, or ergodicity for a measure-preserving mapping?

Thanks and regards!