For a stationary and ergodic discrete stochastic process $\{X[n]\}_t$, $X[n]$'s are Independent and identically distributed?

Question

I can find some materials that say

"if $X[n]$'s are independent and identically distributed (IID), then a random process $\{X[n]\}_n$ is an ergodic process."

I think the converse does not hold (just guess...) That is, "even if a random process $\{X[n]\}_n$ is an ergodic process, $X[n]$'s may be non-IID."

What if a random process $\{X[n]\}_n$ is of ergodic as well as stationary? That is, "if a discrete stochastic process $\{X[n]\}_t$ is stationary and ergodic, then $X[n]$'s are IID?"

Answer 1

This is not true in general. Discrete Markov chains are a popular example: these are often stationary and ergodic, but not iid.

You can try to construct one yourself.