TITLE: Example of a random process which is strictly stationary but not iid
QUESTION [6 upvotes]: I understand $IID\subseteq SSS\subseteq WSS$. What could be an example of a stochastic process which is not iid but is strict sense stationary? I will appreciate examples for $SSS\setminus IID$ and $WSS\setminus SSS$...

REPLY [6 votes]: Try $x_n=x_0$ for every $n$. This is a stationary process but, unless $x_0$ is almost surely constant, not independent.
To be weakly stationary but not strongly stationary, change the distribution without changing the mean nor the variance: for example, assume that each $x_{2n}$ is symmetric Bernoulli, each $x_{2n+1}$ is standard normal, and that $(x_{n})_{n\geqslant0}$ is independent. 
In continuous time, consider $x_t=\cos(t)\xi+\sin(t)\eta$ where $\xi$ and $\eta$ are independent and centered with variance $1$, for example $\xi$ standard Bernoulli and $\eta$ standard normal.