Example of a random process which is strictly stationary but not iid 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 $WSS\setminus SSS$...
 A: 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.
A: Well, any stationary process which has some correlation (an autocorrelation function different from a Dirac delta) would fit the bill. IID is a very special case of a stationary process (white noise, basically; or a subset of white noise, if we are dealing with strict-sense stationary).
A simple example, say $x_n$ is an IID process. Then $y_n=x_n+x_{n+1}$ (or any non trivial LTI filter) is strictly stationary, but not IID.
