Im studying old exams and came across this one
Question:
a. Find a (discrete time or continuous time) random process that is wide-sense
stationary (WSS) but not strict-sense stationary.
b. Also, is it possible for a strict-sense
stationary random process not to be wide-sense stationary?
Answer:
a. A sequence of uncorrelated random variables with common expected values
and common variances constitute a WSS discrete time process, but is not strict-sense stationary if the random variables are not identically distributed.
b. A seqeunce of independent identically distributed random variables with infinite variances constitute a strict-sense stationary discrete time process that is not WSS.
a. Can anyone give a simple example of such a process?
b. Our course litterature says WSS processes are always strict-sense stationary?!?
Best Answer
a. Let $X\sim U(0,2\pi)$ and $Z_n=\sin(nX)$. Then $\{Z_n : n\in \mathbb{N}\}$ is weakly stationary, but not strictly stationary.
b. Strictly stationary $L^2$-process (finite second moments) is always weakly stationary.