# Find a random process that is wide-sense stationary (WSS) but not strict-sense stationary etc.

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?

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?!?

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.

• So "A seqeunce of independent identically distributed random variables with infinite variances constitute a strict-sense stationary discrete time process that is not WSS." is not a Strictly stationary $L^2$-process? Im surprised they asked that question in that case because this is a course in basic stochastic processes...nothing advanced. – JKnecht Dec 29 '15 at 15:34
• @JKnecht Yes. WSS is about expectations and covariances... – d.k.o. Dec 29 '15 at 22:02
• @JKnecht: Yes, it's zero. – d.k.o. Jan 3 '16 at 22:54
• @JKnecht The variance is 1/2 and $Cov(X(n), X(m))=0$ for $n\ne m$. – d.k.o. Jan 4 '16 at 0:25
• Sigh, im struggling :) $$Cov(X(n),X(m))=E(cos(nU)cos(mU))=(1/2)E(cos((n+m)U)+cos((n-m)U)$$=$$(1/2)(1/(2(m+n)\pi)) [sin(m+n)\pi) - sin(-(m+n)\pi)]$$ + $$(1/2)(1/(2(m-n)\pi)) [sin (m-n)\pi) - sin(-(m-n)\pi)] = 0$$ – JKnecht Jan 4 '16 at 17:52