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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?!?

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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.

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  • $\begingroup$ 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. $\endgroup$ – JKnecht Dec 29 '15 at 15:34
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    $\begingroup$ @JKnecht Yes. WSS is about expectations and covariances... $\endgroup$ – d.k.o. Dec 29 '15 at 22:02
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    $\begingroup$ @JKnecht: Yes, it's zero. $\endgroup$ – d.k.o. Jan 3 '16 at 22:54
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    $\begingroup$ @JKnecht The variance is 1/2 and $Cov(X(n), X(m))=0$ for $n\ne m$. $\endgroup$ – d.k.o. Jan 4 '16 at 0:25
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    $\begingroup$ Sigh, im struggling :) $$Cov(X(n),X(m))=E(cos(nU)cos(mU))=(1/2)E(cos((n+m)U)+cos((n-m)U)$$=$$(1/2)(1/(2(m+n)\pi)) [sin(m+n)\pi) - sin(-(m+n)\pi)]$$ + $$(1/2)(1/(2(m-n)\pi)) [sin (m-n)\pi) - sin(-(m-n)\pi)] = 0$$ $\endgroup$ – JKnecht Jan 4 '16 at 17:52

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