# Can two weak stationary time series $X_t, Y_t$ have covariance $\mathrm{Cov}(X_t,Y_t)$ that changes over time?

Can two weak stationary time series $$X_t, Y_t$$ have covariance $$Cov(X_t,Y_t)$$ that changes over time? No solution is needed, please give me some hints. I think it can but I find it is difficult to support my idea.

• If you cannot prove your idea, just sharing your idea is also good. I think Cov(Xt, Yt) may change over time. But I can not find the way to support my idea. Nov 4 '20 at 12:13
• Is that a homework exercise? If so, please add the self-study tag and read its Wiki. Nov 4 '20 at 12:27
• It a question that teacher left us, not a necessary homework. Thanks your comment ! Nov 4 '20 at 12:35
• Hint: let each series consist of iid variables. Does this imply the variables in one series must be independent of those in the other?
– whuber
Nov 5 '20 at 15:05
• Thanks your hint. Yes! It must be independent. But because of independence, the covariance must be 0 which will not change with t. Nov 6 '20 at 8:28

Let $$(X_t),$$ with "times" $$t=0,1,2,\ldots,$$ be variables with independent and identical distributions having finite variance $$\sigma^2$$ -- which implies they form a stationary (whence weakly stationary) stochastic process. Suppose further that this common distribution is symmetric about $$0$$: this means the $$-X_t$$ all have the same distribution as the $$X_t.$$ For instance, the standard Normal distribution is symmetric about $$0.$$

Define

$$Y_t = (-1)^t X_t$$

and notice that

$$\operatorname{Cov}(X_t,Y_t) = \operatorname{Cov}(X_t,(-1)^tX_t) = (-1)^t \operatorname{Cov}(X_t,X_t) = (-1)^t\sigma^2$$

alternates between $$\sigma^2$$ and $$-\sigma^2:$$ that is, it changes over time. Nevertheless, the symmetry of the distribution implies the $$(Y_t)$$ are identically distributed and, since the $$(X_t)$$ are independent, the $$(Y_t)$$ are independent too. Thus $$(Y_t)$$ is a (weakly) stationary process, too. In this example the covariance changes over time.

• Thanks!!! This is really ingenious! Nov 6 '20 at 13:06