find random variables with covariance $>0$ and $=0$

Question

I am currently studying for an exam and found the following question:

Let $(x)_ {n\in\mathbb N}$ be a sequence of random variables. Can you construct a sequence of random variables such that for the covariance we get $$\sigma(x_k,x_l)\begin{cases} =0&\textrm{if }|k-l|>1\\>0& \textrm{if } |k-l|\leq1\end{cases}$$?

I've tried some sequence but all of them didn't fit. So is there any sequence?

Answer 1

Yes, let $\{u_t\}$ be i.i.d. with zero mean and $$x_t=u_t+u_{t-1}.$$