Finite nonzero covariance but infinite variance

Question

Cauchy-Schwarz tells us that it is not possible for random variables $X,Y$ to both have finite variance while having infinite covariance. Now it is easy to think of examples where $X,Y$ have zero covariance and infinite variances (e.g. take $X,Y\overset{\text{iid}}{\sim} t_{\text{df}=2}$). But what is an example of $X,Y$ having finite nonzero covariance and infinite variances?

Answer 1

In your example, change $X$ to $ Z = Y 1( |Y| < 1) + X 1( |Y| > 1)$, this gives a covariance of cov$(Z, Y) = \mathbb E(Y^2 1( |Y| < 1))$.