Covariance of random variables which don't have variances.

Question

1. Whether there is a covariance of two random variables, which both don't have variances?
2. Is existence of the variances of two random variables implies existence of covariance?

Thanks in advance.

Answer 1

The Cauchy--Schwarz inequality implies that $$ |\operatorname{cov}(X,Y)| \le \sqrt{\operatorname{var}(X)\operatorname{var}(Y)}, $$ so existence of the two variancees (meaning both are finite) implies existence of the covariance. Notice that no generality is lost by assuming $\mathbb E(X)=\mathbb E(Y)=0$, so the variances of $X$ and $Y$ are $\mathbb E(X^2)$ and $\mathbb E(Y^2)$. Cauchy--Schwarz says $\mathbb E(|XY|) \le \sqrt{(\mathbb E(X^2))(\mathbb E(Y^2))} $, and that means $\mathbb E(XY)$ exists, and that is the covariance.

As for existence of the covariance when the variances of $X$ and $Y$ are both infinite, I would let $$ X = \begin{cases} W & \text{if some even number} \le W<\text{that even number}+1, \\ 0 & \text{otherwise}, \end{cases} $$ $$ Y = \begin{cases} W & \text{if some odd number} \le W<\text{that odd number}+1, \\ 0 & \text{otherwise}, \end{cases} $$ and then see if I can choose the distribution of $W$ in such a way that $\mathbb E(X)=\mathbb E(Y)=0$ but both have infinite variance. Then the covariance between $X$ and $Y$ would be $\mathbb E(XY)=0$.