In my notes, I have a statement which says that if two random variables $X$ and $Y$ are independent, then they are also uncorrelated provided that they have finite expected values and variances, variances also cannot be equal to $0$.
My question is:
What can be an example of two independent random variables for which we cannot say that they are uncorrelated, because one of them (or both) has infinite expected value?
On
Two random variables are uncorrelated if their covariance is zero. Two RVs being independent is a very strong condition but it does not guarantee that the covariance exists. If you have additional requirements that the first two moments exist, then so does the covariance and if it exists it has to be zero.
A (standard) Cauchy distributed random variable $X$ has density $$f_X(x)=\frac{1}{\pi(1+x^2)},$$ and it's first moment is undefined (and therefore all it's moments are by Lyapunov's inequality). So in that case you might choose a standard Cauchy distributed random variable and some independent other random variable (what particular distribution does not really matter).
If you really want an infinite first moment (although the convention is still that the first moment is undefined in this case), i.e. $\mathbb{E}(X)=\infty$, go for $X':=|X|$.