Are i.i.d. random variables with infinite expectation uncorrelated? If I have a sequence of i.i.d. random variables $(X_i)_{i\in \mathbb{N}}$, with $\mathbb{E}[X_i^+] = \infty$ and $\mathbb{E}[X_i^-] < \infty$, where $X_i^+ = \max(X_i, 0)$ and $X_i^- = \max(-X_i, 0)$ can I say that they are uncorrelated? 
Because I have $$cov(X_i,X_j) = \mathbb{E}(X_iX_j) - \mathbb{E}(X_i)\mathbb{E}(X_j) = 0$$ by independence, but since my expectations are infinite I get  $$cov(X_i,X_j) = \infty - \infty$$ which is undefined.
Thank you for any help! :) 

Covariance of $X$ and $Y$ is defined only when $X$ and $Y$ have finite variance. 
