# 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! :)

• The definition of correlation requires the second moments to exist. If they do not exist, you can't define the correlation coefficient. As such in your case, I would say talking about correlation is not meaningful. – Math-fun Oct 12 '19 at 7:47
• As stated, $X_i^+=-X_i^-$. Did you perhaps mean $X_i^- = \min(X_i, 0)$ or $X_i^- = \max(-X_i, 0)$? – robjohn Oct 12 '19 at 8:31

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