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! :)

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    $\begingroup$ 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. $\endgroup$ – Math-fun Oct 12 '19 at 7:47
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    $\begingroup$ As stated, $X_i^+=-X_i^-$. Did you perhaps mean $X_i^- = \min(X_i, 0)$ or $X_i^- = \max(-X_i, 0)$? $\endgroup$ – robjohn Oct 12 '19 at 8:31

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

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