1
$\begingroup$

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! :)

$\endgroup$
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
2
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.