Could you provide an example of a random variable $X$ such that $|\mathbb{E}(\ln(X))|<\infty$ but $\text{Var}(\ln(X))=\infty$, if such a random variable exists at all?

Related: "Random variable with finite exponential first moment, infinite exponential variance"
Motivated by "Is (covariance) stationarity preserved under log or exponential transformation?".

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    $\begingroup$ Let $Y$ be a positive variable with finite expectation and infinite variance, then set $X=\exp(Y).$ $\endgroup$ – whuber Mar 4 at 18:31
  • $\begingroup$ Richard, you should perhaps include the condition that $Var(X)\leq \infty$. $\endgroup$ – Dayne Mar 5 at 0:36
  • $\begingroup$ Sorry it should strict inequality above. $\endgroup$ – Dayne Mar 5 at 4:25