Is there an easy construction of a sequence of independent real random variables $\{X_n\}$ that satisfy the following conditions?
(1) $\mathbb{E} X_n \to 0$ as $n\to\infty$;
(2) $\mathbb{E} \exp(X_n) = 1$ for all $n$;
(3) $\mathbb{E} |\exp(X_n) - 1|$ does not converge $0$ as $n\to\infty$.
I understand that if one changes (1) to $X_n\to 0$ in probability then $\{X_n\}$ would be uniformly integrable and it would then hold that $\mathbb{E}|\exp(X_n)-1|\to 0$. I think when it's just $\mathbb{E} X_n\to 0$ for (1) we wouldn't expect the convergence in (3). Any idea of a concrete construction?