Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent random variables with the same distribution. The common distribution $\mu$ is such that it is symmetric, that is, $\mu((-\infty,x])=\mu([-x,\infty))$ for every $x\in\mathbb R$, and it has no first moment, that is, $$\int_{-\infty}^{\infty}|x|\,\mathrm d\mu(x)=\infty.$$
Are there any well-known results as to whether $$\frac{X_1+\cdots+X_n}{n}$$ converges in law as $n\to\infty$? My preliminary experimentation (and the fact that the average of Cauchy random variables is Cauchy) seems to me to suggest the possibility that the answer may be positive and the limit distribution is Cauchy, but I’m not sure. Any thoughts would be appreciated.
