Consider this lesser-known fact:
If $X\sim\mathcal N(\mu,\sigma^2)$ then $$ \frac{\frac{1}{n}\sum_{k=1}^n\frac{1}{X_k}-\mathsf E_\mathcal PX^{-1}}{\pi f_X(0)}\overset{d}{\to}\operatorname{Cauchy}(0,1), $$ where $\operatorname{Cauchy}(0,1)$ is the standard Cauchy distribution and $$ \mathsf E_\mathcal PX^{-1}=\mathcal P\int_{-\infty}^\infty\frac{f_X(x)}{x}\,\mathrm dx $$ is the Cauchy principal value integral for the expected value of $1/X$.
This result bears a striking resemblance to the central limit theorem and indeed may be considered part of the generalized central limit theorem. Because suitably normalized sums of $1/X$ converge to the Cauchy distribution we say that $1/X$ lies within the domain of attraction of the Cauchy law.
Question: Are there analogous results for sums containing $1/X^m$ with $m\in\Bbb N$? In other words, what stable distributions do sums of $1/X^m$ converge to and what are the associated location and scaling constants?
Thoughts: For the case $n=2$, I believe $1/X^2$ lies within the domain of attraction of the Levy distribution so that we have potentially something of the form $$ \frac{a\sum_{k=1}^n\frac{1}{X_k^2}-b}{c}\overset{d}{\to}\operatorname{Levy}(0,1), $$ where $\operatorname{Levy}(0,1)$ is the standard Levy distribution. But how to find the constants $a$, $b$, and $c$ is outside my current understanding of the subject.