An unexpected encounter with Cauchy and L\'evy

@article{Pillai2015AnUE,  title={An unexpected encounter with Cauchy and L\'evy},  author={Natesh S. Pillai and Xiao-Li Meng},  journal={arXiv: Statistics Theory},  year={2015},  url={https://api.semanticscholar.org/CorpusID:31582370}}
The Cauchy distribution is usually presented as a mathematical curiosity, an exception to the Law of Large Numbers, or even as an "Evil" distribution in some introductory courses. It therefore surprised us when Drton and Xiao (2016) proved the following result for $m=2$ and conjectured it for $m\ge 3$. Let $X= (X_1,..., X_m)$ and $Y = (Y_1, ...,Y_m)$ be i.i.d $N(0,\Sigma)$, where $\Sigma=\{\sigma_{ij}\}\ge 0$ is an $m\times m$ and \textit{arbitrary} covariance matrix with $\sigma_{jj}>0$ for… 

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