What is the largen limit of a distribution of the following sample statistic:$$x\equiv\displaystyle\frac{\sum^{n}X_{i}}{\,\sqrt{\,\sum^{n}X_{i}^{2}\,}\,}$$ when sampling the Cauchy(0,1) distribution? Monte Carlo simulation indicates that convergence to this limit is quite fast, and that the resulting (symmetric) PDF has very sharp cusps at 1 and +1, but of course does not yield an analytic expression for this PDF  would anyone be able to find it?
Here is how the positive half of the PDF looks like:

$\begingroup$ A somewhat random remark: the limiting distribution is also the distribution of $X_1 / \langle X\rangle_1$ if $X_t$ is the Cauchy process and $\langle X\rangle_t$ is the quadratic variation process, a $\tfrac{1}{2}$stable subordinator. I bet someone has studied the joint law of $X_t$ and $\langle X\rangle_t$, but unfortunately I do not have time now to search for the reference. $\endgroup$ – Mateusz Kwaśnicki Jun 16 '19 at 20:17
This desired large$n$ limit of the distribution $P_n(x)$ is calculated in Limit Distributions of Selfnormalized Sums (1973). The Cauchy distribution is the case $\alpha=1$ on page 798. The singularity in $\lim_{n\rightarrow\infty}P_n(x)$ at $x=\pm 1$ is logarithmic $$P_n(x)\rightarrow\pi^{2}\log1x^2,$$ see equation (5.12) and figure 4 (reproduced below), while for large $x$ the decay is Gaussian.

$\begingroup$ Thanks for your help; the article you mention is as exhaustive as it could be  one has to accept that the asymptotic PDF is not a "pretty" function! $\endgroup$ – Honza Jun 16 '19 at 22:29