Limit distribution of the self-normalized sum of Cauchy random variables

This is something that has come up in my research. I originally posted this question on CrossValidated but realized it might be better suited for this site. I have deleted the question there (in case it annoys someone).

Let $$X_1, X_2, \dots,$$ be a sequence of iid Cauchy$$(x, 1)$$ random variables for some $$x \in \mathbb{R}$$. Define, $$S_n(x) = \frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n |X_i|},\quad n \in \mathbb{N}.$$ What, if it exists, is the limit of $$S_n(x)$$ as $$n \to \infty$$? I don't think an almost sure limit exists. Monte Carlo simulations suggest there should be a non-trivial limit. Perhaps, $$S_n$$ converges in distribution or in probability.

For fixed $$n = 20000$$, below are the density plots of $$S_n(x)$$ for different $$x$$. For each $$x$$, I generated $$20000$$ instances of $$S_n(x)$$ and used them to estimate the density. On the $$x-$$axis is the location parameter $$x$$.

A paper I found in an answer to another question studies the above limit for stable distributions. It conveniently ignores the Cauchy case. They cite Feller's second volume but I haven't been able to find a result there. I wonder if any references consider the above limit for Cauchy random variables.

Another excellent answer I found concerning the median of the sum of half-Cauchy random variables. I think it suggests that the half-Cauchy might somehow belong to the domain of attraction of an asymmetric $$\alpha=1$$ stable law. If this is so, it might help understand the limit of $$S_n$$. But I am not sure.

• Noting the answer of mike: This is one example where simulation is unfortunately of no use. Feb 28 at 19:27

It converges to 0 in probability. $$\frac{\Sigma_1^nX_i}n$$ always has the same cauchy distribution while $$\frac{\Sigma_1^n|X_i|}n \rightarrow \infty$$ by the law of large numbers