TITLE: Large-n limit of the distribution of the normalized sum of Cauchy random variables
QUESTION [13 upvotes]: What is the large-n 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:

REPLY [15 votes]: This desired large-$n$ limit of the distribution $P_n(x)$ is calculated in Limit Distributions of Self-normalized 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
$$P_n(x)\rightarrow-\pi^{-2}\log|1-x^2|,$$
see equation (5.12) and figure 4 (reproduced below), while for large $|x|$ the decay is Gaussian.