Let us consider a simple random walk on $\mathbb{Z}^2$ started at $(x,0)$ and killed upon hitting the origin. Define the total winding number $w_x$ around the origin to be the (signed) number of complete rotations around 0 (up to the last step before the random walk is killed). **What is known about the distribution of $w_x$ as $x\to\infty$?** More precisely, does $w/\log(x)$ converge in distribution? So far I have not been able to find a reference addressing this particular question. Indeed, most references on winding numbers of random walks seem to focus on the distribution after a (large) fixed number of steps.

One would expect the distribution to be related to the analogue for Brownian motion started at $(x,0)$ and killed upon hitting the unit circle. By going to radial coordinates it is easy to see that this total winding is given by the integer part of a Cauchy random variable with scale parameter $\log(x)/(2\pi)$.