I need help understanding the reasoning that the Cauchy distribution can't have a finite mean in this answer: https://stats.stackexchange.com/a/36131/25186. It uses some geometric intuition arising from the fact that a Cauchy distribution is the stereographic projection of a point drawn uniformly on a unit circle. Defining $x$ as a function that does this stereographic mapping, I'm pasting the argument here:
To understand why the mean doesn't exist, think of x as a function on the unit circle. It's quite easy to find an infinite number of disjoint arcs on the unit circle, such that, if one of the arcs has length d, then x > 1/4d on that arc. So each of these disjoint arcs contributes more than 1/4 to the mean, and the total contribution from these arcs is infinite. We can do the same thing again, but with x < -1/4d, with a total contribution minus infinity.
I wish I could pin-point what exactly I'm not following, but am totally lost here. The poster wanted to upload an image but never did. Can someone please explain this argument?
