How to understand that a distribution has no mean?

Question

I've learned that Cauchy distribution doesn't have mean, i.e. the integral $\int_{-\infty}^\infty xf(x,x_0,\gamma)dx$ diverges. But it still has Cauchy principal value equal to location parameter $x_0$.

So from the divergence of the integral I might conclude that sequence of averages of larger and larger samples won't converge to anything in any sense.

But is it really true, or does the existence of Cauchy principal value still allow the sequence of averages to converge to $x_0$?

Answer 1

The Cauchy principal will give you the median. Thus, the median of a sequence of iid Cauchy variables will converge to the principal value. However, the mean of that same sequence will not converge. So, the principal value provides a measure of location, just not in the sense of "average".

What happens with the mean is that no matter how many observations we have made of a Cauchy sequence (iid), there is a large enough probability (there is a precise definition of "large enough" that I will not go into) that the next observation will swamp the sum of all the previous observations.

Answer 2

Try sampling numerically. It's easy to sample from the Cauchy distribution: the CDF is

$$F(x) = \frac{1}{\pi} \int_{-\infty}^x \frac{1}{1+y^2} dy = \frac{1}{2} + \frac{\arctan(x)}{\pi}.$$

So the quantile function is the inverse of that, which is $\tan(\pi x - \pi/2)$. So a sequence of iid Cauchy variables is given by $X_i=\tan(\pi U_i - \pi/2)$, where $U_i$ are iid uniform variables on $(0,1)$.

Most programming languages have $\tan,\pi$ and a $U(0,1)$ random number generator, usually called rand, so you can try this yourself. If you do, you will find that, after some initial fluctuation, the average stays near the same number, and then eventually (but reliably) it changes very abruptly. Thus in the long run the sequence oscillates without decaying amplitude.

This is related to the fact that the sum of iid Cauchy variables has the same distribution as the variables themselves. Thus if you take many averages of samples with a fixed large $N$, their distribution is not any more concentrated around any particular point than it was with a single Cauchy variable.