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The Cauchy distribution is used as an example of a pathological case where the mean blows up. For such a distribution, we can imagine drawing samples and tracking the sample mean as the number of samples increases. For any distribution where the mean doesn't blow up, the sample mean will start converging to it.

For the Cauchy distribution, it isn't clear what to expect. I suspected the sample mean would simply keep growing as the number of samples grew. Per Wikipedia, it seems that it simply varies wildly no matter how large $n$ becomes but there is no trend. See here: https://upload.wikimedia.org/wikipedia/commons/a/aa/Mean_estimator_consistency.gif

The question is, does this behavior apply to all distributions where the mean blows up or just the Cauchy distribution? And can we predict without drawing samples what the behavior will be?

ryu576
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  • "Blows up" to me implies a mean increasing in magnitude toward $\infty$ in the limit (and that seems to be your interpretation also), but the mean of a Cauchy is *undefined* - it's the limit of a difference of two terms each going to $\infty$ ( loosely, "$\infty - \infty$"). I wouldn't say that it " blows up" in that simple sense. There just isn't a mean to converge *to*. – Glen_b Oct 24 '21 at 05:18
  • The mean is an integral going from $-\infty$ to $+\infty$. If we take the integral from $-a$ to $a$ instead, is it an increasing function of $a$ or does it meander between some range never settling on anything? Do you know? – ryu576 Oct 24 '21 at 05:35
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    That integral is 0 for any finite $a$, and in the limit, that's the Cauchy principal value, but that's not how integration works. Expectation follows the ordinary definition of integration where you take the limits separately. – Glen_b Oct 24 '21 at 07:42

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When we say that something "blows up" (in math/stat context, of course) we mean that it grows to infinity, many times also meaning it grows exponentially with $n$. That is, if we discuss sample means, $$\lim_{n\rightarrow\infty}{\bar{X}_n}=\pm\infty$$ This isn't the case with the mean of the Cauchy distribution. It does not blow up but simply doesn't converge (that's why it's undefined). Unlike the limit shown above, the sample mean of the Cauchy does not have any limit. That's what makes it such an annoying distribution coming straight from hell a useful distribution for counterexamples.

More specifically, the pdf of Cauchy is given by $$f(x;\gamma,x_0)=\frac{1}{\pi\gamma\left[ 1+\left(\frac{x-x_0}{\gamma}\right)^2 \right]}$$ and so the mean should be obtained by solving the devilish integral $$\int_{-\infty}^{\infty}{\frac{x}{\pi\gamma\left[ 1+\left(\frac{x-x_0}{\gamma}\right)^2 \right]}dx}$$ but it isn't solvable.

Recall Leo Tolstoy's famous words:

All happy families are alike; each unhappy family is unhappy in its own way.

(Anna Karenina, 1877)

All happy distributions with a defined mean are alike; each distribution with undefined mean is unhappy in its own way. We cannot deduce from Cauchy to other distributions.

Spätzle
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