How to prove the Cauchy distribution has no moments?

[Neothilic]

TL;DR

Proving how the Cauchy distribution has no moments for all n.

How can I prove the Cauchy distribution has no moments?

$E(X^n)=\int_{-\infty}^\infty\frac{x^n}{\pi(1+x^2)}\ dx.$

I can prove myself, letting $n=1$ or $n=2$ that it does not have any moment. However, how would I prove for ALL $n$, that the Cauchy distribution has no moments?

 

[Paul Colby]

Sorry for the dumb I'm not a mathematician question but why isn't

$E_A(X^n) = \int_{-A}^A \frac{x^n}{\pi(1+x^2)}dx = 0$

for all $n$ odd and all $A$?

 

[StoneTemplePython]

so you've proven
$E\Big[\big \vert X\big \vert\Big] = \infty$

in general
$\big \vert X\big \vert= \Big(\big \vert X\big \vert^n \cdot 1\Big)^\frac{1}{n}= \Big(\big \vert X\big \vert^n \cdot \prod_{k=1}^{n-1}1\Big)^\frac{1}{n} \leq \frac{1}{n} \Big(\big \vert X\big \vert^n + \sum_{k=1}^{n-1}1 \Big) =\frac{1}{n} \big \vert X\big \vert^n + \frac{n-1}{n}$
by $\text{GM}\leq \text{AM}$

Now take expectations of each side (and use LOTUS)
$E\Big[\big \vert X\big \vert\Big]\leq \frac{1}{n} E\Big[\big \vert X\big \vert^n\Big] + \frac{n-1}{n}$

thus
$E\Big[\big \vert X\big \vert\Big] = \infty \longrightarrow E\Big[\big \vert X\big \vert^n\Big] = \infty$

 

[Neothilic]

so you've proven
$E\Big[\big \vert X\big \vert\Big] = \infty$

in general
$\big \vert X\big \vert= \Big(\big \vert X\big \vert^n \cdot 1\Big)^\frac{1}{n}= \Big(\big \vert X\big \vert^n \cdot \prod_{k=1}^{n-1}1\Big)^\frac{1}{n} \leq \frac{1}{n} \Big(\big \vert X\big \vert^n + \sum_{k=1}^{n-1}1 \Big) =\frac{1}{n} \big \vert X\big \vert^n + \frac{n-1}{n}$
by $\text{GM}\leq \text{AM}$

Now take expectations of each side (and use LOTUS)
$E\Big[\big \vert X\big \vert\Big]\leq \frac{1}{n} E\Big[\big \vert X\big \vert^n\Big] + \frac{n-1}{n}$

thus
$E\Big[\big \vert X\big \vert\Big] = \infty \longrightarrow E\Big[\big \vert X\big \vert^n\Big] = \infty$

Wow. I have been trying to prove this for a couple weeks now. I have tried doing series test with the integrands etc. However, this is so unique and short and brilliant! How did you come up with this? Btw, what is meant by GM and AM?

 

[mathman]

$\frac{|x^n|}{1+x^2}\to \infty$ as $|x|\to \infty$, for $n\gt 2$. So the integral blows up.

 

[mfb]

$\frac{|x^n|}{1+x^2}\to \infty$ as $|x|\to \infty$, for $n\gt 2$. So the integral blows up.

And even for n=2 the integrand does not go to zero, that means all these cases can be solved without calculating anything. Only n=1 needs some thought.

Btw, what is meant by GM and AM?

Geometric mean, arithmetic mean

You opened three threads for basically the same topic - it would have been better to keep the discussion in one thread. But now merging them would create a mess so I keep them separate.

https://www.physicsforums.com/threa...nts-using-the-characteristic-function.992462/
https://www.physicsforums.com/threads/what-is-the-nth-differential-of-this-equation.992492/