Is there any random variable which has unbounded fourth moment?

In many statements in probability, there is an assumption like bounded fourth moment. So is there any random variable which has unbounded fourth moment?

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That hypothesis means that a certain integral is finite. You can always arrange that integrals diverge! –  Mariano Suárez-Alvarez Jan 7 '10 at 6:49

An example of a random variable having an infinite fourth moment (and finite lower moments) is the student's t-distributionwith 4 degrees of freedom (see for example the Wikipedia page). In general, unless certain conditions are satisfied, ratios of two random variables have infinite moments, the following article by Cedilnik, Katarina , and Blejec addresses the question of the existence of moments of a ratio of two random variables.

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For example, the Cauchy distribution (mentioned by David Speyer above) is the ratio of two normal random variables with zero mean and unit variance. –  Michael Lugo Jan 7 '10 at 15:18

More generally, given $p > 1$, take any bounded function on $\mathbb{R}$ which behaves like $1/|x|^p$ as $x\to \infty$, for example $1/(1+|x|^p)$. After renormalizing, this is will be the density of a random variable which has finite absolute $q$th moments for $0 \le q < p-1$, and infinite $q$th moments for $q \ge p-1$.

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An example of this type that has a name is the Pareto distribution (en.wikipedia.org/wiki/Pareto_distribution) –  user2734 Jan 7 '10 at 15:32

What about just taking the real valued random variable which lands in the interval $(x, x+dx)$ with probability $(1/\pi) dx/(1+x^2)$? The fourth moment is

$$\frac{1}{\pi} \int_{\mathbb{R}} \frac{x^4 dx}{(1+x^2)}$$

which is extremely divergent.

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This even has a name--it's the Cauchy distribution. –  Reid Barton Jan 7 '10 at 14:14
I assumed the question was more specific: infinite fourth moment, and finite third moment. –  Gerald Edgar Jan 7 '10 at 16:47