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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 ( – 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

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