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?
 A: 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.
A: 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 $q \ge p-1$.
A: 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.
