Question

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

Answer 1

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.

Answer 2

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.

Answer 3

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 $q \ge p-1$.