is there any random variable which has unbounded fourth moment?
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.
