My question concerns the boundedness of the absolute moments of a random variable $E|X|^k$ for any $0\leq k<\infty$. Or equivalently, under which conditions holds $X\in L^k(\Omega)$
Are there conditions on the probability distribution under which this holds, or holds at least for $k\leq K$ for some $K$?
Does finiteness of all moments or the existence of the moment generating function imply finiteness of absolute moments?