I need to construct a random variable $X$ such that the $L^p$ norm $\|X\|_p<\infty$ for an arbitrary $p\geq1$ but has infinite $L^q$ norm $\|X\|_q=\infty$ for all $q>p$.
I've tried every combination of elementary functions I can think of and can't quite get something that works. Any help or hints would be appreciated. Even just a hint in the right direction would be nice. I've been stuck thinking about this for over 24 hours now.
Thank you!