I was looking into Doob's upcrossing inequality, which says that for a supermartingale $X$ and real numbers $b > a$, $$(b-a)\mathbb{E}[N_n([a,b], X)] \leq \mathbb{E}[(X_n -a)^-],$$where $N_n([a,b], X)$ is the number of upcrossings $[a,b]$ by the sequence $X$ by time $n$. If $X_n$ were IID random variables, then the right hand side of this inequality would always be constant, but I would imagine that the left hand side would be increasing at a constant rate. This would indicate to me that it is impossible for an IID sequence to be a supermartingale (let alone a martingale). Unfortunately, I can't see why this is impossible just from the raw definitions of a (super)martingale. Is anyone able to help me with this?
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