Let $(X_T)$ be a positive sequence of random variable such that $\mathbb{E}(X_T)=1$ for all $T$. Under which condition can we state that: $$ \mathbb{E}(\limsup X_T) = 1 $$ Would uniform boundedness be enough for example? Fatou Lemma gives the inequality $\geq$, what remains to see is the other inequality.
2026-04-07 00:45:27.1775522727
Reverse Fatou Lemma with uniform boundedness
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