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Is there a version of Hoeffding's inequality for vector valued random variables? This seems to be hard to find and I wonder why. I suppose it is difficult to show Hoeffding's lemma, since the proof ...

This is a follow-up on another question. Can something be said about the image of a weakly converging sequence in $L^1$? More precisely $u_k\ge 0$ $\|u_k\|_{L^1}=\int u_k=1$ $u_k$ converges to $u$ ...