# Hoeffding's inequality for vector valued random variables

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 for the inequality seems to translate relatively easy to a vector space.

• What if I'm interested in bounding $\mathbb{P}\left[\left\|\sum_i^n X_i - \mu_i\right\| > \lambda\right]$ instead? – Thomas Mar 16 '15 at 18:19
• It's worth mentioning that the above is much less general than a first glance might imply. It gives you bounds $\vec{X}\geq \vec{z}$ (where $\vec{X}\geq\vec{z}$ means the partial-order induced by entry-wise comparisons), but the conditions on $\vec{z}$ are rather restrictive. Specifically, the $\lvert \vec{z}\rvert_1 = \mathbb{E}[\lvert \vec{X}\rvert_1]$, where each $z_i$ is a non-negative integer. I was unable to adapt the results in the paper for my required usage, involving Chernoff-type bounds on multinomial random variables. – Mark Apr 5 at 17:51