2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

Concentration Inequalities for Bounded Random Vectors, by Xinjia Chen (2013):

We derive simple concentration inequalities for bounded random vectors, which generalize Hoeffding's inequalities for bounded scalar random variables. As applications, we apply the general results to multinomial and Dirichlet distributions to obtain multivariate concentration inequalities.

$\endgroup$
  • 4
    $\begingroup$ What if I'm interested in bounding $\mathbb{P}\left[\left\|\sum_i^n X_i - \mu_i\right\| > \lambda\right]$ instead? $\endgroup$ – Thomas Mar 16 '15 at 18:19
  • $\begingroup$ 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. $\endgroup$ – Mark Apr 5 at 17:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.