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.
 A: 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.

A: Here is one way to extend it using a simple union bound to control the deviation of every coordinate at once. You could imagine other ways to extend it, but ensuring that every coordinate is near the true value is already very strong.
For $X_1, ..., X_n \in [0,1]$ be iid from some distribution with mean $\mu$, Hoefdding
 says : $P ( | \bar{X} - \mu| > \epsilon) \leq 2\exp( - 2 n \epsilon^2) $.
Suppose we have iid vectors $Y_1, Y_2, \ldots, Y_n \in [0,1]^m$. For a vector $Z \in [0,1]^m$ let $Z(j)$ denote the $j$th coordinate. In particular, $(Y_k)(j)$ denotes the $jth$ coordinate of the $k$th vector in the sample. 
Suppose $\mathbb{E}[Y_1(j)] = \mu_j$. We have $\mathbb{E}[Y](j) = \mathbb{E}[Y_1(j)]$ by linearity of expectation, and $\overline{Y}(j) = \overline {Y(j)}$ similarly, where the overline denotes the empirical mean.
We define the event that the $j$th coordinate of $\overline{Y}$ is far from the true mean of that coordinate: $A_j = \{ | \bar{Y}(j) - \mu_j| > \epsilon \}$. Hoeffding's tells us that $P(A_j) \leq 2\exp( - 2 n \epsilon^2) $.
We can bound the probability that any coordinate is $\epsilon$ away from the true mean of that coordinate by using the union bound. That is, $P ( \bigcup_{j = 1}^m  A_j ) \leq \sum_{j = 1}^m P(A_j) \leq 2m \exp( -2n \epsilon^2)$. 
This is useful as we only need to increase $n$ by adding $\log(m)/(2\epsilon^2)$ in order to eliminate the multiplier.
A: As far as I know, currently, I have not found Hoeffding's inequality for vector-valued random variables. However, we can use Bernstein Inequality to describe how the concentration of the vectors sum. As indicated in 1, we have the following bound:

This bound is derived from 2 which is

We can have a more generalized version of the Bernstein Inequality for matrices. You can refer to 3:

1 Jonas Moritz Kohler. Sub-sampled Cubic Regularization for Non-convex optimization. 2017
2 David Gross. Recovering Low-Rank Matrices From Few Coefficients In Any Basis. 2010
3 Joel A. Tropp. An Introduction to Matrix Concentration. 2015
