In one dimension, if $X_i\in\left[a_i,b_i\right]$, $i\in\left\{1,\ldots{},n\right\}$, are i.i.d. then \begin{align*} P\left(|\bar{X}-E\left[\bar{X}\right]|\geq{}t\right)\leq{}2\exp\left(-\frac{2n^2t^2}{\sum_{i=1}^{n}\left(a_i-b_i\right)^2}\right), \end{align*} where $\bar{X}=\frac{1}{n}\sum_{i=1}^{n}$ and $E\left[\bar{X}\right]$ is the expected value of $\bar{X}$.
For multidimensional $X_i$, if $\left\Vert{}X_i\right\Vert\in\left[a_i,b_i\right]$ and $p<<n$, I'm hoping for something like \begin{align*} P\left(\left\Vert\bar{X}-E\left[\bar{X}\right]\right\Vert\geq{}t\right)\leq{}2\exp\left(-\frac{2n^2t^2}{\sum_{i=1}^{n}\left(a_i-b_i\right)^2}\right), \end{align*} where $\left\Vert\right\Vert$ is the Euclidean norm.
Is there a version of Hoeffding's inequality (or some other concentration inequality) for high dimensional data, specifically when $p>>n$?
Thanks.
