Suppose that $\mathbb H$ is a separable Hilbert space and $X\_1,\ldots,X\_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X\_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|$ is the norm of $\mathbb H$. Denote $S\_n=X\_1+\ldots+X\_n$.



> 
> Is it true that
>  $$
> P(\|S\_n\|\ge t)\le Ce^{-c\frac{t^2}{ns^2}}
> $$
>  for $t>0$, where $C$ and $c$ are positive constants?
> 
> 
> 


This reduces to Hoeffding's inequality if $\mathbb H=\mathbb R$. So basically what I am asking is if there exists a generalization of Hoeffding's inequality to Hilbert spaces. Unfortunately, I could not find a result of this sort.


Any help is much appreciated.



Hoeffding's inequality for Hilbert space valued random elements
This is a special case of more general [Theorem 3.5](https://projecteuclid.org/euclid.aop/1176988477), which holds for martingales in $2$-smooth Banach spaces. Note that (i) any Hilbert space is $2$-smooth, with the $2$-smoothness constant $D=1$ and (ii) the consecutive sums of independent zero-mean random vectors constitute a martingale. 


The inequality you need can also be easily deduced from the earlier result: [Theorem 3](https://www.researchgate.net/publication/235412539_Remarks_on_Inequalities_for_Large_Deviation_Probabilities) specifically for sums of independent zero-mean random vectors in a Hilbert space. 


