Hoeffding's inequality for Hilbert space valued random elements

Question

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.

Answer 1

This is a special case of more general Theorem 3.5, 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 specifically for sums of independent zero-mean random vectors in a Hilbert space.