# Hoeffding's inequality for Hilbert space valued random elements

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.

## 1 Answer

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.