I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A_{\Omega},P)$ a probability space. Suppose that $Y_1,Y_2,…$ is a sequence of independent $V$-valued random variables. If $E\left(\Vert Y_i\Vert_{V}^m\right)\leq \frac{1}{2} m! B^2 L^{m-2}$ $\forall m\geq 2$, then, $\forall n\in N$ and $\epsilon>0$
$$ P\big(\big\Vert \frac{1}{n} \sum_{i=1}^n Y_i\big\Vert_{V}>\epsilon\big)\leq 2 \exp\Big(-\frac{n\epsilon^2}{B^2+L\varepsilon+B\sqrt{B^2+2L\epsilon}}\Big).$$
There was no proof given but it was mentioned that the result is well known. I think I found some literature that shows the result if we would assume that $E(Y_i)=0$ for all $i\in N$ holds (see Theorem 3.3 of Pinelis). But I don't know much about martingales so maybe it wouldn't follow. Does anyone know if the result is actually true and if $E(Y_i)=0$ is a necessary condition?
Best Answer
The condition $EY_i=0$ cannot be dropped.
Indeed, if e.g. the $Y_i$'s are iid with $\mu=EY_i\ne0$ and $n\to\infty$, then, by the law of large numbers, the left-hand side of the inequality in question will go to $1$ for each $\epsilon\in(0,\|\mu\|_V)$, whereas the right-hand side of the inequality will go to $0$.