Exercise 2.16: Deviation Inequalities in Hilbert Space
(a)
Let \(\Omega_i \subseteq \Omega\) be a set of probability one on which \(\| X_i \|_{\mathcal{H}} \leq b_i\), and set \(D_i = \lbrace X_i (\omega) \in \mathcal{H} \colon \omega \in \Omega_i \}\). Define \(f \colon \mathcal{H}^n \to \R\) such that \begin{align} f(x_1, \ldots, x_n) = \begin{cases} \| x_1 + \ldots + x_n \|_{\mathcal{H}} & \text{if } x_1 \in D_1 , \ldots , x_n \in D_n \, , \newline 0 & \text{otherwise.} \end{cases} \end{align} Then \(S_n = f (X_1, \ldots , X_n)\) almost surely. Moreover, by the reverse triangle inequality, \(f\) satisfies the bounded difference property with parameters \(L_i = 2 b_i\). Since \(X_1, \ldots , X_n\) are independent by assumption, we can use the Bounded differences inequality (Corollary 2.21) \begin{align} \P (|S_n - \E [S_n]| \geq n \delta) \leq 2 \exp \left\lbrace - \frac{n^2 \delta^2}{2 \sum_i b_i^2 } \right\rbrace = 2 \exp \left\lbrace - \frac{n \delta^2}{2 b^2} \right\rbrace \, , \end{align} which improves upon the bound in the book.
(b)
Assume \(\E [ \langle X_i , X_j \rangle_{\mathcal{H}} ] = 0\) for all \(i \neq j\). Then \begin{align} \sqrt{\E \biggl[ \sum_{i=1}^n \| X_i \|_{\mathcal{H}}^2 \biggr]} = \sqrt{\E \biggl[ \biggl\| \sum_{i=1}^n X_i \biggr\|_{\mathcal{H}}^2 \biggr]} \geq \E \biggl[ \biggl\| \sum_{i=1}^n X_i \biggr\|_{\mathcal{H}} \biggr] \, , \end{align} in which the case the result follows from the one-sided analogue of (a).
The assumption \(\E [ \langle X_i , X_j \rangle_{\mathcal{H}} ] = 0\) for all \(i \neq j\) does not follow from the assumed independence of \(\{ X_i \}_i\) (e.g., take \(X_i = x\) a.s. for all \(i\) and some fixed \(x \in \mathcal{H}\)). It is unclear how to derive the result in the general case without further assumptions, such as that the Bochner integral with respect to \(\P\) of each \(X_i\) is \(0 \in \mathcal{H}\).