Tail bounds on random series in Hilbert space

Question

Tail bounds on random series in Hilbert space

Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$, $n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ such that $\sum_n \|a_n\|^2 < \infty$, say $= 1$. Set $Z = \| \sum_n a_n X_n \|$ so that $E(Z^2) = \sum_n \|a_n\|^2 = 1$. Since $E(Z) \leq 1$, it is of interest to bound the probability $ P( Z > 1 + t)$, $t >0$. What would be a (sharp, exponential?) tail estimate in terms of the coefficients $a_n$, and in which range of $t >0$?

Answer 1

You can use e.g. Theorem 3.5, whereby $$P(Z\ge t)\le2e^{-t^2/2}\tag{1}$$ for all real $t\ge0$.

The coefficient $1/2$ in the exponent is of course sharp, in view of the standard central limit theorem (in $\mathbb R$), whereby $\sum_{j=1}^nX_j/\sqrt n$ converges to a standard normal random variable in distribution.

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$\newcommand\si\sigma$Write $$Z=\Big\|\sum_n a_n X_n\Big\|=\sup_{\|x\|\le1}\sum_n X_n\langle a_n,x\rangle $$ and let $$\si^2:=4\sup_{\|x\|\le1}\sum_n \langle a_n,x\rangle^2 \le4\sum_n \|a_n\|^2=4. $$ Then, by Talagrand's concentration inequality for product measures (see e.g. Section 2.2), $$P(|Z-m_Z|\ge t)\le4\exp\Big\{-\frac{t^2}{4\si^2}\Big\}$$ for all real $t\ge0$, where $m_Z$ is a median of $Z$ and hence $$P(|Z-EZ|\ge t)\le P(|Z-m_Z|\ge t-\sqrt{8\pi}\,\si) \le4\exp\Big\{-\frac{(t-\sqrt{8\pi}\,\si)^2}{4\si^2}\Big\}\tag{2}$$ for all real $t\ge\sqrt{8\pi}\,\si$. The bound (2) will be better than (1) if $\si^2<1/8$ and $t$ is large enough.