# Tail bounds on random series in Hilbert space

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 $$t >0$$?

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.
$$\newcommand\si\sigma$$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.