It is known that for independent sub-exponential random variables, the following Bernstein-type inequality holds:
\begin{align}
\mathbb{P}\biggl(\biggl| \sum_{i=1}^N a_i X_i\biggr| >t \biggr) \leq 2 \exp\left[ -c\min \left(\frac{t^2}{K^2 \| \vec{a}\|_2^2}, \frac{t}{ K \| \vec{a}\|_{\infty}} \right)\right],
\end{align}
where $K = \max \| X_1\|_{\psi_1}$ and $\vec{a}\in\mathbb{R}^N$.
I wonder if similar concentration inequality holds for heavy-tailed random variables where $X_i$ satisfies $\mathbb{P}(X_i > t) \leq C\exp(c t^{-\alpha}) $ for $\alpha\in(0,1)$.
Best Answer
Yes, see Theorem 6.21 of [LT13], Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: isoperimetry and processes, volume 23. Springer Science & Business Media, 2013.
For simplicity you may also look at section 8 of my paper.
http://arxiv.org/pdf/1507.06370v2.pdf
(I just summarized those theorems -- the purpose of the paper is completely different)