# Bernstein's inequality for heavy-tailed random variables

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 $\alpha\in(0,1)$.

• It seems like you're defining $\psi_\alpha$ somewhat differently from talagrand when $alpha<1$? He says $e^{x^\alpha}-1$ (linear near the origin). You say $x^\alpha-1$. I don't see how your definition can give tail bounds, since we're only considering very tiny moments, but probably I misunderstood something.. – Thomas Ahle Dec 27 '18 at 9:30