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


1 Answer 1


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.


(I just summarized those theorems -- the purpose of the paper is completely different)

  • $\begingroup$ 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.. $\endgroup$ Dec 27, 2018 at 9:30
  • $\begingroup$ I guess you just forgot an exp() in the second part of definition 8.1? Then you also match the definition in [vdVW00]. Btw, do you a more precise reference for your lemma 8.3? The vdVW book is long 😉 $\endgroup$ Jan 2, 2019 at 12:25
  • $\begingroup$ It is lemma 2.2.2 on p96 of vdv-W. $\endgroup$ Apr 20, 2022 at 3:50

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