Sharp Concentration Results for Heavy-Tailed Distributions. (arXiv:2003.13819v3 [math.PR] UPDATED)

We obtain concentration and large deviation for the sums of independent and
identically distributed random variables with heavy-tailed distributions. Our
concentration results are concerned with random variables whose distributions
satisfy $\mathbb{P}(X>t) \leq {\rm e}^{- I(t)}$, where $I: \mathbb{R}
\rightarrow \mathbb{R}$ is an increasing function and $I(t)/t \rightarrow
\alpha \in [0, \infty)$ as $t \rightarrow \infty$. Our main theorem can not
only recover some of the existing results, such as the concentration of the sum
of subWeibull random variables, but it …

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