Sharp Concentration Results for Heavy-Tailed Distributions
This provides foundational theoretical tools for probability and statistics, applicable in fields like machine learning and risk analysis, but it is incremental as it builds on standard truncation methods.
The paper tackles the problem of deriving concentration and large deviation results for sums of independent heavy-tailed random variables, achieving sharp inequalities that generalize and unify existing results, such as for subWeibull variables.
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 $I(t)/t \rightarrow α\in [0, \infty)$ 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 can also produce new results for the sum of random variables with heavier tails. We show that the concentration inequalities we obtain are sharp enough to offer large deviation results for the sums of independent random variables as well. Our analyses which are based on standard truncation arguments simplify, unify and generalize the existing results on the concentration and large deviation of heavy-tailed random variables.