  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 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.
