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 P(X>t)β€eβI(t), where I:RβR is an increasing function and I(t)/tβΞ±β[0,β) as tββ. 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.Comment: 16 page