Concentration for independent random variables with heavy tails
  If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of $n$ independent copies, with good dependence in $n$. 
Franck Barthe (LSProba), Patrick Cattiaux (MODAL'X, CMAP), Cyril   Roberto (LAMA)