Moment free deviation inequalities for linear combinations of independent random variables with power-type tails

We present order of magnitude estimates for the quantiles of non-negative linear combinations of non-negative random variables, as well as deviation inequalities for general linear combinations of independent random variables, under the assumption that all random variables satisfy the same power-type tail bound on $\mathbb{P}\{\mid X_i\mid >t\}$ of the form $t^{-q}$, $t^{-q\mathrm{/}2}$ or $t^{-q\mathrm{/}2}(\ln t{)}^{q\mathrm{/}2}$, for $q>2$. The third type is applicable in the nonlinear setting. In the situations we consider, these results improve on classical estimates of Nagaev.