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We present order of magnitude estimates for the quantiles of non-negativelinear combinations of non-negative random variables, as well as deviationinequalities for general linear combinations of independent random variables,under the assumption that all random variables satisfy the same power-type tailbound on $\mathbb{P}\{\left\vert X_i\right\vert>t\}$ of the form $t^{-q}$,$t^{-q/2}$ or $t^{-q/2}(\ln t)^{q/2}$, for $q>2$. The third type is applicablein the nonlinear setting. In the situations we consider, these results improveon classical estimates of Nagaev.

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