Some stochastic inequalities for weighted sums

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let Yi be i.i.d. random variables on R+. Assuming that log Yi has a log-concave density, we show that ∑ aiYi is stochastically smaller than ∑ biYi, if (log a1, . . . , log an) is majorized by (log b1, . . . , log bn). On the other hand, assuming that Y p i has a log-concave density for some p > 1, we show that ∑ aiYi is stochastically larger than ∑ biYi, if (a q 1 , . . . , aqn) is majorized by (b q 1 , . . . , bqn), where p −1 + q = 1. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko (1998) on Weibull variables is proved. Some applications in reliability and wireless communications are mentioned.