Some stochastic inequalities for weighted sums

Abstract

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 Y i be i.i.d. random variables on R+. Assuming that logYi has a log-concave density, we show that Σa iYi is stochastically smaller than Σ b iYi, if (loga1,..., logan) is majorized by (logb1,..., logbn). On the other hand, assuming that Yip has a log-concave density for some p > 1, we show that ΣaiYi is stochastically larger than ΣbiYi, if (a1q,..., anq) is majorized by (b1q,...,b nq), where p-1 +q-1 = 1. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko [Sankhyā A 60 (1998) 171-175] on Weibull variables is proved. Potential applications in reliability and wireless communications are mentioned. © 2011 ISI/BS.