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August 2011 Some stochastic inequalities for weighted sums
Yaming Yu
Bernoulli 17(3): 1044-1053 (August 2011). DOI: 10.3150/10-BEJ302


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 $\log Y_i$ has a log-concave density, we show that $∑ a_iY_i$ is stochastically smaller than ∑ biYi, if ($\log a_1$, …, $\log a_n$) is majorized by ($\log b_1$, …, $\log b_n$). On the other hand, assuming that $Y_i^p$ has a log-concave density for some $p > 1$, we show that $∑a_iY_i$ is stochastically larger than $∑ b_iY_i$, if ($a_1^q$, …, $a_n^q$) is majorized by ($b_1^q$, …, $b_n^q$), 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.


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Yaming Yu. "Some stochastic inequalities for weighted sums." Bernoulli 17 (3) 1044 - 1053, August 2011.


Published: August 2011
First available in Project Euclid: 7 July 2011

zbMATH: 1225.60035
MathSciNet: MR2817616
Digital Object Identifier: 10.3150/10-BEJ302

Keywords: gamma distribution , Log-concavity , majorization , Prékopa–Leindler inequality , Rayleigh distribution , tail probability , usual stochastic order , Weibull distribution , weighted sum

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability


Vol.17 • No. 3 • August 2011
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