Some stochastic inequalities for weighted sums

TL;DR

This paper establishes new stochastic inequalities for weighted sums of i.i.d. positive random variables using majorization and log-concavity assumptions, unifying and extending previous results with applications in reliability and wireless communications.

Contribution

It introduces novel stochastic inequalities for weighted sums under log-concavity conditions, proving a conjecture on Weibull variables and unifying various existing results.

Findings

Proved stochastic ordering for sums with log-concave log-densities.

Extended inequalities to cases where $Y_i^p$ has a log-concave density.

Confirmed a conjecture related to Weibull distributions.

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 $\mathbf{R}_+$. Assuming that $\log Y_i$ has a log-concave density, we show that $\sum a_iY_i$ is stochastically smaller than $\sum b_iY_i$, 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 $\sum a_iY_i$ is stochastically larger than $\sum b_iY_i$, 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 [Sankhy\={a} A 60 (1998) 171--175] on Weibull variables is proved. Potential applications in reliability and wireless communications are mentioned.