Schur properties of convolutions of gamma random variables
The authors consider Schur convexity and concavity properties for tail probabilities of linear combinations of i.i.d. gamma random variables. That is, letting \(\boldsymbol{\lambda}=(\lambda_1,\ldots,\lambda_n)\) and \(\boldsymbol{\mu}=(\mu_1,\ldots,\mu_n)\) be elements of \(\mathbb{R}_+^n\) such that \(\boldsymbol{\lambda}\) majorizes \(\boldsymbol{\mu}\), the authors give conditions on \(x\in\mathbb{R}_+\) under which \[ P\left(\sum_{i=1}^n\lambda_iX_i<x\right)\leq P\left(\sum_{i=1}^n\mu_iX_i<x\right), \] where \(X_1,X_2,\ldots\) are i.i.d. gamma random variables. Similar conditions are also given for the reverse inequality to hold. These results improve similar inequalities in the literature, and are also extended to the case where \(\boldsymbol{\lambda}\) only weakly majorizes \(\boldsymbol{\mu}\), and to the infinite-dimensional setting. The proofs make use of Laplace transforms. Applications to signal detection and matrix trace estimation are also discussed.
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