论文标题
非负高斯二次形式的尖锐方差 - 注入比较
Sharp variance-entropy comparison for nonnegative Gaussian quadratic forms
论文作者
论文摘要
在本文中，我们研究了$ n $ i.i.d.的加权总和。伽玛（$α$）随机变量具有非负重。我们表明，对于$ n \ geq 1/α$，当固定方差时，具有相等系数的总和可以最大化差分熵。结果，我们证明，在$ n $ n $独立的标准高斯随机变量中，在固定方差下，具有相等系数的对角线形式在$ n $ n $独立的标准高斯随机变量中，具有相等系数的对角性形式。这为非负二次形式和高斯随机变量之间的相对熵提供了锐利的下限。还会得出符合$ n $独立添加剂噪声的传输通道能力的界限。
In this article we study weighted sums of $n$ i.i.d. Gamma($α$) random variables with nonnegative weights. We show that for $n \geq 1/α$ the sum with equal coefficients maximizes differential entropy when variance is fixed. As a consequence, we prove that among nonnegative quadratic forms in $n$ independent standard Gaussian random variables, a diagonal form with equal coefficients maximizes differential entropy, under a fixed variance. This provides a sharp lower bound for the relative entropy between a nonnegative quadratic form and a Gaussian random variable. Bounds on capacities of transmission channels subject to $n$ independent additive gamma noises are also derived.
