We establish several convexity properties for the entropy and Fisher information of mixtures of centered Gaussian distributions. First, we prove that if $X_1, X_2$ are independent scalar Gaussian mixtures, then the entropy of $\sqrt{t}X_1 + \sqrt{1-t}X_2$ is concave in $t \in [0,1]$, thus confirming a conjecture of Ball, Nayar and Tkocz (2016) for this class of random variables. In fact, we prove a generalisation of this assertion which also strengthens a result of Eskenazis, Nayar and Tkocz (2018). For the Fisher information, we extend a convexity result of Bobkov (2022) by showing that the Fisher information matrix is operator convex as a matrix-valued function acting on densities of mixtures in $\mathbb{R}^d$. As an application, we establish rates for the convergence of the Fisher information matrix of the sum of weighted i.i.d. Gaussian mixtures in the operator norm along the central limit theorem under mild moment assumptions.

翻译：我们建立了中心高斯分布混合的熵与费舍尔信息的若干凸性性质。首先，我们证明若$X_1, X_2$是独立标量高斯混合，则$\sqrt{t}X_1 + \sqrt{1-t}X_2$的熵在$t \in [0,1]$上是凹的，从而证实了Ball、Nayar和Tkocz（2016）关于此类随机变量的猜想。事实上，我们证明了该断言的一个推广形式，该推广也强化了Eskenazis、Nayar和Tkocz（2018）的一个结果。对于费舍尔信息，我们将Bobkov（2022）的一个凸性结果进行推广，通过证明费舍尔信息矩阵作为作用于$\mathbb{R}^d$中混合密度的矩阵值函数是算子凸的。作为应用，我们在温和矩假设下，建立了沿中心极限定理的算子范数下加权独立同分布高斯混合之和的费舍尔信息矩阵收敛速率。