Wishart和Kummer随机矩阵的独立性刻画
Independence characterization for Wishart and Kummer random matrices
摘要：
本文将Kummer分布和Gamma分布的一元刻画推广到对称正定矩阵锥上：设$X$和$Y$是独立的、非退化的随机变量，其值在$(0, \infty)$中，则$U= Y/(1+X)$和$V = X(1+U)$是独立的当且仅当$X$服从Kummer分布，$Y$服从Γ分布，且具有适当的参数.我们在对称正定矩阵的锥上求解了一个相关的函数方程，这是我们的第一个主要结果，并应用它的解证明了Wishart分布和矩阵-Kummer分布的特征，这是我们的第二个主要结果。
英文摘要：
We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let $X$ and $Y$ be independent, non-degenerate random variables valued in $(0, \infty)$, then $U= Y/(1+X)$ and $V = X(1+U)$ are independent if and only if $X$ follows the Kummer distribution and $Y$ follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization of Wishart and matrix-Kummer distributions, which is our second main result.
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