Pursuing the study initiated in Hassairi and Roula (2019), we show in the present paper that the reliability function of a probability distribution on the cone $\Omega$ of positive definite symmetric matrices characterizes the distribution without any invariance condition. We also show that the characterization of the exponential probability distribution on $\Omega$ by a memoryless property holds without assuming an invariance condition. We then study the connection between the exponential distribution on $\Omega$ and the uniform distribution on a bounded interval of $\Omega$. A notion of matrix Pareto distribution is introduced, and it is shown that this distribution possesses the long tail property.

根据 Hassairi 和 Roula (2019) 发起的研究，我们在本文中展示了圆锥上概率分布的可靠性函数$\Omega$正定对称矩阵的特征描述了没有任何不变性条件的分布。我们还表明，指数概率分布的表征$\Omega$在不假设不变条件的情况下，无记忆属性成立。然后我们研究指数分布之间的联系$\Omega$和有界区间上的均匀分布$\Omega$. 引入矩阵Pareto分布的概念，证明该分布具有长尾性质。