运算体、运算体oids和非交换zonoids
Operatopes, Operanoids, and Noncommutative Zonoids
AI总结：
本文提出运算体和非交换zonoids的概念，通过自由概率理论研究其性质及在统计学中的应用。
摘要：
我们研究了一类称为运算体的凸体，它们是通过取仿射图像的算子范数球的Minkowski和得到的。这一概念扩展了zonotopes的概念，即线段的Minkowski和。当线段数量趋于无穷时，得到的凸体称为zonoids，也可以视为随机线段的期望。在此解释基础上，我们类比地定义运算体oids为随机仿射图像的算子范数球的期望。在研究运算体oids的性质时，当算子范数球的维度增长时，我们得到一个新的凸体极限渐进行为。这导致了更一般的凸体类称为非交换zonoids，我们利用自由概率理论的框架来说明其基本性质和例子。最后，我们讨论了运算体oids和非交换zonoids在统计学和随机过程中的应用。
英文摘要：
We study a class of convex bodies called operatopes that are obtained by taking Minkowski sums of affine images of an operator norm ball. This notion generalizes that of zonotopes which are Minkowksi sums of line segments. Taking the limit of the number of line segments to infinity yields the class of convex bodies called zonoids, which can also be viewed as the expectation of a random line segment. Expanding on this interpretation, we analogously define operanoids as the expectation of a random affine image of an operator norm ball. In studying the properties of operanoids when the dimension of the operator norm ball grows, we arrive at a new asymptotic regime for limits of convex bodies. This leads to the more general class of convex bodies called noncommutative zonoids, and we use the framework of free probability theory to illustrate basic properties and examples. Finally, we discuss applications of operanoids and noncommutative zonoids in statistics and stochastic processes.
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