论文标题
在运营商价值无限的布尔和单调独立性上
On operator-valued infinitesimal Boolean and monotone independence
论文作者
论文摘要
我们介绍了布尔和单调病例的操作员价值无限（OVI）独立性的概念。然后证明OVI Boolean（分别单调）独立性等于操作员价值布尔（分别单调）独立性，超过了$ 2 \ times 2 $上三角矩阵的代数。此外，我们得出公式以将其减少到操作员值情况下，以获取OVI Boolean（分别单调）添加剂卷积。 我们还定义了OVI布尔和单调累积剂，并研究其基本特性。此外，对于OVI独立性的每个概念，我们构建了相应的OVI中心极限定理。自由，布尔和单调累积物之间的关系扩展到了这种环境。此外，在布尔案例中，我们推断出混合累积物的消失仍然等同于独立性，并使用它与矩阵值的无限无限布尔独立性连接标量。最终，我们研究了两个随机矩阵模型，它们在渐近散布的独立性，但事实并非无限散布。
We introduce the notion of operator-valued infinitesimal (OVI) independence for the Boolean and monotone cases. Then show that OVI Boolean (resp. monotone) independence is equivalent to the operator-valued Boolean (resp. monotone) independence over an algebra of $2\times 2$ upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (resp. monotone) additive convolution by reducing it to the operator-valued case. We also define OVI Boolean and monotone cumulants and study its basic properties. Moreover, for each notion of OVI independence, we construct the corresponding OVI Central Limit Theorem. The relations among free, Boolean and monotone cumulants are extended to this setting. Besides, in the Boolean case we deduce that the vanishing of mixed cumulants is still equivalent to independence, and use this to connect scalar-valued with matrix-valued infinitesimal Boolean independence. Finally we study two random matrix models that are asymptotically Boolean independent but turn out to not be infinitesimally Boolean independent.
