R-Transforms of Free Joint Distributions and Non-crossing Partitions
R-Transforms of Free Joint Distributions and Non-crossing Partitions
复制标题
自由联合分布和非交叉分区的 R 变换
DOI：
10.1006/jfan.1996.0011
复制
发表时间：
1996
影响因子：
1.7
通讯作者：
A. Nica
中科院分区：
文献类型：
--
作者：
A. Nica
本文对D. Voiculescu，（自由）联合分布，即，多元非交换多项式代数上的正规线性泛函。指出R变换对于联合分布的自由积和自由卷积运算具有良好的性能。我们证明了联合分布的逆R变换的显式计算是通过非交叉划分格上的求和公式来完成的，这表明R变换是R的组合方法中考虑的“自由累积量”的算子理论对应物。斯派克此外，在连接的多维R -变换下的旋转的等方差，我们表明，一个自然的自由模拟的经典结果旋转的独立随机变量是持有。
Abstract We introduce an extension of the notion of R-transform , defined by D. Voiculescu, to ( free ) joint distributions , i.e., normalized linear functionals on algebras of non-commutative polynomials in several indeterminates. We point out that the R -transform has good behavior with respect to the operations of free product and free convolution of joint distributions. We prove that the explicit computation of the inverse- R -transform of a joint distribution is done via a formula of summation over the lattice of non-crossing partitions, which shows that the R -transform is the operator-theoretic counterpart of the “free cumulants” considered in the combinatorial approach of R. Speicher. Moreover, in connection to the equivariance of the multidimensional R -transform under rotations, we show that a natural free analogue of a classical result about rotations of independent random variables is holding.