Some weak laws of large numbers in noncommutative probability
Some weak laws of large numbers in noncommutative probability
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非交换概率中大数的一些弱定律
DOI：
10.1007/pl00004356
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发表时间：
1997
期刊：
影响因子：
--
通讯作者：
V. Pata
中科院分区：
文献类型：
--
作者：
J. Lindsay;V. Pata
经典概率的随机变量可以看作是一个阿贝尔冯诺依曼代数中具有特定状态的元素。然后事件对应于代数中的（正交）投影，事件的概率由相应投影上的状态值给出。一个系统的发展概率和积分理论的非交换冯诺依曼代数开始由IE西格尔在50年代（[赛格]），但发展量子理论从这一点来看，可以追溯到J.冯诺依曼（[Neu]）。事实上，这是一个主要的动机，他的重要工作与FJ穆雷环的运营商（现称为冯诺依曼代数）。非对易概率的独立性问题是一个尚未解决的深层次问题。一方面，基本的因式分解条件（在状态上，在子代数上）已经被研究并用于建立强大数定律（[Bat]，[Jaj]）。另一方面，特定类型的独立性已用于中心极限结果（[CuH]，[Wal]，[VDN]，[Spe]）和不变性原理（[Sc 1]）。这些类型包括玻色、费米和自由独立性，它们分别与代数的张量积、分次张量积和自由积有关。基本的因式分解属性是补充（反）交换关系的元素之间的各个子代数，在玻色和费米的情况下，更广泛的因式分解在自由的情况下。M. Schürmann的分析表明，上述三种类型基本上是非交换独立性的唯一可能形式（[Sc 2]）。在经典概率中，独立随机变量的相加对应于它们各自定律的卷积。Schürmann开发了一种双代数方法来处理独立性，其中涉及状态的卷积
The random variables of classical probability may be viewed as elements affiliated to an abelian von Neumann algebra with specified state. Events then correspond to (orthogonal) projections in the algebra, and the probability of an event is given by the value of the state on the corresponding projection. A systematic development of probability and integration theory for non-abelian von Neumann algebras was begun by IE Segal in the fifties ([Seg]), but the development of quantum theory from this point of view may be traced back to J. von Neumann ([Neu]). Indeed, it was a major motivation for his important work with FJ Murray on Rings of Operators (now called von Neumann algebras). The general problem of independence in noncommutative probability is a deep one which remains unresolved. On the one hand, basic factorisation conditions (on a state, over the subalgebras) have been investigated and used for establishing strong laws of large numbers ([Bat],[Jaj]). On the other hand, specific types of independence have been used for central limit results ([CuH],[Wal],[VDN],[Spe]), and invariance principles ([Sc 1]). These types include Bose, Fermi and free independence which are, respectively, related to the tensor product, the graded tensor product and the free product of algebras. The basic factorisation property is supplemented by (anti-) commutation relations between elements of the respective sub-algebras, in the Bose and Fermi cases, and more extensive factorisation in the free case. A recent paper of M. Schürmann contains an analysis which suggests that the above three types are essentially the only possible forms of noncommutative independence ([Sc 2]). Addition of independent random variables, in classical probability, corresponds to the convolution of their respective laws. Schürmann has developed a bialgebra approach to independence which involves convolutions of states