Noncommutative analogs of probabilistic notions and results

Abstract

Considering a random variable as a multiplication operator by a measurable function, a natural generalization consists in allowing noncommuting and unbounded operators defined on a common invariant domain with cyclic vector φ. By multiplication and addition, these operators generate a ∗-algebra which in turn can be considered as a representation π of an abstract ∗-(tensor) algebra. Moments are replaced by m( a 1 ··· a n ) = 〈 φ, π( a 1) ··· π( a n ) φ〉. In analogy to the classical case the notions of cumulants, addition of independent random variables, and infinite divisibility are introduced, as well as Gaussianness as a generalization of normal random variables. Previous results are briefly reviewed, including a characterization of infinite divisibility. Among the new results are noncommutative analogs of versions of the central limit theorem and of Cramér's theorem. All results have direct applications to representations of Lie algebras, to quantum field theory, and to statistical mechanics.