An algebraic version of the central limit theorem

Abstract

A non-commutative analogue of the central limit theorem and the weak law of large numbers has been derived, the analogues of integrable functions being non-commutative polynomials. Without the assumption of positivity higher central limit theorems hold which have no analogy in the classical probabilistic case. The treatment includes this classical case and the convergence to so-called "quasi-free states" in the quantum mechanics of bosons [3, 4]. © 1978 Springer-Verlag.