CONSTRUCTIVE UNIVERSAL CENTRAL LIMIT THEOREMS BASED ON INTERACTING FOCK SPACES

Abstract

Cabana-Duvillard and lonescu11 have proved that any symmetric probability measure with moments of any order can be obtained as central limit theorem of self-adjoint, weakly independent and symmetrically distributed (in a quantum souse) random variables. Results of this type will be called "universal central limit theorem". Using Interacting Fock Space (IFS) techniques we extend this result in two directions: (i) we prove that the random variables can be taken to be generalized Gaussian in the sense of Accardi and Bożejko3 and we give a realization of such random variables as sums of creation, annihilation and preservation operators acting on an appropriate IFS; (ii) we extend the above-mentioned result to the nonsymmetric case. The nontrivial difference between the symmetric and the nonsymmetric case is explained at the end of the introduction below.