A central limit theorem for Bose Z -independent quantum random variables

Stéphane Attal, Nadine Guillotin‐Plantard, Christophe Sabot

Chul Ki Ko, Norio Konno, Etsuo Segawa +1 more

Michael Skeide

We present a central limit theorem for Bose [Formula: see text]-independent operator-valued random variables. Furthermore, we show that the central limit distribution may be represented by an algebra of creators and annihilators on a symmetric Fock module. As an example we recover the distribution of creators and annihilators on the truncated Fock space, i.e. the central limit distribution of Boolean independence.

Michael Skeide

Michael J. Hartmann, G. Mahler, Ortwin G. Hess

Luigi Accardi, Vitonofrio Crismale, Yun Gang Lu

Cabana-Duvillard and lonescu11have 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 Bożejko3and 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.