A non-commutative central limit theorem. (English) Zbl 0724.60023

Speicher, Roland

Math. Z. 209, No. 1, 55-66 (1992).

The recently investigated relations \[ c(f)c^*(g)-\mu c^*(g)c(f)=<f,g>1\quad (f,g\in L^ 2({\mathbb{R}})) \] give for \(\mu\in [- 1,1]\) rise to interpolations between the bosonic and fermionic Gaussian distributions and Brownian motions. Here we show how these generalized Gaussian distributions and Brownian motions arise from central limit theorems and invariance principles, respectively. This yields also a stochastic interpretation for the distribution function of the orthogonal continuous q-Hermite polynomials.

Reviewer: R.Speicher

MSC:

60F05	Central limit and other weak theorems
46L51	Noncommutative measure and integration
46L53	Noncommutative probability and statistics
46L54	Free probability and free operator algebras

Keywords:

generalized Gaussian distribution; Brownian motions; central limit theorems; q-Hermite polynomials

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