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