Open Access
January 2012 De Finetti theorems for easy quantum groups
Teodor Banica, Stephen Curran, Roland Speicher
Ann. Probab. 40(1): 401-435 (January 2012). DOI: 10.1214/10-AOP619


We study sequences of noncommutative random variables which are invariant under “quantum transformations” coming from an orthogonal quantum group satisfying the “easiness” condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups Sn, On, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of Köstler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.


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Teodor Banica. Stephen Curran. Roland Speicher. "De Finetti theorems for easy quantum groups." Ann. Probab. 40 (1) 401 - 435, January 2012.


Published: January 2012
First available in Project Euclid: 3 January 2012

zbMATH: 1242.46073
MathSciNet: MR2917777
Digital Object Identifier: 10.1214/10-AOP619

Primary: 46L53
Secondary: 46L54 , 46L65 , 60G09

Keywords: Gaussian distribution , Quantum invariance , Rayleigh distribution , semicircle law

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • January 2012
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