The Annals of Probability

De Finetti theorems for easy quantum groups

Teodor Banica, Stephen Curran, and Roland Speicher

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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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Ann. Probab., Volume 40, Number 1 (2012), 401-435.

First available in Project Euclid: 3 January 2012

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Zentralblatt MATH identifier

Primary: 46L53: Noncommutative probability and statistics
Secondary: 46L54: Free probability and free operator algebras 60G09: Exchangeability 46L65: Quantizations, deformations

Quantum invariance Gaussian distribution Rayleigh distribution semicircle law


Banica, Teodor; Curran, Stephen; Speicher, Roland. De Finetti theorems for easy quantum groups. Ann. Probab. 40 (2012), no. 1, 401--435. doi:10.1214/10-AOP619.

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