The Annals of Probability

De Finetti theorems for easy quantum groups

Teodor Banica, Stephen Curran, and Roland Speicher

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1325605007

Digital Object Identifier
doi:10.1214/10-AOP619

Mathematical Reviews number (MathSciNet)
MR2917777

Zentralblatt MATH identifier
1242.46073

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

Keywords
Quantum invariance Gaussian distribution Rayleigh distribution semicircle law

Citation

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. https://projecteuclid.org/euclid.aop/1325605007


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References

  • [1] Aldous, D. J. (1981). Representations for partially exchangeable arrays of random variables. J. Multivariate Anal. 11 581–598.
  • [2] Banica, T. and Collins, B. (2007). Integration over compact quantum groups. Publ. Res. Inst. Math. Sci. 43 277–302.
  • [3] Banica, T. and Collins, B. (2007). Integration over quantum permutation groups. J. Funct. Anal. 242 641–657.
  • [4] Banica, T., Collins, B. and Zinn-Justin, P. (2009). Spectral analysis of the free orthogonal matrix. Int. Math. Res. Not. 17 3286–3309.
  • [5] Banica, T., Curran, S. and Speicher, R. (2010). Classification results for easy quantum groups. Pacific J. Math. 247 1–26.
  • [6] Banica, T., Curran, S. and Speicher, R. (2011). Stochastic aspects of easy quantum groups. Probab. Theory Related Fields 149 435–462.
  • [7] Banica, T. and Speicher, R. (2009). Liberation of orthogonal Lie groups. Adv. Math. 222 1461–1501.
  • [8] Banica, T. and Vergnioux, R. (2010). Invariants of the half-liberated orthogonal group. Ann. Inst. Fourier (Grenoble) 60 2137–2164.
  • [9] Bercovici, H. and Voiculescu, D. (1995). Superconvergence to the central limit and failure of the Cramér theorem for free random variables. Probab. Theory Related Fields 103 215–222.
  • [10] Birkhoff, G. (1967). Lattice Theory, 3rd ed. American Mathematical Society Colloquium Publications XXV. Amer. Math. Soc., Providence, RI.
  • [11] Collins, B. (2003). Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17 953–982.
  • [12] Collins, B. and Śniady, P. (2006). Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264 773–795.
  • [13] Curran, S. (2009). Quantum exchangeable sequences of algebras. Indiana Univ. Math. J. 58 1097–1125.
  • [14] Curran, S. (2010). Quantum rotatability. Trans. Amer. Math. Soc. 362 4831–4851.
  • [15] Curran, S. (2011). A characterization of freeness by invariance under quantum spreading. J. Reine Angew. Math. To appear.
  • [16] Curran, S. and Speicher, R. (2011). Quantum invariant families of matrices in free probability. J. Funct. Anal. 261 897–933.
  • [17] Diaconis, P. and Freedman, D. (1987). A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 23 397–423.
  • [18] Diaconis, P. and Freedman, D. A. (1988). Conditional limit theorems for exponential families and finite versions of de Finetti’s theorem. J. Theoret. Probab. 1 381–410.
  • [19] Diaconis, P. and Shahshahani, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31A 49–62.
  • [20] Freedman, D. A. (1962). Invariants under mixing which generalize de Finetti’s theorem. Ann. Math. Statist 33 916–923.
  • [21] Goldstein, M. S. and Grabarnik, G. Y. (1991). Almost sure convergence theorems in von Neumann algebras. Israel J. Math. 76 161–182.
  • [22] Goldstein, S. (1987). Norm convergence of martingales in Lp-spaces over von Neumann algebras. Rev. Roumaine Math. Pures Appl. 32 531–541.
  • [23] Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. Springer, New York.
  • [24] Köstler, C. and Speicher, R. (2009). A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation. Comm. Math. Phys. 291 473–490.
  • [25] Nica, A. and Speicher, R. (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge.
  • [26] Ryll-Nardzewski, C. (1957). On stationary sequences of random variables and the de Finetti’s equivalence. Colloq. Math. 4 149–156.
  • [27] Speicher, R. (1997). On universal products. In Free Probability Theory (Waterloo, ON, 1995). Fields Inst. Commun. 12 257–266. Amer. Math. Soc., Providence, RI.
  • [28] Speicher, R. (1998). Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Amer. Math. Soc. 132 x+88.
  • [29] Voiculescu, D. (1985). Symmetries of some reduced free product C*-algebras. In Operator Algebras and Their Connections with Topology and Ergodic Theory (BuŞteni, 1983). Lecture Notes in Math. 1132 556–588. Springer, Berlin.
  • [30] Voiculescu, D. V., Dykema, K. J. and Nica, A. (1992). Free Random Variables. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI.
  • [31] Wang, S. (1995). Free products of compact quantum groups. Comm. Math. Phys. 167 671–692.
  • [32] Wang, S. (1998). Quantum symmetry groups of finite spaces. Comm. Math. Phys. 195 195–211.
  • [33] Weingarten, D. (1978). Asymptotic behavior of group integrals in the limit of infinite rank. J. Mathematical Phys. 19 999–1001.
  • [34] Woronowicz, S. L. (1987). Compact matrix pseudogroups. Comm. Math. Phys. 111 613–665.