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Invariance principles for homogeneous sums of free random variables. (English) Zbl 1292.60041

Summary: We extend, in the free probability framework, an invariance principle for multilinear homogeneous sums with low influences recently established by E. Mossel et al. [Ann. Math. (2) 171, No. 1, 295–341 (2010; Zbl 1201.60031)]. We then deduce several universality phenomenons, in the spirit of the paper of I. Nourdin et al. [Ann. Probab. 38, No. 5, 1947–1985 (2010; Zbl 1246.60039)].

MSC:

60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
46L54 Free probability and free operator algebras

References:

[1] Biane, P. and Speicher, R. (1998). Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Related Fields 112 373-409. · Zbl 0919.60056 · doi:10.1007/s004400050194
[2] Deya, A. and Nourdin, I. (2012). Convergence of Wigner integrals to the tetilla law. ALEA Lat. Am. J. Probab. Math. Stat. 9 101-127. · Zbl 1285.46053
[3] Kargin, V. (2007). A proof of a non-commutative central limit theorem by the Lindeberg method. Electron. Commun. Probab. 12 36-50 (electronic). · Zbl 1133.46037 · doi:10.1214/ECP.v12-1250
[4] Kemp, T., Nourdin, I., Peccati, G. and Speicher, R. (2012). Wigner chaos and the fourth moment. Ann. Probab. 40 1577-1635. · Zbl 1277.46033 · doi:10.1214/11-AOP657
[5] Kemp, T. and Speicher, R. (2007). Strong Haagerup inequalities for free \(\mathscr{R}\)-diagonal elements. J. Funct. Anal. 251 141-173. · Zbl 1128.46024 · doi:10.1016/j.jfa.2007.03.011
[6] Lindeberg, J.W. (1922). Eine neue Herleitung des exponential-Gesetzes in der Warscheinlichkeitsrechnung. Math. Z. 15 211-235. · JFM 48.0602.04 · doi:10.1007/BF01494395
[7] .1 Mossel, E., O’Donnell, R. and Oleszkiewicz, K. (2010). Noise stability of functions with low influences: Invariance and optimality. Ann. of Math. (2) 171 295-341. · Zbl 1201.60031 · doi:10.4007/annals.2010.171.295
[8] Nica, A. and Speicher, R. (1998). Commutators of free random variables. Duke Math. J. 92 553-592. · Zbl 0968.46053 · doi:10.1215/S0012-7094-98-09216-X
[9] Nica, A. and Speicher, R. (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335 . Cambridge: Cambridge Univ. Press. · Zbl 1133.60003
[10] Nourdin, I. (2011). Yet another proof of the Nualart-Peccati criterion. Electron. Commun. Probab. 16 467-481. · Zbl 1253.60068 · doi:10.1214/ECP.v16-1642
[11] Nourdin, I., Peccati, G. and Reinert, G. (2010). Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab. 38 1947-1985. · Zbl 1246.60039 · doi:10.1214/10-AOP531
[12] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177-193. · Zbl 1097.60007 · doi:10.1214/009117904000000621
[13] Peccati, G. and Zheng, C. (2014). Universal Gaussian fluctuations on the discrete Poisson chaos. Bernoulli . · Zbl 1302.60059
[14] Voiculescu, D. (1985). Symmetries of some reduced free product \(C^{\ast}\)-algebras. In Operator Algebras and Their Connections with Topology and Ergodic Theory ( Buşteni , 1983). Lecture Notes in Math. 1132 556-588. Berlin: Springer. · Zbl 0618.46048 · doi:10.1007/BFb0074909
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