The Annals of Probability

Wigner chaos and the fourth moment

Todd Kemp, Ivan Nourdin, Giovanni Peccati, and Roland Speicher

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We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer–Major theorem.

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Ann. Probab., Volume 40, Number 4 (2012), 1577-1635.

First available in Project Euclid: 4 July 2012

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Primary: 46L54: Free probability and free operator algebras 60H07: Stochastic calculus of variations and the Malliavin calculus 60H30: Applications of stochastic analysis (to PDE, etc.)

Free probability Wigner chaos central limit theorem Malliavin calculus


Kemp, Todd; Nourdin, Ivan; Peccati, Giovanni; Speicher, Roland. Wigner chaos and the fourth moment. Ann. Probab. 40 (2012), no. 4, 1577--1635. doi:10.1214/11-AOP657.

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  • [1] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
  • [2] Anshelevich, M. (2000). Free stochastic measures via noncrossing partitions. Adv. Math. 155 154–179.
  • [3] Anshelevich, M. (2001). Partition-dependent stochastic measures and $q$-deformed cumulants. Doc. Math. 6 343–384 (electronic).
  • [4] Anshelevich, M. (2002). Free stochastic measures via noncrossing partitions. II. Pacific J. Math. 207 13–30.
  • [5] Biane, P. (1997). Free hypercontractivity. Comm. Math. Phys. 184 457–474.
  • [6] Biane, P. (1998). Processes with free increments. Math. Z. 227 143–174.
  • [7] Biane, P., Capitaine, M. and Guionnet, A. (2003). Large deviation bounds for matrix Brownian motion. Invent. Math. 152 433–459.
  • [8] 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.
  • [9] Biane, P. and Speicher, R. (2001). Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincaré Probab. Statist. 37 581–606.
  • [10] Biane, P. and Voiculescu, D. (2001). A free probability analogue of the Wasserstein metric on the trace-state space. Geom. Funct. Anal. 11 1125–1138.
  • [11] Capitaine, M. and Donati-Martin, C. (2005). Free Wishart processes. J. Theoret. Probab. 18 413–438.
  • [12] Demni, N. (2008). Free Jacobi process. J. Theoret. Probab. 21 118–143.
  • [13] Gao, M. (2006). Free Ornstein–Uhlenbeck processes. J. Math. Anal. Appl. 322 177–192.
  • [14] Guionnet, A. and Shlyakhtenko, D. (2009). Free diffusions and matrix models with strictly convex interaction. Geom. Funct. Anal. 18 1875–1916.
  • [15] Haagerup, U. (1978/79). An example of a nonnuclear $C^{\ast}$-algebra, which has the metric approximation property. Invent. Math. 50 279–293.
  • [16] Hiai, F. and Petz, D. (2000). The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs 77. Amer. Math. Soc., Providence, RI.
  • [17] Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129. Cambridge Univ. Press, Cambridge.
  • [18] Kümmerer, B. and Speicher, R. (1992). Stochastic integration on the Cuntz algebra $O_\infty $. J. Funct. Anal. 103 372–408.
  • [19] Kuo, H.-H. (2006). Introduction to Stochastic Integration. Springer, New York.
  • [20] Major, P. (1981). Multiple Wiener–Itô Integrals: With Applications to Limit Theorems. Lecture Notes in Math. 849. Springer, Berlin.
  • [21] Nica, A. and Speicher, R. (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge.
  • [22] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
  • [23] Nourdin, I. and Peccati, G. (2010). Stein’s method meets Malliavin calculus: A short survey with new estimates. Interdiscip. Math. Sci. 8 207–236.
  • [24] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Univ. Press, Cambridge.
  • [25] Nourdin, I., Peccati, G. and Podolskij, M. (2011). Quantitative Breuer–Major theorems. Stochastic Process. Appl. 121 793–812.
  • [26] Nourdin, I., Peccati, G. and Reinert, G. (2010). Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab. 38 1947–1985.
  • [27] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [28] Nualart, D. and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614–628.
  • [29] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.
  • [30] Parthasarathy, K. R. (1992). An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics 85. Birkhäuser, Basel.
  • [31] Peccati, G. and Taqqu, M. S. (2010). Wiener Chaos: Moments, Cumulants and Diagrams. A Survey with Computer Implementation. Bocconi Univ. Press, Milano.
  • [32] Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw-Hill, New York.
  • [33] Rüschendorf, L. (2001). Wasserstein metric. In Encyclopaedia of Mathematics (M. Hazewinkel, ed.). Springer, Berlin.
  • [34] Surgailis, D. (2003). CLTs for polynomials of linear sequences: Diagram formula with illustrations. In Theory and Applications of Long-range Dependence 111–127. Birkhäuser, Boston, MA.
  • [35] Takesaki, M. (1972). Conditional expectations in von Neumann algebras. J. Funct. Anal. 9 306–321.
  • [36] Talagrand, M. (2003). Mean field models for spin glasses: A first course. In Lectures on Probability Theory and Statistics (Saint-Flour, 2000). Lecture Notes in Math. 1816 181–285. Springer, Berlin.
  • [37] 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. Springer, Berlin.
  • [38] Voiculescu, D. (1991). Limit laws for random matrices and free products. Invent. Math. 104 201–220.
  • [39] Voiculescu, D. (1993). The analogues of entropy and of Fisher’s information measure in free probability theory. I. Comm. Math. Phys. 155 71–92.
  • [40] Voiculescu, D. (2000). The coalgebra of the free difference quotient and free probability. Int. Math. Res. Not. 2 79–106.
  • [41] Voiculescu, D. V., Dykema, K. J. and Nica, A. (1992). Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups. CRM Monograph Series 1. Amer. Math. Soc., Providence, RI.