## Electronic Communications in Probability

### Vector-valued semicircular limits on the free Poisson chaos

Solesne Bourguin

#### Abstract

In this note, we prove a multidimensional counterpart of the central limit theorem on the free Poisson chaos recently proved by Bourguin and Peccati (2014). A noteworthy property of convergence toward the semicircular distribution on the free Poisson chaos is obtained as part of the limit theorem: component-wise convergence of sequences of vectors of multiple integrals with respect to a free Poisson random measure toward the semicircular distribution implies joint convergence. This result complements similar findings for the Wiener chaos by Peccati and Tudor (2005), the classical Poisson chaos by Peccati and Zheng (2010) and the Wigner chaos by Nourdin, Peccati and Speicher (2013).

#### Article information

Source
Electron. Commun. Probab. Volume 21 (2016), paper no. 55, 11 pp.

Dates
Accepted: 1 August 2016
First available in Project Euclid: 2 September 2016

http://projecteuclid.org/euclid.ecp/1472830237

Digital Object Identifier
doi:10.1214/16-ECP12

Zentralblatt MATH identifier
06635857

#### Citation

Bourguin, Solesne. Vector-valued semicircular limits on the free Poisson chaos. Electron. Commun. Probab. 21 (2016), paper no. 55, 11 pp. doi:10.1214/16-ECP12. http://projecteuclid.org/euclid.ecp/1472830237.

#### References

• [1] Bourguin, S., Durastanti, C., Marinucci, D. and Peccati, G.: Gaussian approximation of nonlinear statistics on the sphere. J. Math. Anal. Appl. 436, (2016), 1121–1148.
• [2] Bourguin, S. and Peccati, G.: Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering. Electron. J. Probab. 19, (2014), 42 pp.
• [3] Bourguin, S. and Peccati, G.: Semicircular limits on the free Poisson chaos: Counterexamples to a transfer principle. J. Funct. Anal. 267, (2014), 963–997.
• [4] Kemp, T., Nourdin, I., Peccati, G. and Speicher, R.: Wigner chaos and the fourth moment. Ann. Probab. 40, (2012), 1577–1635.
• [5] Last, G., Penrose, M., Schulte, M. and Thäle, C.: Moments and central limit theorems for some multivariate Poisson functionals. Adv. in Appl. Probab. 46, (2014), 348–364.
• [6] Nica, A. and Speicher, R.: Lectures on the combinatorics of free probability. London Mathematical Society Lecture Note Series, 335. Cambridge University Press, Cambridge, 2006. xvi+417 pp.
• [7] Nourdin, I., Peccati, G. and Speicher, R.: Multi-dimensional semicircular limits on the free Wigner chaos. Seminar on Stochastic Analysis, Random Fields and Applications VII, Progr. Probab. 67, (2013), 211–221.
• [8] Nualart, D. and Peccati, G: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33, (2005), 177–193.
• [9] Peccati, G. and Tudor, C.A.: Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII, Lecture Notes in Math., 1857, Springer, Berlin, (2005), 247–262.
• [10] Peccati, G. and Zheng, C.: Multidimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15, (2010), 1487–1527.
• [11] Reitzner, M. and Schulte, M.: Central limit theorems for $U$-statistics of Poisson point processes. Ann. Probab. 41, (2013), 3879–3909.
• [12] Schulte, M.: A central limit theorem for the Poisson-Voronoi approximation. Adv. in Appl. Math. 49, (2012), 285–306.
• [13] Schulte, M. and Thäle, C.: The scaling limit of Poisson-driven order statistics with applications in geometric probability. Stochastic Process. Appl. 122, (2012), 4096–4120.
• [14] Schulte, M. and Thäle, C.: Distances between Poisson $k$-flats. Methodol. Comput. Appl. Probab. 16, (2014), 311–329.