• Bernoulli
  • Volume 20, Number 2 (2014), 697-715.

Universal Gaussian fluctuations on the discrete Poisson chaos

Giovanni Peccati and Cengbo Zheng

Full-text: Open access


We prove that homogeneous sums inside a fixed discrete Poisson chaos are universal with respect to normal approximations. This result parallels some recent findings, in a Gaussian context, by Nourdin, Peccati and Reinert (Ann. Probab. 38 (2010) 1947–1985). As a by-product of our analysis, we provide some refinements of the CLTs for random variables on the Poisson space proved by Peccati, Solé, Taqqu and Utzet (Ann. Probab. 38 (2010) 443–478) and by Peccati and Zheng (Electron. J. Probab. 15 (2010) 1487–1527).

Article information

Bernoulli, Volume 20, Number 2 (2014), 697-715.

First available in Project Euclid: 28 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Central Limit Theorems contractions discrete Poisson chaos universality


Peccati, Giovanni; Zheng, Cengbo. Universal Gaussian fluctuations on the discrete Poisson chaos. Bernoulli 20 (2014), no. 2, 697--715. doi:10.3150/12-BEJ503.

Export citation


  • [1] de Jong, P. (1989). Central Limit Theorems for Generalized Multilinear Forms. CWI Tract 61. Amsterdam: Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica.
  • [2] de Jong, P. (1990). A central limit theorem for generalized multilinear forms. J. Multivariate Anal. 34 275–289.
  • [3] Decreusefond, L., Ferraz, E. and Randriam, H. (2011). Simplicial homology of random configurations. Preprint.
  • [4] Dudley, R.M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge: Cambridge Univ. Press. Revised reprint of the 1989 original.
  • [5] Ferraz, E. and Vergne, A. (2011). Statistics of geometric random simplicial complexes. Preprint.
  • [6] Kabanov, J.M. (1975). Extended stochastic integrals. Teor. Verojatnost. i Primenen. 20 725–737.
  • [7] Ledoux, M. and Talagrand, M. (1990). Probability on Banach Spaces. Berlin: Springer.
  • [8] 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.
  • [9] Nourdin, I. and Peccati, G. (2010). Universal Gaussian fluctuations of non-Hermitian matrix ensembles: From weak convergence to almost sure CLTs. ALEA Lat. Am. J. Probab. Math. Stat. 7 341–375.
  • [10] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus. Cambridge Tracts in Mathematics 192. Cambridge: Cambridge Univ. Press.
  • [11] Nourdin, I., Peccati, G. and Reinert, G. (2010). Stein’s method and stochastic analysis of Rademacher functionals. Electron. J. Probab. 15 1703–1742.
  • [12] Nourdin, I., Peccati, G. and Reinert, G. (2010). Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab. 38 1947–1985.
  • [13] Nourdin, I., Peccati, G. and Réveillac, A. (2010). Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat. 46 45–58.
  • [14] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Berlin: Springer.
  • [15] Nualart, D. and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614–628.
  • [16] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.
  • [17] Nualart, D. and Vives, J. (1990). Anticipative calculus for the Poisson process based on the Fock space. In Séminaire de Probabilités, XXIV, 1988/89. Lecture Notes in Math. 1426 154–165. Berlin: Springer.
  • [18] Peccati, G. and Lachièze-Rey, R. (2011). Fine Gaussian fluctuations on the Poisson space: Cumulants contractions and random graphs. Preprint.
  • [19] Peccati, G., Solé, J.L., Taqqu, M.S. and Utzet, F. (2010). Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38 443–478.
  • [20] Peccati, G. and Taqqu, M.S. (2008). Central limit theorems for double Poisson integrals. Bernoulli 14 791–821.
  • [21] Peccati, G. and Taqqu, M.S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi & Springer Series 1. Milan: Springer.
  • [22] Peccati, G. and Tudor, C.A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 247–262. Berlin: Springer.
  • [23] Peccati, G. and Zheng, C. (2010). Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15 1487–1527.
  • [24] Privault, N. (2009). Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Lecture Notes in Math. 1982. Berlin: Springer.
  • [25] Reitzner, M. and Schulte, M. (2013). Central limit theorems for U-statistics of Poisson point processes. Ann. Probab. 41 3879–3909.
  • [26] Schulte, M. and Thaele, C. (2010). Exact and asymptotic results for intrinsic volumes of Poisson k-flat processes. Preprint.
  • [27] Surgailis, D. (1984). On multiple Poisson stochastic integrals and associated Markov semigroups. Probab. Math. Statist. 3 217–239.