Electronic Journal of Probability

Fourth moment theorems on the Poisson space in any dimension

Christian Döbler, Anna Vidotto, and Guangqu Zheng

Full-text: Open access

Abstract

We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. Döbler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case.

Finally, a transfer principle “from-Poisson-to-Gaussian” is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 36, 27 pp.

Dates
Received: 28 September 2017
Accepted: 13 April 2018
First available in Project Euclid: 3 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1525312960

Digital Object Identifier
doi:10.1214/18-EJP168

Zentralblatt MATH identifier
1387.60045

Subjects
Primary: 60F05: Central limit and other weak theorems 60H07: Stochastic calculus of variations and the Malliavin calculus 60H05: Stochastic integrals

Keywords
multivariate Poisson functionals multiple Wiener-Itô integrals fourth moment theorems carré du champ operator Gaussian approximation Stein’s method exchangeable pairs Peccati-Tudor theorem

Rights
Creative Commons Attribution 4.0 International License.

Citation

Döbler, Christian; Vidotto, Anna; Zheng, Guangqu. Fourth moment theorems on the Poisson space in any dimension. Electron. J. Probab. 23 (2018), paper no. 36, 27 pp. doi:10.1214/18-EJP168. https://projecteuclid.org/euclid.ejp/1525312960


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References

  • [1] E. Azmoodeh, S. Campese, and G. Poly. Fourth Moment Theorems for Markov diffusion generators. J. Funct. Anal., 266(4):2341–2359, 2014.
  • [2] S. Bai and M. S. Taqqu. The universality of homogeneous polynomial forms and critical limits. J. Theoret. Probab., 29(4):1710–1727, 2016.
  • [3] N. Bouleau and L. Denis. Dirichlet forms methods for Poisson point measures and Lévy processes, With emphasis on the creation-annihilation techniques, volume 76 of Probability Theory and Stochastic Modelling. Springer, 2015.
  • [4] S. Bourguin and G. Peccati. Semicircular limits on the free poisson chaos: counterexamples to a transfer principle. J. Funct. Anal., 267(4):963–997, 2014.
  • [5] S. Campese, I. Nourdin, G. Peccati, and G. Poly. Multivariate Gaussian approximations on Markov chaoses. Electron. Commun. Probab., 21:Paper No. 48, 9, 2016.
  • [6] S. Chatterjee, J. Fulman, and A. Röllin. Exponential approximation by Stein’s method and spectral graph theory. ALEA Lat. Am. J. Probab. Math. Stat., 8:197–223, 2011.
  • [7] S. Chatterjee and E. Meckes. Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat., 4:257–283, 2008.
  • [8] S. Chatterjee and Q.-M. Shao. Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie-Weiss model. Ann. Appl. Probab., 21(2):464–483, 2011.
  • [9] P. Diaconis. The distribution of leading digits and uniform distribution $\rm mod$ $1$. Ann. Probability, 5(1):72–81, 1977.
  • [10] C. Döbler. Stein’s method of exchangeable pairs for the Beta distribution and generalizations. Electron. J. Probab., 20:no. 109, 1–34, 2015.
  • [11] C. Döbler and K. Krokowski. On the fourth moment condition for Rademacher chaos. arXiv:1706.00751, to appear in: Ann. Inst. Henri Poincaré Probab. Stat., 2017.
  • [12] C. Döbler and G. Peccati. Quantiative de Jong theorems in any dimension. Electron. J. Probab., 22:no. 2, 1–35, 2017.
  • [13] C. Döbler and G. Peccati. The fourth moment theorem on the Poisson space. arXiv:1701.03120, to appear in: Ann. Probab., 2017.
  • [14] C. Döbler and M. Stolz. Stein’s method and the multivariate CLT for traces of powers on the classical compact groups. Electron. J. Probab., 16:no. 86, 2375–2405, 2011.
  • [15] P. Eichelsbacher and M. Löwe. Stein’s method for dependent random variables occurring in statistical mechanics. Electron. J. Probab., 15:no. 30, 962–988, 2010.
  • [16] P. Eichelsbacher and C. Thäle. New Berry-Esseen bounds for non-linear functionals of Poisson random measures. Electron. J. Probab., 19:no. 102, 25, 2014.
  • [17] K. Krokowski, A. Reichenbachs, and C. Thäle. Berry-Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences. Ann. Inst. Henri Poincaré Probab. Stat., 52(2):763–803, 2016.
  • [18] K. Krokowski, A. Reichenbachs, and C. Thäle. Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation. Ann. Probab., 45(2):1071–1109, 2017.
  • [19] G. Last. Stochastic analysis for Poisson processes. In G. Peccati and M. Reitzner, editors, Stochastic analysis for Poisson point processes, Mathematics, Statistics, Finance and Economics, chapter 1, pages 1–36. Bocconi University Press and Springer, 2016.
  • [20] G. Last, G. Peccati, and M. Schulte. Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Related Fields, 165(3-4):667–723, 2016.
  • [21] G. Last and M. Penrose. Lectures on the Poisson Process. IMS Textbooks. Cambridge University Press, Cambridge, 2017.
  • [22] M. Ledoux. Chaos of a Markov operator and the fourth moment condition. Ann. Probab., 40(6):2439–2459, 2012.
  • [23] E. Meckes. An infinitesimal version of stein’s method of exchangeable pairs. Ph.d Dissertation, Stanford University, 2006.
  • [24] E. Meckes. On Stein’s method for multivariate normal approximation. In High dimensional probability V: the Luminy volume, volume 5 of Inst. Math. Stat. Collect., pages 153–178. Inst. Math. Statist., Beachwood, OH, 2009.
  • [25] S. Noreddine and I. Nourdin. On the Gaussian approximation of vector-valued multiple integrals. J. Multivariate Anal., 102(6):1008–1017, 2011.
  • [26] I Nourdin and G. Peccati. Stein’s method on wiener chaos. Probab. Theory Related Fields, 145(1):75–118, 2009.
  • [27] I. Nourdin and G. Peccati. Normal approximations with Malliavin calculus, volume 192 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2012. From Stein’s method to universality.
  • [28] I. Nourdin, G. Peccati, G. Poly, and R. Simone. Classical and free fourth moment theorems: universality and thresholds. J. Theoret. Probab., 29(2):653–680, 2016.
  • [29] I. Nourdin, G. Peccati, and G. Reinert. Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann. Probab., 38(5):1947–1985, 2010.
  • [30] I. Nourdin, G. Peccati, and G. Reinert. Stein’s method and stochastic analysis of Rademacher functionals. Electron. J. Probab., 15:no. 55, 1703–1742, 2010.
  • [31] I. Nourdin and J. Rosiński. Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws. Ann. Probab., 42(2):497–526, 2014.
  • [32] I. Nourdin and G. Zheng. Exchangeable pairs on Wiener chaos. arXiv:1704.02164, 2017.
  • [33] D. Nualart and G. Peccati. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab., 33(1):177–193, 2005.
  • [34] G. Peccati, J. L. Solé, M. S. Taqqu, and F. Utzet. Stein’s method and normal approximation of Poisson functionals. Ann. Probab., 38(2):443–478, 2010.
  • [35] G. Peccati and C. A. Tudor. Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII, volume 1857 of Lecture Notes in Math., pages 247–262. Springer, Berlin, 2005.
  • [36] G. Peccati and C. Zheng. Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab., 15:no. 48, 1487–1527, 2010.
  • [37] G. Peccati and C. Zheng. Universal Gaussian fluctuations on the discrete Poisson chaos. Bernoulli, 20(2):697–715, 2014.
  • [38] G. Reinert and A. Röllin. Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab., 37(6):2150–2173, 2009.
  • [39] Y. Rinott and V. Rotar. On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted $U$-statistics. Ann. Appl. Probab., 7(4):1080–1105, 1997.
  • [40] A. Röllin. A note on the exchangeability condition in Stein’s method. Statist. Probab. Lett., 78(13):1800–1806, 2008.
  • [41] M. Schulte. Normal approximation of Poisson functionals in Kolmogorov distance. J. Theoret. Probab., 29(1):96–117, 2016.
  • [42] Q.-M. Shao and Z.-G. Su. The Berry-Esseen bound for character ratios. Proc. Amer. Math. Soc., 134(7):2153–2159 (electronic), 2006.
  • [43] Ch. Stein. Approximate computation of expectations. In Institute of Mathematical Statistics Lecture Notes - Monograph Series, volume 7. Institute of Mathematical Statistics, 1986.
  • [44] G. Zheng. Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals. Stochastic Process. Appl., 127(5):1622–1636, 2017.