## The fourth moment theorem on the Poisson space.(English)Zbl 1431.60022

Summary: We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result – that has been elusive for several years – shows that the so-called ‘fourth moment phenomenon’, first discovered by D. Nualart and the second author [Ann. Probab. 33, No. 1, 177–193 (2005; Zbl 1097.60007)] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein’s method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by M. Ledoux [Ann. Probab. 40, No. 6, 2439–2459 (2012; Zbl 1266.60042)] and E. Azmoodeh et al. [J. Funct. Anal. 266, No. 4, 2341–2359 (2014; Zbl 1292.60078)]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.

### MSC:

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

### Citations:

Zbl 1097.60007; Zbl 1266.60042; Zbl 1292.60078
Full Text:

### References:

 [1] Azmoodeh, E., Campese, S. and Poly, G. (2014). Fourth Moment Theorems for Markov diffusion generators. J. Funct. Anal.266 2341-2359. · Zbl 1292.60078 [2] Azmoodeh, E., Malicet, D., Mijoule, G. and Poly, G. (2016). Generalization of the Nualart-Peccati criterion. Ann. Probab.44 924-954. · Zbl 1342.60026 [3] Bachmann, S. and Peccati, G. (2016). Concentration bounds for geometric Poisson functionals: Logarithmic Sobolev inequalities revisited. Electron. J. Probab.21 no. 6, 21. · Zbl 1337.60011 [4] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Cham. · Zbl 1376.60002 [5] Bouleau, N. and Denis, L. (2015). Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes. Springer, Cham. · Zbl 1369.60002 [6] Bourguin, S. and Peccati, G. (2014). Semicircular limits on the free Poisson chaos: Counterexamples to a transfer principle. J. Funct. Anal.267 963-997. · Zbl 1308.46071 [7] Bourguin, S. and Peccati, G. (2016). The Malliavin-Stein method on the Poisson space. In Stochastic Analysis for Poisson Point Processes. Bocconi Springer Ser.7 185-228. Bocconi Univ. Press. [8] Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg. · Zbl 1213.62027 [9] Chen, L. H. Y. and Poly, G. (2015). Stein’s method, Malliavin calculus, Dirichlet forms and the fourth moment theorem. In Festschrift Masatoshi Fukushima. Interdiscip. Math. Sci.17 107-130. World Sci. Publ., Hackensack, NJ. · Zbl 1343.60012 [10] de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields75 261-277. · Zbl 0596.60022 [11] de Jong, P. (1989). Central Limit Theorems for Generalized Multilinear Forms. CWI Tract61. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam. [12] de Jong, P. (1990). A central limit theorem for generalized multilinear forms. J. Multivariate Anal.34 275-289. · Zbl 0709.60019 [13] Döbler, C. and Krokowski, K. (2017). On the fourth moment condition for Rademacher chaos. Ann. Inst. Henri Poincaré Probab. Stat. To appear. arXiv:1706.00751. [14] Döbler, C. and Peccati, G. (2017a). Quantiative de Jong theorems in any dimension. Electron. J. Probab.22 1-35. [15] Döbler, C. and Peccati, G. (2017b). The Gamma Stein equation and non-central de Jong theorems. Bernoulli. To appear. arXiv:1612.02279. [16] Döbler, C., Vidotto, A. and Zheng, G. (2017). Fourth moment theorems on the Poisson space in any dimension. arXiv:1707.01889. [17] Eichelsbacher, P. and Thäle, C. (2014). New Berry-Esseen bounds for non-linear functionals of Poisson random measures. Electron. J. Probab.19 no. 102, 25. · Zbl 1307.60066 [18] Fissler, T. and Thäle, C. (2016). A four moments theorem for gamma limits on a Poisson chaos. ALEA Lat. Am. J. Probab. Math. Stat.13 163-192. · Zbl 1342.60028 [19] Lachièze-Rey, R. and Peccati, G. (2013). Fine Gaussian fluctuations on the Poisson space, I: Contractions, cumulants and geometric random graphs. Electron. J. Probab.18 no. 32, 32. · Zbl 1295.60015 [20] Lachièze-Rey, R. and Reitzner, M. (2016). $$U$$-statistics in stochastic geometry. In Stochastic Analysis for Poisson Point Processes (G. Peccati and M. Reitzner, eds.). 229-253. Bocconi Univ. Press. [21] Lachièze-Rey, R., Schulte, M. and Yukich, J. (2016). Normal approximation for sums of stabilizing functionals. Preprint. arXiv:1702.00726. [22] Last, G. (2016). Stochastic analysis for Poisson processes. In Stochastic Analysis for Poisson Point Processes. Bocconi Springer Ser.7 1-36. Bocconi Univ. Press. [23] Last, G., Peccati, G. and Schulte, M. (2016). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Related Fields165 667-723. · Zbl 1347.60012 [24] Last, G. and Penrose, M. D. (2011). Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Related Fields150 663-690. · Zbl 1233.60026 [25] Last, G. and Penrose, M. (2017). Lectures on the Poisson Process. Cambridge Univ. Press, Cambridge. · Zbl 1392.60004 [26] Ledoux, M. (2012). Chaos of a Markov operator and the fourth moment condition. Ann. Probab.40 2439-2459. · Zbl 1266.60042 [27] Ledoux, M., Nourdin, I. and Peccati, G. (2015). Stein’s method, logarithmic Sobolev and transport inequalities. Geom. Funct. Anal.25 256-306. · Zbl 1350.60013 [28] Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebiete9 36-58. · Zbl 0164.46601 [29] Nourdin, I. and Peccati, G. (2009a). Stein’s method on Wiener chaos. Probab. Theory Related Fields145 75-118. · Zbl 1175.60053 [30] Nourdin, I. and Peccati, G. (2009b). Noncentral convergence of multiple integrals. Ann. Probab.37 1412-1426. · Zbl 1171.60323 [31] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics192. Cambridge Univ. Press, Cambridge. · Zbl 1266.60001 [32] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab.33 177-193. · Zbl 1097.60007 [33] Peccati, G. and Reitzner, M. (2016). Stochastic Analysis for Poisson Point Processes. Springer, Cham. · Zbl 1350.60005 [34] Peccati, G. and Taqqu, M. S. (2008). Central limit theorems for double Poisson integrals. Bernoulli14 791-821. · Zbl 1165.60014 [35] Peccati, G. and Thäle, C. (2013). Gamma limits and $$U$$-statistics on the Poisson space. ALEA Lat. Am. J. Probab. Math. Stat.10 525-560. · Zbl 1277.60052 [36] Peccati, G. and Zheng, C. (2010). Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab.15 1487-1527. · Zbl 1228.60031 [37] Peccati, G. and Zheng, C. (2014). Universal Gaussian fluctuations on the discrete Poisson chaos. Bernoulli20 697-715. · Zbl 1302.60059 [38] 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. · Zbl 1195.60037 [39] Reitzner, M. and Schulte, M. (2013). Central limit theorems for U-statistics of Poisson point processes. Ann. Probab.41 3879-3909. · Zbl 1293.60061 [40] Schulte, M. (2016). Normal approximation of Poisson functionals in Kolmogorov distance. J. Theoret. Probab.29 96-117. · Zbl 1335.60027 [41] Schulte, M. and Thäle, C. (2016). Poisson point process convergence and extreme values in stochastic geometry. In Stochastic Analysis for Poisson Point Processes. Bocconi Springer Ser.7 255-294.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.