Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs

Raphael Lachieze-Rey (Université Paris Descartes)
Giovanni Peccati (Université de Luxembourg)

Abstract


We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit theorems. These conditions can always be expressed in terms of contraction operators or, equivalently, fourth cumulants. Our findings are specifically tailored to deal with the normal approximation of the geometric $U$-statistics introduced by Reitzner and Schulte (2011). In particular, we shall provide a new analytic characterization of geometric random graphs whose edge-counting statistics exhibit asymptotic Gaussian fluctuations, and describe a new form of Poisson convergence for stationary random graphs with sparse connections. In a companion paper, the above analysis is extended to general $U$-statistics of marked point processes with possibly rescaled kernels.

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Pages: 1-32

Publication Date: March 5, 2013

DOI: 10.1214/EJP.v18-2104

References

  • Baryshnikov, Yu.; Yukich, J. E. Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005), no. 1A, 213--253. MR2115042
  • Bhattacharya, Rabi N.; Ghosh, Jayanta K. A class of $U$-statistics and asymptotic normality of the number of $k$-clusters. J. Multivariate Anal. 43 (1992), no. 2, 300--330. MR1193616
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1989. xx+494 pp. ISBN: 0-521-37943-1 MR1015093
  • Blei, R.; Janson, S. Rademacher chaos: tail estimates versus limit theorems. Ark. Mat. 42 (2004), no. 1, 13--29. MR2056543
  • Clark, B. N.; Colbourn, C. J.; Johnson, David S. Unit disk graphs. Discrete Math. 86 (1990), no. 1-3, 165--177. MR1088569
  • Decreusefond, L.; Ferraz, E. On the one dimensional Poisson random geometric graph. J. Probab. Stat. 2011, Art. ID 350382, 21 pp. MR2842939
  • Ferraz E. ; Vergne A. (2011). Statistics of geometric random simplicial complexes. Preprint.
  • Heinrich, L.; Schmidt, H.; Schmidt, V. Central limit theorems for Poisson hyperplane tessellations. Ann. Appl. Probab. 16 (2006), no. 2, 919--950. MR2244437
  • Jammalamadaka, S. R.; Janson, S. Limit theorems for a triangular scheme of $U$-statistics with applications to inter-point distances. Ann. Probab. 14 (1986), no. 4, 1347--1358. MR0866355
  • Lachièze-Rey, R. Concave majorant of stochastic processes and Burgers turbulence. J. Theoret. Probab. 25 (2012), no. 2, 313--332. MR2914431 http://arxiv.org/pdf/1205.0632v2.pdf
  • Last, G.; Penrose, M. D. Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Related Fields 150 (2011), no. 3-4, 663--690. MR2824870
  • Last G., Penrose M. D., Schulte M. and Thaele C. (2012). Moments and central limit theorems for some multivariate Poisson functionals. Preprint.
  • New perspectives in stochastic geometry. Edited by Wilfrid S. Kendall and Ilya Molchanov. Oxford University Press, Oxford, 2010. xx+585 pp. ISBN: 978-0-19-923257-4 MR2668353
  • McDiarmid, C. Random channel assignment in the plane. Random Structures Algorithms 22 (2003), no. 2, 187--212. MR1954610
  • N.T. Minh . Malliavin-Stein method for multi-dimensional U-statistics of Poisson point processes (2011). Preprint.
  • Müller, T. Two-point concentration in random geometric graphs. Combinatorica 28 (2008), no. 5, 529--545. MR2501248
  • Nourdin, I.; Peccati, G. Stein's method on Wiener chaos. Probab. Theory Related Fields 145 (2009), no. 1-2, 75--118. MR2520122
  • Nourdin, I.; Peccati, G. Normal approximations with Malliavin calculus. From Stein's method to universality. Cambridge Tracts in Mathematics, 192. Cambridge University Press, Cambridge, 2012. xiv+239 pp. ISBN: 978-1-107-01777-1 MR2962301
  • Nourdin, I.; Peccati, G.; Reinert, G. Stein's method and stochastic analysis of Rademacher functionals. Electron. J. Probab. 15 (2010), no. 55, 1703--1742. MR2735379
  • Nualart, D.; Peccati, G. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005), no. 1, 177--193. MR2118863
  • Nualart, D.; Vives, J. Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités, XXIV, 1988/89, 154--165, Lecture Notes in Math., 1426, Springer, Berlin, 1990. MR1071538
  • Peccati, G.; Taqqu, M. S. Central limit theorems for double Poisson integrals. Bernoulli 14 (2008), no. 3, 791--821. MR2537812
  • Peccati, G.; Taqqu, M. S. Wiener chaos: moments, cumulants and diagrams. A survey with computer implementation. Supplementary material available online. Bocconi & Springer Series, 1. Springer, Milan; Bocconi University Press, Milan, 2011. xiv+274 pp. ISBN: 978-88-470-1678-1 MR2791919
  • Peccati G. (2011). The Chen-Stein method for Poisson functionals. Preprint.
  • Peccati, G.; Solé, J. L.; Taqqu, M. S.; Utzet, F. Stein's method and normal approximation of Poisson functionals. Ann. Probab. 38 (2010), no. 2, 443--478. MR2642882
  • Peccati, G.; Zheng, C. Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15 (2010), no. 48, 1487--1527. MR2727319
  • Peccati G. and Zheng C. (2011). Universal Gaussian fluctuations on the discrete Poisson chaos. To appear in Bernoulli.
  • Penrose, M. Random geometric graphs. Oxford Studies in Probability, 5. Oxford University Press, Oxford, 2003. xiv+330 pp. ISBN: 0-19-850626-0 MR1986198
  • Penrose, M. D.; Yukich, J. E. Normal approximation in geometric probability. Stein's method and applications, 37--58, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, Singapore Univ. Press, Singapore, 2005. MR2201885
  • Reitzner M. ; Schulte M. (2011). Central Limit Theorems for U-Statistics of Poisson Point Processes. To appear in: Ann. Probab.
  • Schneider, R.; Weil, W. Stochastic and integral geometry. Probability and its Applications (New York). Springer-Verlag, Berlin, 2008. xii+693 pp. ISBN: 978-3-540-78858-4 MR2455326
  • Schulte M.(2011). A Central Limit Theorem for the Poisson-Voronoi Approximation. Advances in Applied Mathematics 49, 285-306.
  • Schulte M. ; Thaele and C. (2010). Exact and asymptotic results for intrinsic volumes of Poisson k-flat processes. Preprint.
  • Schulte, M.; Thäle, C. The scaling limit of Poisson-driven order statistics with applications in geometric probability. Stochastic Process. Appl. 122 (2012), no. 12, 4096--4120. MR2971726
  • Silverman, B.; Brown, T. Short distances, flat triangles and Poisson limits. J. Appl. Probab. 15 (1978), no. 4, 815--825. MR0511059
  • Surgailis, D. On multiple Poisson stochastic integrals and associated Markov semigroups. Probab. Math. Statist. 3 (1984), no. 2, 217--239. MR0764148
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