Electronic Journal of Probability

Concentration bounds for geometric Poisson functionals: Logarithmic Sobolev inequalities revisited

Sascha Bachmann and Giovanni Peccati

Full-text: Open access

Abstract

We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a powerful logarithmic Sobolev inequality proved by L. Wu [43], as well as on several variations of the so-called Herbst argument. We provide several applications, in particular to edge counting and more general length power functionals in random geometric graphs, as well as to the convex distance for random point measures recently introduced by M. Reitzner [30].

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 6, 44 pp.

Dates
Received: 14 April 2015
Accepted: 3 January 2016
First available in Project Euclid: 5 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1454682887

Digital Object Identifier
doi:10.1214/16-EJP4235

Mathematical Reviews number (MathSciNet)
MR3485348

Zentralblatt MATH identifier
1337.60011

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G57: Random measures 60C05: Combinatorial probability

Keywords
concentration of measure convex distance Herbst argument logarithmic Sobolev inequalities Poisson measure random graphs stochastic geometry

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bachmann, Sascha; Peccati, Giovanni. Concentration bounds for geometric Poisson functionals: Logarithmic Sobolev inequalities revisited. Electron. J. Probab. 21 (2016), paper no. 6, 44 pp. doi:10.1214/16-EJP4235. https://projecteuclid.org/euclid.ejp/1454682887


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References

  • [1] C. Ané and M. Ledoux, On logarithmic Sobolev inequalities for continuous time random walks on graphs, Probab. Theory Related Fields 116 (2000), no. 4, 573–602.
  • [2] S. G. Bobkov and M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures, J. Funct. Anal. 156 (1998), no. 2, 347–365.
  • [3] S. Boucheron, G. Lugosi, and P. Massart, Concentration inequalities using the entropy method, Ann. Probab. 31 (2003), no. 3, 1583–1614.
  • [4] S. Boucheron, G. Lugosi, and P. Massart, On concentration of self-bounding functions, Electron. J. Probab. 14 (2009), no. 64, 1884–1899.
  • [5] S. Boucheron, G. Lugosi, and P. Massart, Concentration inequalities, Oxford University Press, Oxford, 2013, A nonasymptotic theory of independence, With a foreword by Michel Ledoux.
  • [6] S. Bourguin and G. Peccati, Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering, Electron. J. Probab. 19 (2014), no. 66, 42 pp.
  • [7] J.-Ch. Breton, Ch. Houdré, and N. Privault, Dimension free and infinite variance tail estimates on Poisson space, Acta Appl. Math. 95 (2007), no. 3, 151–203.
  • [8] D. Chafaï, Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue, ESAIM Probab. Stat. 10 (2006), 317–339 (electronic).
  • [9] L. Decreusefond, E. Ferraz, H. Randriambololona, and A. Vergne, Simplicial homology of random configurations, Adv. in Appl. Probab. 46 (2014), no. 2, 325–347.
  • [10] D. P. Dubhashi and A. Panconesi, Concentration of measure for the analysis of randomized algorithms, Cambridge University Press, Cambridge, 2009.
  • [11] P. Eichelsbacher, M. Raič, and T. Schreiber, Moderate deviations for stabilizing functionals in geometric probability, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 1, 89–128.
  • [12] P. Eichelsbacher and Ch. Thäle, New Berry-Esseen bounds for non-linear functionals of Poisson random measures, Electron. J. Probab. 19 (2014), no. 102, 25 pp.
  • [13] E. N. Gilbert, Random plane networks, J. Soc. Indust. Appl. Math. 9 (1961), 533–543.
  • [14] M. Heveling and M. Reitzner, Poisson-Voronoi approximation, Ann. Appl. Probab. 19 (2009), no. 2, 719–736.
  • [15] Ch. Houdré and N. Privault, Concentration and deviation inequalities in infinite dimensions via covariance representations, Bernoulli 8 (2002), no. 6, 697–720.
  • [16] Ch. Houdré and P. Reynaud-Bouret, Exponential inequalities, with constants, for U-statistics of order two, Stochastic inequalities and applications, Progr. Probab., vol. 56, Birkhäuser, Basel, 2003, pp. 55–69.
  • [17] S. Janson, Bounds on the distributions of extremal values of a scanning process, Stochastic Process. Appl. 18 (1984), no. 2, 313–328.
  • [18] O. Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002.
  • [19] R. Lachièze-Rey and G. Peccati, Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs, Electron. J. Probab. 18 (2013), no. 32, 32 pp.
  • [20] R. Lachièze-Rey and G. Peccati, Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric $U$-statistics, Stochastic Process. Appl. 123 (2013), no. 12, 4186–4218.
  • [21] R. Lachièze-Rey and M. Reitzner, $U$-statistics in stochastic geometry. 2015. arXiv:1503.00110v2 [math.PR].
  • [22] G. Last and M. D. Penrose, Poisson process Fock space representation, chaos expansion and covariance inequalities, Probab. Theory Related Fields 150 (2011), no. 3–4, 663–690.
  • [23] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités, XXXIII, Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 120–216.
  • [24] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001.
  • [25] P. Massart, About the constants in Talagrand’s concentration inequalities for empirical processes, Ann. Probab. 28 (2000), no. 2, 863–884.
  • [26] A. Maurer, Concentration inequalities for functions of independent variables, Random Structures Algorithms 29 (2006), no. 2, 121–138.
  • [27] G. Peccati and M. S. Taqqu, Wiener chaos: moments, cumulants and diagrams, Bocconi & Springer Series, vol. 1, Springer, Milan; Bocconi University Press, Milan, 2011, A survey with computer implementation, Supplementary material available online.
  • [28] G. Peccati and Ch. Thäle, Gamma limits and $U$-statistics on the Poisson space, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 525–560.
  • [29] M. D. Penrose, Random geometric graphs, Oxford Studies in Probability, vol. 5, Oxford University Press, Oxford, 2003.
  • [30] M. Reitzner, Poisson point processes: large deviation inequalities for the convex distance, Electron. Commun. Probab. 18 (2013), no. 96, 7 pp.
  • [31] M. Reitzner and M. Schulte, Central limit theorems for $U$-statistics of Poisson point processes, Ann. Probab. 41 (2013), no. 6, 3879–3909.
  • [32] M. Reitzner, M. Schulte, and Ch. Thäle, Limit theory for the gilbert graph. 2013. arXiv:1312.4861v1 [math.PR].
  • [33] P. Reynaud-Bouret, Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities, Probab. Theory Related Fields 126 (2003), no. 1, 103–153.
  • [34] I. Rivin, Counting cycles and finite dimensional $L^p$ norms, Adv. in Appl. Math. 29 (2002), no. 4, 647–662.
  • [35] R. Schneider and W. Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008.
  • [36] M. Schulte, Malliavin-stein method in stochastic geometry, Ph.D. thesis, Universität Osnabrück, 2013.
  • [37] M. Schulte and Ch. Thäle, The scaling limit of Poisson-driven order statistics with applications in geometric probability, Stochastic Process. Appl. 122 (2012), no. 12, 4096–4120.
  • [38] M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958), 171–176.
  • [39] J. M. Steele, Probability theory and combinatorial optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.
  • [40] D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probab. Math. Statist. 3 (1984), no. 2, 217–239.
  • [41] M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. (1995), no. 81, 73–205.
  • [42] T. Tao, Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132, American Mathematical Society, Providence, RI, 2012.
  • [43] L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, Probab. Theory Related Fields 118 (2000), no. 3, 427–438.