## The Annals of Probability

### Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry

#### Abstract

A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein’s method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived, and examples from stochastic geometry are investigated.

#### Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 2147-2197.

Dates
Revised: March 2015
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.aop/1463410041

Digital Object Identifier
doi:10.1214/15-AOP1020

Mathematical Reviews number (MathSciNet)
MR3502603

Zentralblatt MATH identifier
1347.60027

#### Citation

Decreusefond, Laurent; Schulte, Matthias; Thäle, Christoph. Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry. Ann. Probab. 44 (2016), no. 3, 2147--2197. doi:10.1214/15-AOP1020. https://projecteuclid.org/euclid.aop/1463410041

#### References

• [1] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
• [2] Barbour, A. D. (1988). Stein’s method and Poisson process convergence: A celebration of applied probability. J. Appl. Probab. 25A 175–184.
• [3] Barbour, A. D. (1990). Stein’s method for diffusion approximations. Probab. Theory Related Fields 84 297–322.
• [4] Barbour, A. D. and Brown, T. C. (1992). Stein’s method and point process approximation. Stochastic Process. Appl. 43 9–31.
• [5] Barbour, A. D., Chen, L. H. Y. and Loh, W.-L. (1992). Compound Poisson approximation for nonnegative random variables via Stein’s method. Ann. Probab. 20 1843–1866.
• [6] Barbour, A. D. and Chryssaphinou, O. (2001). Compound Poisson approximation: A user’s guide. Ann. Appl. Probab. 11 964–1002.
• [7] Barbour, A. D. and Eagleson, G. K. (1984). Poisson convergence for dissociated statistics. J. Roy. Statist. Soc. Ser. B 46 397–402.
• [8] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Univ. Press, New York.
• [9] Barbour, A. D. and Utev, S. (1998). Solving the Stein equation in compound Poisson approximation. Adv. in Appl. Probab. 30 449–475.
• [10] Barbour, A. D. and Xia, A. (2006). On Stein’s factors for Poisson approximation in Wasserstein distance. Bernoulli 12 943–954.
• [11] Brown, T. C., Weinberg, G. V. and Xia, A. (2000). Removing logarithms from Poisson process error bounds. Stochastic Process. Appl. 87 149–165.
• [12] Brown, T. C. and Xia, A. (1995). On metrics in point process approximation. Stochastics Stochastics Rep. 52 247–263.
• [13] Brown, T. C. and Xia, A. (2001). Stein’s method and birth–death processes. Ann. Probab. 29 1373–1403.
• [14] Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3 534–545.
• [15] Chen, L. H. Y. and Xia, A. (2004). Stein’s method, Palm theory and Poisson process approximation. Ann. Probab. 32 2545–2569.
• [16] Coutin, L. and Decreusefond, L. (2013). Stein’s method for Brownian approximations. Commun. Stoch. Anal. 7 349–372.
• [17] Dabrowski, A. R., Dehling, H. G., Mikosch, T. and Sharipov, O. (2002). Poisson limits for $U$-statistics. Stochastic Process. Appl. 99 137–157.
• [18] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd ed. Springer, New York.
• [19] Decreusefond, L. (2008). Wasserstein distance on configuration space. Potential Anal. 28 283–300.
• [20] Decreusefond, L., Joulin, A. and Savy, N. (2010). Upper bounds on Rubinstein distances on configuration spaces and applications. Commun. Stoch. Anal. 4 377–399.
• [21] Eichelsbacher, P. and Roos, M. (1999). Compound Poisson approximation for dissociated random variables via Stein’s method. Combin. Probab. Comput. 8 335–346.
• [22] Heinrich, L. and Wolf, W. (1993). On the convergence of $U$-statistics with stable limit distribution. J. Multivariate Anal. 44 266–278.
• [23] Hug, D., Schneider, R. and Schuster, R. (2008). Integral geometry of tensor valuations. Adv. in Appl. Math. 41 482–509.
• [24] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
• [25] Kingman, J. F. C. (1993). Poisson Processes. Oxford Univ. Press, New York.
• [26] 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 1–35.
• [27] Lao, W. and Mayer, M. (2008). $U$-max-statistics. J. Multivariate Anal. 99 2039–2052.
• [28] Last, G., Peccati, G. and Schulte, M. (2014). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Available at arXiv:1401.7568.
• [29] Last, G. and Penrose, M. D. (2011). Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Related Fields 150 663–690.
• [30] Malevich, T. L. and Abdalimov, B. (1977). Stable limit distributions for $U$-statistics. Theory Probab. Appl. 22 370–377.
• [31] Mayer, M. and Molchanov, I. (2007). Limit theorems for the diameter of a random sample in the unit ball. Extremes 10 129–150.
• [32] Peccati, G. (2011). The Chen–Stein method for Poisson functionals. Available at arXiv:1112.5051.
• [33] 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.
• [34] Penrose, M. (2003). Random Geometric Graphs. Oxford Univ. Press, Oxford.
• [35] Preston, C. (1975). Spatial birth-and-death processes. In Proceedings of the 40th Session of the International Statistical Institute (Warsaw, 1975), Vol. 2. 371–391.
• [36] Reinert, G. (2005). Three general approaches to Stein’s method. In An Introduction to Stein’s Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4 183–221. Singapore Univ. Press, Singapore.
• [37] Reitzner, M., Schulte, M. and Thäle, C. (2013). Limit theory for the Gilbert graph. Available at arXiv:1312.4861.
• [38] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
• [39] Schuhmacher, D. (2005). Upper bounds for spatial point process approximations. Ann. Appl. Probab. 15 615–651.
• [40] Schuhmacher, D. (2009). Stein’s method and Poisson process approximation for a class of Wasserstein metrics. Bernoulli 15 550–568.
• [41] Schuhmacher, D. and Stucki, K. (2014). Gibbs point process approximation: Total variation bounds using Stein’s method. Ann. Probab. 42 1911–1951.
• [42] Schuhmacher, D. and Xia, A. (2008). A new metric between distributions of point processes. Adv. in Appl. Probab. 40 651–672.
• [43] Schulte, M. and Thäle, C. (2012). The scaling limit of Poisson-driven order statistics with applications in geometric probability. Stochastic Process. Appl. 122 4096–4120.
• [44] Schulte, M. and Thäle, C. (2014). Distances between Poisson $k$-flats. Methodol. Comput. Appl. Probab. 16 311–329.
• [45] Shih, H.-H. (2011). On Stein’s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces. J. Funct. Anal. 261 1236–1283.
• [46] Villani, C. (2009). Optimal Transport: Old and New. Springer, Berlin.
• [47] Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. Amer. Math. Soc., Providence, RI.