Annals of Probability

Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation

Kai Krokowski, Anselm Reichenbachs, and Christoph Thäle

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Abstract

A new Berry–Esseen bound for nonlinear functionals of nonsymmetric and nonhomogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin–Stein method and an analysis of the discrete Ornstein–Uhlenbeck semigroup. The result is applied to sub-graph counts and to the number of vertices having a prescribed degree in the Erdős–Rényi random graph. A further application deals with a percolation problem on trees.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 1071-1109.

Dates
Received: March 2015
Revised: November 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947314

Digital Object Identifier
doi:10.1214/15-AOP1081

Mathematical Reviews number (MathSciNet)
MR3630293

Zentralblatt MATH identifier
1372.05203

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60F05: Central limit and other weak theorems 60H07: Stochastic calculus of variations and the Malliavin calculus 82B43: Percolation [See also 60K35]

Keywords
Berry–Esseen bound central limit theorem Malliavin–Stein method Mehler’s formula percolation Rademacher functional random graph sub-graph count tree

Citation

Krokowski, Kai; Reichenbachs, Anselm; Thäle, Christoph. Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation. Ann. Probab. 45 (2017), no. 2, 1071--1109. doi:10.1214/15-AOP1081. https://projecteuclid.org/euclid.aop/1490947314


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References

  • [1] Azaïs, J.-M. and León, J. R. (2013). CLT for crossings of random trigonometric polynomials. Electron. J. Probab. 18 1–17.
  • [2] Barbour, A. D., Karoński, M. and Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47 125–145.
  • [3] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
  • [4] Decreusefond, L., Schulte, M. and Thäle, C. (2016). Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to $U$-statistics and stochastic geometry. Ann. Probab. 44 2147–2197.
  • [5] Durastanti, C., Marinucci, D. and Peccati, G. (2014). Normal approximations for wavelet coefficients on spherical Poisson fields. J. Math. Anal. Appl. 409 212–227.
  • [6] 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.
  • [7] Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 290–297.
  • [8] Féray, V., Méliot, P.-L. and Nikeghbali, A. (2013). Mod-phi convergence and precise deviations. Available at arXiv:1304.2934.
  • [9] Goldstein, L. (2013). A Berry–Esseen bound with applications to vertex degree counts in the Erdős–Rényi random graph. Ann. Appl. Probab. 23 617–636.
  • [10] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley-Interscience, New York.
  • [11] Janson, S. and Nowicki, K. (1991). The asymptotic distributions of generalized $U$-statistics with applications to random graphs. Probab. Theory Related Fields 90 341–375.
  • [12] Kordecki, W. (1990). Normal approximation and isolated vertices in random graphs. In Random Graphs ’87 (Poznań, 1987) 131–139. Wiley, Chichester.
  • [13] Krokowski, K. (2015). Poisson approximation of Rademacher functionals by the Chen–Stein method and Malliavin calculus. Available at arXiv:1505.01417.
  • [14] Krokowski, K., Reichenbachs, A. and Thäle, C. (2016). Berry–Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences. Ann. Inst. H. Poncaré Probab. Statist. To appear.
  • [15] 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.
  • [16] Lachièze-Rey, R. and Peccati, G. (2013). Fine Gaussian fluctuations on the Poisson space II: Rescaled kernels, marked processes and geometric $U$-statistics. Stochastic Process. Appl. 123 4186–4218.
  • [17] 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 Fields 165 667–723.
  • [18] Last, G., Penrose, M. D., Schulte, M. and Thäle, C. (2014). Moments and central limit theorems for some multivariate Poisson functionals. Adv. in Appl. Probab. 46 348–364.
  • [19] Marinucci, D. and Peccati, G. (2011). Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications. 389. Cambridge Univ. Press, Cambridge.
  • [20] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
  • [21] Nourdin, I. and Peccati, G. (2010). Universal Gaussian fluctuations of non-Hermitian matrix ensembles: From weak convergence to almost sure CLTs. ALEA Lat. Am. J. Probab. Math. Stat. 7 341–375.
  • [22] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.
  • [23] Nourdin, I., Peccati, G. and Reinert, G. (2009). Second order Poincaré inequalities and CLTs on Wiener space. J. Funct. Anal. 257 593–609.
  • [24] Nourdin, I., Peccati, G. and Reinert, G. (2010). Stein’s method and stochastic analysis of Rademacher functionals. Electron. J. Probab. 15 1703–1742.
  • [25] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [26] 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.
  • [27] Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi & Springer Series 1. Springer, Milan.
  • [28] 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.
  • [29] Pike, J. and Ren, H. (2014). Stein’s method and the Laplace distribution. ALEA Lat. Am. J. Probab. Math. Stat. 11 571–587.
  • [30] Privault, N. (2008). Stochastic analysis of Bernoulli processes. Probab. Surv. 5 435–483.
  • [31] Privault, N. and Torrisi, G. L. (2015). The Stein and Chen–Stein methods for functionals of non-symmetric Bernoulli processes. ALEA Lat. Am. J. Probab. Math. Stat. 12 309–356.
  • [32] Raič, M. (2003). Normal approximation by Stein’s method. In Proceedings of the Seventh Young Statisticians Meeting. Metodoloski Zvezki 21. FDV, Ljubljana.
  • [33] Reitzner, M. and Schulte, M. (2013). Central limit theorems for $U$-statistics of Poisson point processes. Ann. Probab. 41 3879–3909.
  • [34] Ruciński, A. (1988). When are small subgraphs of a random graph normally distributed? Probab. Theory Related Fields 78 1–10.
  • [35] Saulis, L. and Statulevičius, V. A. (1991). Limit Theorems for Large Deviations. Mathematics and Its Applications (Soviet Series) 73. Kluwer Academic, Dordrecht.
  • [36] 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.
  • [37] Sugimine, N. and Takei, M. (2006). Remarks on central limit theorems for the number of percolation clusters. Publ. Res. Inst. Math. Sci. 42 101–116.