• Bernoulli
  • Volume 25, Number 2 (2019), 1076-1104.

Error bounds in local limit theorems using Stein’s method

A.D. Barbour, Adrian Röllin, and Nathan Ross

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We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erdős–Rényi random graph, and of the Curie–Weiss model of magnetism, where we provide optimal or near optimal rates of convergence in the local limit metric. In the Hoeffding example, even the discrete normal approximation bounds seem to be new. The general result follows from Stein’s method, and requires a new bound on the Stein solution for the Poisson distribution, which is of general interest.

Article information

Bernoulli, Volume 25, Number 2 (2019), 1076-1104.

Received: July 2017
Revised: November 2017
First available in Project Euclid: 6 March 2019

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approximation error Curie–Weiss model Erdős–Rényi random graph Hoeffding combinatorial statistic local limit theorem


Barbour, A.D.; Röllin, Adrian; Ross, Nathan. Error bounds in local limit theorems using Stein’s method. Bernoulli 25 (2019), no. 2, 1076--1104. doi:10.3150/17-BEJ1013.

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  • [1] Arratia, R. and Baxendale, P. (2015). Bounded size bias coupling: A Gamma function bound, and universal Dickman-function behavior. Probab. Theory Related Fields 162 411–429.
  • [2] Barbour, A.D. (1980). Equilibrium distributions for Markov population processes. Adv. in Appl. Probab. 12 591–614.
  • [3] Barbour, A.D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. New York: The Clarendon Press, Oxford Univ. Press.
  • [4] 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.
  • [5] Barbour, A.D. and Xia, A. (1999). Poisson perturbations. ESAIM Probab. Stat. 3 131–150.
  • [6] Bartroff, J., Goldstein, L. and Işlak, Ü. (2015). Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models. Bernoulli. To appear. Available at arXiv:1402.6769v2.
  • [7] Bolthausen, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrsch. Verw. Gebiete 66 379–386.
  • [8] Chatterjee, S. (2007). Stein’s method for concentration inequalities. Probab. Theory Related Fields 138 305–321.
  • [9] Chatterjee, S. and Dey, P.S. (2010). Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 2443–2485.
  • [10] Chatterjee, S. and Shao, Q.-M. (2011). Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Ann. Appl. Probab. 21 464–483.
  • [11] Chen, L.H.Y. and Fang, X. (2015). On the error bound in a combinatorial central limit theorem. Bernoulli 21 335–359.
  • [12] Chen, L.H.Y., Fang, X. and Shao, Q.-M. (2013). From Stein identities to moderate deviations. Ann. Probab. 41 262–293.
  • [13] Chen, L.H.Y. and Röllin, A. (2010). Stein couplings for normal approximation. Preprint. Available at arXiv:1003.6039v2.
  • [14] Dembo, A. and Montanari, A. (2010). Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24 137–211.
  • [15] Eichelsbacher, P. and Löwe, M. (2010). Stein’s method for dependent random variables occurring in statistical mechanics. Electron. J. Probab. 15 962–988.
  • [16] Ellis, R.S. (2006). Entropy, Large Deviations, and Statistical Mechanics. Classics in Mathematics. Berlin: Springer. Reprint of the 1985 original.
  • [17] Ellis, R.S. and Newman, C.M. (1978). Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 117–139.
  • [18] Ellis, R.S., Newman, C.M. and Rosen, J.S. (1980). Limit theorems for sums of dependent random variables occurring in statistical mechanics. II. Conditioning, multiple phases, and metastability. Z. Wahrsch. Verw. Gebiete 51 153–169.
  • [19] Esseen, C.-G. (1945). Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law. Acta Math. 77 1–125.
  • [20] Fang, X. (2014). Discretized normal approximation by Stein’s method. Bernoulli 20 1404–1431.
  • [21] Goldstein, L. (2005). Berry–Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing. J. Appl. Probab. 42 661–683.
  • [22] 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.
  • [23] Goldstein, L. and Işlak, Ü. (2014). Concentration inequalities via zero bias couplings. Statist. Probab. Lett. 86 17–23.
  • [24] Goldstein, L. and Xia, A. (2006). Zero biasing and a discrete central limit theorem. Ann. Probab. 34 1782–1806.
  • [25] Hoeffding, W. (1951). A combinatorial central limit theorem. Ann. Math. Stat. 22 558–566.
  • [26] Kordecki, W. (1990). Normal approximation and isolated vertices in random graphs. In Random Graphs ’87 (Poznań, 1987) 131–139. Chichester: Wiley.
  • [27] Krokowski, K., Reichenbachs, A. and Thäle, C. (2017). Discrete Malliavin–Stein method: Berry–Esseen bounds for random graphs and percolation. Ann. Probab. 45 1071–1109.
  • [28] McDonald, D.R. (1979). On local limit theorem for integer valued random variables. Teor. Veroyatn. Primen. 24 607–614.
  • [29] Petrov, V.V. (1975). Sums of Independent Random Variables. Ergebnisse der Mathematik und ihrer Grenzgebiete 82. New York: Springer. Translated from the Russian by A.A. Brown.
  • [30] Röllin, A. (2005). Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli 11 1115–1128.
  • [31] Röllin, A. (2007). Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Probab. 17 1596–1614.
  • [32] Röllin, A. (2008). Symmetric and centered binomial approximation of sums of locally dependent random variables. Electron. J. Probab. 13 756–776.
  • [33] Röllin, A. (2017). Kolmogorov bounds for the normal approximation of the number of triangles in the Erdős–Rényi random graph. Preprint. Available at arXiv:1704.00410v1.
  • [34] Röllin, A. and Ross, N. (2015). Local limit theorems via Landau–Kolmogorov inequalities. Bernoulli 21 851–880.
  • [35] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8 210–293.
  • [36] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes – Monograph Series 7. Hayward, CA: IMS.
  • [37] Wald, A. and Wolfowitz, J. (1944). Statistical tests based on permutations of the observations. Ann. Math. Stat. 15 358–372.