Error bounds in local limit theorems using Stein’s method. (English) Zbl 07049400

Summary: 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.


60F05 Central limit and other weak theorems
Full Text: DOI arXiv Euclid


[1] Arratia, R. and Baxendale, P. (2015). Bounded size bias coupling: A Gamma function bound, and universal Dickman-function behavior. Probab. Theory Related Fields162 411–429. · Zbl 1323.60034
[2] Barbour, A.D. (1980). Equilibrium distributions for Markov population processes. Adv. in Appl. Probab.12 591–614. · Zbl 0434.60084
[3] Barbour, A.D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability2. New York: The Clarendon Press, Oxford Univ. Press. · Zbl 0746.60002
[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. B47 125–145.
[5] Barbour, A.D. and Xia, A. (1999). Poisson perturbations. ESAIM Probab. Stat.3 131–150. · Zbl 0949.62015
[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. Gebiete66 379–386. · Zbl 0563.60026
[8] Chatterjee, S. (2007). Stein’s method for concentration inequalities. Probab. Theory Related Fields138 305–321. · Zbl 1116.60056
[9] Chatterjee, S. and Dey, P.S. (2010). Applications of Stein’s method for concentration inequalities. Ann. Probab.38 2443–2485. · Zbl 1203.60023
[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. · Zbl 1216.60018
[11] Chen, L.H.Y. and Fang, X. (2015). On the error bound in a combinatorial central limit theorem. Bernoulli21 335–359. · Zbl 1354.60011
[12] Chen, L.H.Y., Fang, X. and Shao, Q.-M. (2013). From Stein identities to moderate deviations. Ann. Probab.41 262–293. · Zbl 1275.60029
[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. · Zbl 1205.05209
[15] Eichelsbacher, P. and Löwe, M. (2010). Stein’s method for dependent random variables occurring in statistical mechanics. Electron. J. Probab.15 962–988. · Zbl 1225.60042
[16] Ellis, R.S. (2006). Entropy, Large Deviations, and Statistical Mechanics. Classics in Mathematics. Berlin: Springer. Reprint of the 1985 original. · Zbl 1102.60087
[17] Ellis, R.S. and Newman, C.M. (1978). Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete44 117–139. · Zbl 0364.60120
[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. Gebiete51 153–169. · Zbl 0404.60096
[19] Esseen, C.-G. (1945). Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law. Acta Math.77 1–125. · Zbl 0060.28705
[20] Fang, X. (2014). Discretized normal approximation by Stein’s method. Bernoulli20 1404–1431. · Zbl 1310.62021
[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. · Zbl 1087.60021
[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. · Zbl 1278.60048
[23] Goldstein, L. and Işlak, Ü. (2014). Concentration inequalities via zero bias couplings. Statist. Probab. Lett.86 17–23. · Zbl 1292.60030
[24] Goldstein, L. and Xia, A. (2006). Zero biasing and a discrete central limit theorem. Ann. Probab.34 1782–1806. · Zbl 1111.60015
[25] Hoeffding, W. (1951). A combinatorial central limit theorem. Ann. Math. Stat.22 558–566. · Zbl 0044.13702
[26] Kordecki, W. (1990). Normal approximation and isolated vertices in random graphs. In Random Graphs ’87 (Poznań, 1987) 131–139. Chichester: Wiley. · Zbl 0741.05063
[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. · Zbl 1372.05203
[28] McDonald, D.R. (1979). On local limit theorem for integer valued random variables. Teor. Veroyatn. Primen.24 607–614. · Zbl 0411.60027
[29] Petrov, V.V. (1975). Sums of Independent Random Variables. Ergebnisse der Mathematik und ihrer Grenzgebiete82. New York: Springer. Translated from the Russian by A.A. Brown. · Zbl 0322.60042
[30] Röllin, A. (2005). Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli11 1115–1128.
[31] Röllin, A. (2007). Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Probab.17 1596–1614. · Zbl 1143.60020
[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. Bernoulli21 851–880. · Zbl 1320.60065
[35] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv.8 210–293. · Zbl 1245.60033
[36] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes – Monograph Series7. Hayward, CA: IMS. · Zbl 0721.60016
[37] Wald, A. and Wolfowitz, J. (1944). Statistical tests based on permutations of the observations. Ann. Math. Stat.15 358–372. · Zbl 0063.08124
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.