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KOLMOGOROV BOUNDS FOR THE NORMAL APPROXIMATION OF THE NUMBER OF TRIANGLES IN THE ERDŐS–RÉNYI RANDOM GRAPH

Published online by Cambridge University Press:  08 March 2021

Adrian Röllin Open the ORCID record for Adrian Röllin [Opens in a new window]
Adrian Röllin*
Affiliation:
Department of Statistics and Applied Probability, National University of Singapore, Singapore E-mail: adrian.roellin@nus.edu.sg
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Abstract

We bound the error for the normal approximation of the number of triangles in the Erdős–Rényi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein bounds obtained by Barbour et al. [(1989). A central limit theorem for decomposable random variables with applications to random graphs. Journal of Combinatorial Theory, Series B 47: 125–145], resolving a long-standing open problem. The proofs are based on a new variant of the Stein–Tikhomirov method—a combination of Stein's method and characteristic functions introduced by Tikhomirov [(1976). The rate of convergence in the central limit theorem for weakly dependent variables. Vestnik Leningradskogo Universiteta 158–159, 166].

Keywords

random graphsStein's methodStein–Tikhomirov methodsubgraph counts
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 3 , July 2022 , pp. 747 - 773
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

Arras, B., Mijoule, G., Poly, G., & Swan, Y. (2016). A new approach to the Stein-Tikhomirov method: with applications to the second Wiener chaos and Dickman convergence. Preprint arXiv:1605.06819.Google Scholar
Baldi, P. & Rinott, Y. (1989). On normal approximations of distributions in terms of dependency graphs. Annals of Applied Probability 17: 1646–1650.Google Scholar
Barbour, A.D., Karoński, M., & Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. Journal of Combinatorial Theory, Series B 47: 125–145.CrossRefGoogle Scholar
Bloznelis, M. (2004). On combinatorial Hoeffding decomposition and asymptotic normality of subgraph count statistics. In M. Drmota, P. Flajolet, D. Gardy & B. Gittenberger (eds.), Mathematics and computer science III: Algorithms, trees, combinatorics and probabilities. Basel: Springer Basel AG, pp, 73–79.Google Scholar
Bulinskii, A.V. (1996). Rate of convergence in the central limit theorem for fields of associated random variables. Theory of Probability and Its Applications 40: 136–144.CrossRefGoogle Scholar
Chen, L.H.Y. (1978). Two central limit problems for dependent random variables. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiet 43: 223–243.CrossRefGoogle Scholar
Chen, L.H.Y. & Röllin, A. (2010). Stein couplings for normal approximation. Available at: arxiv.org/abs/1003.6039.Google Scholar
Chen, L.H.Y. & Shao, Q.-M. (2004). Normal approximation under local dependence. Annals of Applied Probability 32: 1985–2028.Google Scholar
Diaconis, P. (1977). The distribution of leading digits and uniform distribution mod 1. Annals of Applied Probability 5: 72–81.Google Scholar
Esary, J.D., Proschan, F., & Walkup, D.W. (1967). Association of random variables, with applications. Annals of Mathematical Statistics 38: 1466–1474.CrossRefGoogle Scholar
Ibragimov, I.A. & Linnik, Y.V. (1971). Independent and stationary sequences of random variables. Groningen: Wolters-Noordhoff Publishing.Google Scholar
Krokowski, K., Reichenbachs, A., & Thäle, C. (2017). Discrete Malliavin-Stein method: Berry-Esseen bounds for random graphs and percolation. Annals of Applied Probability 45: 1071–1109.Google Scholar
Newman, C.M. (1980). Normal fluctuations and the FKG inequalities. Communications in Mathematical Physics 74: 119–128.CrossRefGoogle Scholar
Privault, N. & Serafin, G. (2020). Normal approximation for sums of weighted U-statistics—application to Kolmogorov bounds in random subgraph counting. Bernoulli 26: 587–615.CrossRefGoogle Scholar
Röllin, A. & Ross, N. (2015). Local limit theorems via Landau-Kolmogorov inequalities. Bernoulli 21: 851–880.CrossRefGoogle Scholar
Ross, N. (2011). Fundamentals of Stein's method. Probability Surveys 8: 210–293.CrossRefGoogle Scholar
Ruciński, A. (1988). When are small subgraphs of a random graph normally distributed? Probability Theory and Related Fields 78: 1–10.CrossRefGoogle Scholar
Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory. Berkeley, CA: University of California Press, pp. 583–602.Google Scholar
Stein, C. (1986). Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7. Institute of Mathematical Statistics, Hayward, California.Google Scholar
Tikhomirov, A.N. (1976). The rate of convergence in the central limit theorem for weakly dependent variables. Vestnik Leningradskogo Universiteta 158–159: 166.Google Scholar
Tikhomirov, A.N. (1980). Convergence rate in the central limit theorem for weakly dependent random variables. Theory of Probability and Its Applications 25: 790–809.CrossRefGoogle Scholar