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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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