## The Annals of Applied Probability

### A Berry–Esseen bound with applications to vertex degree counts in the Erdős–Rényi random graph

Larry Goldstein

#### Abstract

Applying Stein’s method, an inductive technique and size bias coupling yields a Berry–Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erdős–Rényi random graph of a given degree.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 2 (2013), 617-636.

Dates
First available in Project Euclid: 12 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1360682024

Digital Object Identifier
doi:10.1214/12-AAP848

Mathematical Reviews number (MathSciNet)
MR3059270

Zentralblatt MATH identifier
1278.60048

#### Citation

Goldstein, Larry. A Berry–Esseen bound with applications to vertex degree counts in the Erdős–Rényi random graph. Ann. Appl. Probab. 23 (2013), no. 2, 617--636. doi:10.1214/12-AAP848. https://projecteuclid.org/euclid.aoap/1360682024

#### References

• Baldi, P., Rinott, Y. and Stein, C. (1989). A normal approximation for the number of local maxima of a random function on a graph. In Probability, Statistics, and Mathematics (T. W. Anderson, K. B. Athreya and D. L. Iglehart, eds.) 59–81. Academic Press, Boston, MA.
• Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. The Clarendon Press Oxford Univ. Press, New York.
• 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.
• Bollobás, B. (1985). Random Graphs. Academic Press, London.
• Bolthausen, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrsch. Verw. Gebiete 66 379–386.
• Chen, L. H. Y., Goldstein, L. and Shao, Q. M. (2010). Normal Approximation by Stein’s Method. Springer, Berlin.
• Chen, L. H. Y. and Röllin, A. (2010). Stein couplings for normal approximation. Preprint.
• Chen, L. H. Y. and Shao, Q.-M. (2004). Normal approximation under local dependence. Ann. Probab. 32 1985–2028.
• Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 290–297.
• 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.
• Goldstein, L. and Penrose, M. D. (2010). Normal approximation for coverage models over binomial point processes. Ann. Appl. Probab. 20 696–721.
• Goldstein, L. and Rinott, Y. (1996). Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33 1–17.
• Goldstein, L. and Zhang, H. (2011). A Berry–Esseen bound for the lightbulb process. Adv. in Appl. Probab. 43 875–898.
• 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.
• Karoński, M. and Ruciński, A. (1987). Poisson convergence and semi-induced properties of random graphs. Math. Proc. Cambridge Philos. Soc. 101 291–300.
• Kordecki, W. (1990). Normal approximation and isolated vertices in random graphs. In Random Graphs ’87 (Poznań, 1987) 131–139. Wiley, Chichester.
• Neammanee, K. and Suntadkarn, A. (2009). On the normal approximation of the number of vertices in a random graph. ScienceAsia 25 203–210.
• Palka, Z. (1984). On the number of vertices of given degree in a random graph. J. Graph Theory 8 167–170.
• Riordan, J. (1937). Moment recurrence relations for binomial, Poisson and hypergeometric frequency distributions. Ann. Math. Statist. 8 103–111.
• 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 583–602. Univ. California Press, Berkeley, CA.
• Stein, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA.