## Electronic Journal of Probability

### A Berry-Esseen bound for the uniform multinomial occupancy model

#### Abstract

The inductive size bias coupling technique and Stein's method yield a Berry-Esseen theorem for the number of urns having occupancy $d \geq 2$ when $n$ balls are uniformly distributed over $m$ urns. In particular, there exists a constant $C$ depending only on $d$ such that $$\sup_{z \in \mathbb{R}}\left|P\left( W_{n,m} \le z \right) -P(Z \le z)\right| \le C \frac{\sigma_{n,m}}{1+(\frac{n}{m})^3} \quad\mbox{for all n \ge d and m \ge 2,}$$ where $W_{n,m}$ and $\sigma_{n,m}^2$ are the standardized count and variance, respectively, of the number of urns with $d$ balls, and $Z$ is a standard normal random variable. Asymptotically, the bound is optimal up to constants if $n$ and $m$ tend to infinity together in a way such that $n/m$ stays bounded.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 27, 29 pp.

Dates
Accepted: 17 February 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064252

Digital Object Identifier
doi:10.1214/EJP.v18-1983

Mathematical Reviews number (MathSciNet)
MR3035755

Zentralblatt MATH identifier
1287.60031

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Rights

#### Citation

Bartroff, Jay; Goldstein, Larry. A Berry-Esseen bound for the uniform multinomial occupancy model. Electron. J. Probab. 18 (2013), paper no. 27, 29 pp. doi:10.1214/EJP.v18-1983. https://projecteuclid.org/euclid.ejp/1465064252

#### References

• Baldi, P.; Rinott, Y.; Stein, C. A normal approximation for the number of local maxima of a random function on a graph. Probability, statistics, and mathematics, 59–81, Academic Press, Boston, MA, 1989.
• Barbour, A. D.; Gnedin, A. V. Small counts in the infinite occupancy scheme. Electron. J. Probab. 14 (2009), no. 13, 365–384.
• Barbour, A. D.; Holst, Lars; Janson, Svante. Poisson approximation. Oxford Studies in Probability, 2. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. x+277 pp. ISBN: 0-19-852235-5
• Barbour, A. D.; Karonski, Michal; Rucinski, Andrzej. A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47 (1989), no. 2, 125–145.
• Bollobás, Béla. Random graphs. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. xvi+447 pp. ISBN: 0-12-111755-3; 0-12-111756-1.
• Bolthausen, E. An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrsch. Verw. Gebiete 66 (1984), no. 3, 379–386.
• Chao, Anne; Yip, Paul; Lin, Huey-Shyan. Estimating the number of species via a martingale estimating function. Statist. Sinica 6 (1996), no. 2, 403–418.
• Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2010). Normal approximation by Stein's method. Springer Verlag.
• Chen, Louis H. Y.; Shao, Qi-Man. Normal approximation under local dependence. Ann. Probab. 32 (2004), no. 3A, 1985–2028.
• Chen, L.H. Y. and Röllin, A. (2010). Stein couplings for Normal Approximation. Preprint http://arxiv.org/abs/1003.6039
• Efron, B.; Stein, C. The jackknife estimate of variance. Ann. Statist. 9 (1981), no. 3, 586–596.
• Efron, B. and Thisted, R. (1976). Estimating the Number of Unseen Species: How Many Words Did Shakespeare Know? phBiometrika 63: 435–448.
• Englund, Gunnar. A remainder term estimate for the normal approximation in classical occupancy. Ann. Probab. 9 (1981), no. 4, 684–692.
• Erdős, P.; Rényi, A. On random graphs. I. Publ. Math. Debrecen 6 1959 290–297.
• Gnedin, Alexander; Hansen, Ben; Pitman, Jim. Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws. Probab. Surv. 4 (2007), 146–171.
• Goldstein, Larry. Berry-Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing. J. Appl. Probab. 42 (2005), no. 3, 661–683.
• Goldstein, L. (2012). A Berry-Esseen bound with applications to vertex degree counts in the Erdős-Rényi random graph Ann. Appl. Probab. to appear
• Goldstein, Larry; Penrose, Mathew D. Normal approximation for coverage models over binomial point processes. Ann. Appl. Probab. 20 (2010), no. 2, 696–721.
• Goldstein, Larry; Reinert, Gesine. Zero biasing in one and higher dimensions, and applications. Stein's method and applications, 1–18, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, Singapore Univ. Press, Singapore, 2005.
• Goldstein, Larry; Rinott, Yosef. Multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Probab. 33 (1996), no. 1, 1–17.
• Goldstein, Larry; Zhang, Haimeng. A Berry-Esseen bound for the lightbulb process. Adv. in Appl. Probab. 43 (2011), no. 3, 875–898.
• Hoeffding, Wassily. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 1963 13–30.
• Hwang, Hsien-Kuei; Janson, Svante. Local limit theorems for finite and infinite urn models. Ann. Probab. 36 (2008), no. 3, 992–1022.
• Janson, Svante; Nowicki, Krzysztof. The asymptotic distributions of generalized $U$-statistics with applications to random graphs. Probab. Theory Related Fields 90 (1991), no. 3, 341–375.
• Johnson, Norman L.; Kotz, Samuel. Urn models and their application. An approach to modern discrete probability theory. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-London-Sydney, 1977. xiii+402 pp.
• Karlin, Samuel. Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17 1967 373–401.
• Karoński, Michał; Ruciński, Andrzej. Poisson convergence and semi-induced properties of random graphs. Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 2, 291–300.
• Kolchin, Valentin F.; Sevast'yanov, Boris A.; Chistyakov, Vladimir P. Random allocations. Translated from the Russian. Translation edited by A. V. Balakrishnan. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.; distributed by Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London, 1978. xi+262 pp. ISBN: 0-470-99394-4
• Kordecki, Wojciech. Normal approximation and isolated vertices in random graphs. Random graphs '87 (Poznań, 1987), 131–139, Wiley, Chichester, 1990.
• Palka, Zbigniew. On the number of vertices of given degree in a random graph. J. Graph Theory 8 (1984), no. 1, 167–170.
• Penrose, Mathew D. Normal approximation for isolated balls in an urn allocation model. Electron. J. Probab. 14 (2009), no. 74, 2156–2181.
• Quine, M. P.; Robinson, J. Normal approximations to sums of scores based on occupancy numbers. Ann. Probab. 12 (1984), no. 3, 794–804.
• Riordan, J. (1937) Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions Ann. Math. Statist. 8(2):103–111.
• Robbins, Herbert E. Estimating the total probability of the unobserved outcomes of an experiment. Ann. Math. Statist. 39 1968 256–257.
• Starr, Norman. Linear estimation of the probability of discovering a new species. Ann. Statist. 7 (1979), no. 3, 644–652.
• Stein, Charles. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pp. 583–602. Univ. California Press, Berkeley, Calif., 1972.
• Stein, Charles. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes – Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. iv+164 pp. ISBN: 0-940600-08-0
• Thisted, Ronald; Efron, Bradley. Did Shakespeare write a newly-discovered poem? Biometrika 74 (1987), no. 3, 445–455.