The Abstract

	Annals of Combinatorics 10 (2006) Total Variation Distance for Poisson Subset Numbers Larry Goldstein and Gesine Reinert Department of Mathematics, University of Southern California, 3620 Vermont Avenue, Los Angeles, CA 90089-2532, USA larry@math.usc.edu Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK reinert@stats.ox.ac.uk Received February 24, 2005 AMS Subject Classification: 60C05, 62E17 Abstract. Let n be an integer and A₀, ... , A_k random subsets of {1,... , n} of fixed sizes a₀, ... , a_k, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable W = \|∩^k_j=0 A_j\|, the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and a_j→ ∞, for all j = 0, ... ,k, showing that W has an asymptotic Poisson distribution in this regime. Keywords: Poisson approximation, Stein's method, size biasing, surprisology References 1. R. Arratia, L. Goldstein, and L. Gordon, Two moments suffice for Poisson approximation: the Chen-Stein method, Ann. Probab. 17 (1989) 9–25. 2. R. Arratia, L. Goldstein, and L. Gordon, Poisson approximation and the Chen-Stein method, Statist. Sci. 5 (1990) 403–434. 3. A.D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford University Press, Oxford, 1992. 4. L.H.Y. Chen, Poisson approximation for dependent trials, Ann. Probab. 3 (1975) 534–545. 5. A. Dress, T. Lokot, L.D. Pustyl'nilov, and W. Schubert, Poisson numbers and Poisson distribution in subset surprisology, Ann. Combin. 8 (2004) 473–485. 6. W. Feller, An Introduction to Probability Theory and Its Applications, Vol.1, Wiley, New York, 1968. 7. L. Goldstein and G. Reinert, Distributional transformations, orthogonal polynomials, and Stein characterizations, J. Theoret. Probab. 18 (2005) 185–208. 8. L. Goldstein and Y. Rinott, On multivariate normal approximations by Stein's method and size bias couplings, J. Appl. Probab. 33 (1996) 1–17. 9. W. Schubert, Topological proteomics, toponomics, MELK-technology, In: Proteomics of Microorganisms: Fundamental Aspects and Application, Advances in Biochemical Engineering/ Biotechnology 83, M. Hecker and S. M¨ullner, eds., Springer Verlag, Berlin, (2003) pp. 189–209. 10. C. Stein, 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 2, University California Press, Berkeley, (1972) pp. 583–602. 11. C. Stein, Approximate Computation of Expectations, Institute of Mathematical Statistics, Hayward, CA, 1986. Get the DVI \| PS \| PDF file of this abstract. back

Annals of Combinatorics 10 (2006)

Total Variation Distance for Poisson Subset Numbers

Larry Goldstein and Gesine Reinert

Department of Mathematics, University of Southern California, 3620 Vermont Avenue, Los Angeles, CA 90089-2532, USA
larry@math.usc.edu

Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK
reinert@stats.ox.ac.uk

Received February 24, 2005

AMS Subject Classification: 60C05, 62E17

Abstract. Let n be an integer and A₀, ... , A_k random subsets of {1,... , n} of fixed sizes a₀, ... , a_k, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable W = |∩^k_j=0 A_j|, the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and a_j→ ∞, for all j = 0, ... ,k, showing that W has an asymptotic Poisson distribution in this regime.

Keywords: Poisson approximation, Stein's method, size biasing, surprisology

References

1. R. Arratia, L. Goldstein, and L. Gordon, Two moments suffice for Poisson approximation: the Chen-Stein method, Ann. Probab. 17 (1989) 9–25.

2. R. Arratia, L. Goldstein, and L. Gordon, Poisson approximation and the Chen-Stein method, Statist. Sci. 5 (1990) 403–434.

3. A.D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford University Press, Oxford, 1992.

4. L.H.Y. Chen, Poisson approximation for dependent trials, Ann. Probab. 3 (1975) 534–545.

5. A. Dress, T. Lokot, L.D. Pustyl'nilov, and W. Schubert, Poisson numbers and Poisson distribution in subset surprisology, Ann. Combin. 8 (2004) 473–485.

6. W. Feller, An Introduction to Probability Theory and Its Applications, Vol.1, Wiley, New York, 1968.

7. L. Goldstein and G. Reinert, Distributional transformations, orthogonal polynomials, and Stein characterizations, J. Theoret. Probab. 18 (2005) 185–208.

8. L. Goldstein and Y. Rinott, On multivariate normal approximations by Stein's method and size bias couplings, J. Appl. Probab. 33 (1996) 1–17.

9. W. Schubert, Topological proteomics, toponomics, MELK-technology, In: Proteomics of Microorganisms: Fundamental Aspects and Application, Advances in Biochemical Engineering/ Biotechnology 83, M. Hecker and S. M¨ullner, eds., Springer Verlag, Berlin, (2003) pp. 189–209.

10. C. Stein, 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 2, University California Press, Berkeley, (1972) pp. 583–602.

11. C. Stein, Approximate Computation of Expectations, Institute of Mathematical Statistics, Hayward, CA, 1986.

Get the DVI | PS | PDF file of this abstract.

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