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Probability Approximations and Beyond pp 31–41Cite as

Clubbed Binomial Approximation for the Lightbulb Process

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Part of the book series: Lecture Notes in Statistics ((LNSP,volume 205))

Abstract

In the so called lightbulb process, on days \(r=1,\ldots,n,\) out of n lightbulbs, all initially off, exactly r bulbs selected uniformly and independent of the past have their status changed from off to on, or vice versa. With \(W_n\) the number of bulbs on at the terminal time n and \(C_n\) a suitable clubbed binomial distribution,

$$ d_{{{\rm TV}}}(W_n,C_n) \leqslant 2.7314 \sqrt{n} e^{-(n+1)/3} \quad \hbox{for all}\,n \geqslant 1. $$

The result is shown using Stein’s method.

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References

  1. Barbour AD, Holst L, Janson S (1992) Poisson approximation. Oxford University Press, Oxford

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  2. Goldstein L, Zhang H (2010) A Berry-Esseen theorem for the lightbulb process (preprint)

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  3. Rao C, Rao M, Zhang H (2007) One Bulb? Two Bulbs? How many bulbs light up? A discrete probability problem involving dermal patches. Sanky ? 69:137–161

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  4. Zhou H, Lange K (2009) Composition Markov chains of multinomial type. Adv Appl Probab 41:270–291

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Acknowledgments

The authors would like to thank the organizers of the conference held at the National University of Singapore in honor of Louis Chen’s birthday for the opportunity to collaborate on the present work.

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Authors and Affiliations

  1. Department of Mathematics KAP 108, University of Southern California, Los Angeles, CA, 90089-2532, USA

    Larry Goldstein

  2. Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Victoria, Vic 3010, Australia

    Aihua Xia

Authors
  1. Larry Goldstein
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  2. Aihua Xia
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Corresponding author

Correspondence to Larry Goldstein .

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Editors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, Zürich, 8057, Switzerland

    Andrew Barbour

  2. , Department of Statistics and Applied Pro, National University of Singapore, Singapore, 119260, Singapore

    Hock Peng Chan

  3. Dept. Statistics, Stanford University, Serra Mall 390, Stanford, 94305, California, USA

    David Siegmund

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Goldstein, L., Xia, A. (2012). Clubbed Binomial Approximation for the Lightbulb Process. In: Barbour, A., Chan, H., Siegmund, D. (eds) Probability Approximations and Beyond. Lecture Notes in Statistics(), vol 205. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1966-2_3

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