Clubbed Binomial Approximation for the Lightbulb Process

Conference paper
Part of the Lecture Notes in Statistics book series (LNS, 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.

Keywords

Stein 

Notes

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.

References

  1. 1.
    Barbour AD, Holst L, Janson S (1992) Poisson approximation. Oxford University Press, OxfordMATHGoogle Scholar
  2. 2.
    Goldstein L, Zhang H (2010) A Berry-Esseen theorem for the lightbulb process (preprint)Google Scholar
  3. 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 ? 69:137–161MATHMathSciNetGoogle Scholar
  4. 4.
    Zhou H, Lange K (2009) Composition Markov chains of multinomial type. Adv Appl Probab 41:270–291CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics KAP 108University of Southern CaliforniaLos AngelesUSA
  2. 2.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia

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