Clubbed Binomial Approximation for the Lightbulb Process

  • Larry GoldsteinEmail author
  • Aihua Xia
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 205)


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.


Binomial Distribution Spectral Decomposition Hadamard Matrix Binomial Coefficient Terminal Time 
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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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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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